The one-loop six-dimensional hexagon integral with three massive corners

Transcription

1 HU-EP-/22 CERN PH TH/2-5 SLAC PUB 4458 LAPTH-6/ IPPP//2 DCPT//42 NSF-KITP--72 The one-loop six-dimensional hexagon integal with thee massive ones Vittoio Del Dua (2) Lane J. Dixon (34) James M. Dummond (45) Claude Duh (62) Johannes M. Henn (72) Vladimi A. Sminov (8) () INFN Laboatoi Nazionali Fasati 44 Fasati (Roma) Italy (2) Kavli Institute fo Theoetial Physis Univesity of Califonia Santa Babaa CA 936 USA (3) SLAC National Aeleato Laboatoy Stanfod Univesity Stanfod CA 9439 USA (4) PH-TH Division CERN Geneva Switzeland (5) LAPTH Univesité de Savoie CNRS B.P. F-7494 Anney-le-Vieux Cedex Fane (6) Institute fo Patile Physis Phenomenology Univesity of Duham Duham DH 3LE U.K. (7) Institut fü Physik Humboldt-Univesität zu Belin Newtonstaße 5 D-2489 Belin Gemany (8) Nulea Physis Institute of Mosow State Univesity Mosow 9992 Russia s: deldua@lnf.infn.it lane@sla.stanfod.edu dummond@lapp.in2p3.f laude.duh@duham.a.uk henn@physik.hu-belin.de sminov@theoy.sinp.msu.u Abstat We ompute the six-dimensional hexagon integal with thee non-adjaent extenal masses analytially. Afte a simple esaling it is given by a funtion of six dual onfomally invaiant oss-atios. The esult an be expessed as a sum of 24 tems involving only one basi funtion whih is a simple linea ombination of logaithms dilogaithms and tilogaithms of unifom degee thee tansendentality. Ou method uses diffeential equations to detemine the symbol of the funtion and an algoithm to eonstut the latte fom its symbol. It is known that six-dimensional hexagon integals ae losely elated to satteing amplitudes in N = 4 supe Yang-Mills theoy and we theefoe expet ou esult to be helpful fo undestanding the stutue of satteing amplitudes in this theoy in patiula at two loops. Wok suppoted in pat by US Depatment of Enegy ontat DE-AC2-76SF55.

2 Intodution Sala n point integals in dimensions D > 4 ae inteesting objets fo a numbe of easons. They appea in the O(ǫ) pat of (D = 4 2ǫ)-dimensional one-loop amplitudes [] whih ae equied fo omputations at highe loop odes. Quite geneally highe-dimensional sala integals ae elated to tenso integals in D = 4 dimensions [2]. In patiula the D = 6 dimensional hexagons ae elated to finite tenso integals [3] that appea in N = 4 supe Yang-Mills (SYM). Moe peisely they appea as deivatives of fou-dimensional two-loop tenso integals. Moeove applying a futhe diffeential opeato the integals edue to fou-dimensional one-loop tenso integals [4]. See Ref. [5] fo elated wok on diffeential equations elevant fo integals in N = 4 SYM. Finite dual onfomal invaiant funtions [6 7] ae also pototypes of funtions that an appea in the emainde funtion of MHV amplitudes and the atio funtion of non-mhv amplitudes in N = 4 SYM [8 9 ]. Reently the massless and one-mass hexagon integals in D = 6 dimensions wee omputed in Refs. [4 2]. It was noted that the massless hexagon integal in D = 6 esembles vey losely the analytial esult of the two-loop emainde funtion fo n = 6 points [3 4 5]. In this note we extend the omputations of hexagon integals in D = 6 dimensions to the ase of thee non-adjaent extenal masses. Ou stategy is the following. We deive simple diffeential equations that elate the theemass hexagon to known pentagon integals. These diffeential equations togethe with a bounday ondition ompletely detemine the answe in piniple. We find it onvenient to fist ompute the symbol [6] of the answe and then eonstut the funtion fom that symbol. 2 Integal epesentation and diffeential equations We onside the hexagon integal with thee massive ones d 6 x i H 9 := () iπ 3 x 2 i x2 2i x2 4i x2 5i x2 7i x2 8i whee we used dual (o egion) oodinates p µ j = xµ j xµ j+ (with indies being defined modulo 9) and x µ ij = xµ i xµ j. The on-shell onditions ead x2 2 = x 2 45 = and x 2 78 =. As a sala integal H 9 is a funtion of the (non-zeo) extenal Loentz invaiants x 2 jk. We wok in signatue ( +++) so that the Eulidean egion has all (non-zeo) x 2 jk positive. Dual onfomal ovaiane [6 7] of H 9 in patiula unde the invesion of all dual oodinates x µ x µ /x 2 allows us to wite H 9 =: x 2 5x 2 27x 2 48 Φ 9 (u...u 6 ) (2) 2

3 whee the oss atios u := x2 25 x2 7 u x 2 5x 2 2 := x2 58 x2 4 u 27 x 2 48x 2 3 := x2 82 x x 2 27x 2 48 u 4 := x2 24x 2 5 u x 2 5 := x2 57x 2 48 u 4 x2 25 x 2 6 := x2 8x x2 58 x 2 82 x2 7 ae invaiant unde dual onfomal tansfomations. Futhemoe the one-loop hexagon integal with thee non-adjaent masses is invaiant unde the ation of the dihedal symmety goup D 3 S 3 geneated by the yli otation and the efletion ating on the dual oodinates via x µ j x µ j+3 and xµ j (3) x µ 9 j (4) whee as usual all indies ae undestood modulo 9. It is easy to see that unde the symmety the six onfomal oss atios goup into two obits of thee elements u u 2 u 3 u u 4 u 5 u 6 u 4 u u 3 u 4 u 5 u 2 u 6 u 6. u 2 (5) One an easily deive a diffeential equation fo H 9 by noting that (x 2 x2 + ) x 2 i x2 2i =. (6) (x 2 2i )2 Applying this diffeential opeato to Eq. () we find d 6 x i (x 2 x2 + )H 9 = =: P iπ 3 (x 2 2i )2 x 2 8. (7) 4i x2 5i x2 7i x2 8i The one-loop pentagon integal P 8 appeaing as an inhomogeneous tem in this equation is equivalent to a known fou-dimensional pentagon integal [4] P 8 =: x 2 25x 2 27x 2 48 Ψ 8 (u 3 u 4 u 2 u 5 ). (8) The latte is given by Ψ 8 (u v w) = [ log u log v + Li2 ( u) + Li 2 ( v) + Li 2 ( w) u v + uvw Li 2 ( uw) Li 2 ( vw) ]. (9) We an ewite Eq. (7) as a diffeential equation fo the esaled hexagon integal Φ 9 (u...u 6 ) that depends on oss-atios only In Refs. [4 5] the notation Ψ was used fo Ψ 8. D Φ 9 (u...u 6 ) = Ψ 8 (u 3 u 4 u 2 u 5 ) () 3

4 x 5 x x 5 7 x 7 y 4 y 7 y 4 y 7 x 8 x 8 x 4 x 4 x 2 y x x 2 (a) (b) Figue : (a) depits the epesentation of H 9 as a line integal see Eqs. (2) and (3). The diffeential opeato in Eq. (7) loalizes the y integation to x 2 yielding P 8 (x 2 x 4 x 5 x 7 x 8 ) see (b). whee D := u + u u 6 (u 6 ) 6 + (u 4 ) 4 + u (u ) + u ( u 6 )u 3 3 () with i := / u i. By yli and efletion symmety we have a total of six diffeential equations. It tuns out that only five of them ae independent. The emaining feedom an be fixed e.g. by the bounday ondition H 9 (u u 2 u 3 ) = H 6 (u u 2 u 3 ) with H 6 given expliitly in Refs. [4 ]. (Altenatively one ould deive futhe diffeential equations as in Ref. [4]). Theefoe the set of equations and the bounday ondition ompletely detemine H 9. In the next setion we will use this set of diffeential equations to detemine the symbol S( Φ 9 ) whee Φ 9 is obtained fom Φ 9 by a simple esaling see Eq. (6). Then we will eonstut the funtion Φ 9 (and equivalently H 9 ) fom its symbol. We note that thee is a simple line integal epesentation of H 9 [4] see Fig. (a) H 9 = dξ dξ 4 dξ 7 (y y 4 ) 2 (y 4 y 7 ) 2 (y 7 y ) 2 (2) whee y µ = xµ + ξ x µ 2 yµ 4 = xµ 4 + ξ 4x µ 54 and yµ 7 = xµ 7 + ξ 7x µ 87. The pentagon integal P 8 an be expessed in a simila way whih allows us to wite H 9 = dξ P 8 (y (ξ ) x 4 x 5 x 7 x 8 ). (3) In this fom the diffeential equation (7) has the intepetation of loalizing one of the line integals in this ase y (ξ ) x 2 see Fig. (b). It is inteesting that simila integals whee etain popagatos ae loalized at usp points have also appeaed in omputations of two-loop Wilson loops [7]. Fom this disussion it is also lea that the integal edues futhe in degee unde the ation of othe diffeential opeatos until one eventually obtains a ational funtion. Moe expliitly the opeato (x 54 x5 + ) ating on P 8 similaly gives a fist-ode diffeential equation elating 4

5 Ψ 8 to a single-log funtion namely a 3-mass box integal with two doubled popagatos X 7 := d 6 x i iπ 3 (x 2 2i )2 (x 2 4i )2 x 2 7i x2 8i =: x 2 25x 2 27x 2 58 χ 7 (u 3 u 5 ) (4) whee χ 7 (y) = log(y)/(y ). Ating futhe on X 7 with (x 87 x8 + ) gives the 3-mass tiangle with thee doubled popagatos whih is a onstant up to the usual pefatos /(x 2 25 x2 58 x2 82 ). The epesentation (2) may also be useful fo numeial heks. Fo futue efeene it an be ewitten as Φ 9 (u...u 6 ) = whee ξ i := ξ i. dξ dξ 4 dξ 7 (u 2 ξ ξ4 + u 4 u 2 ξ ξ4 + ξ 4 )(u 3 ξ4 ξ7 + u 5 u 3 ξ 4 ξ7 + ξ 7 )(u ξ7 ξ + u 6 u ξ 7 ξ + ξ ) (5) 3 Symbols fom diffeential equations We find that the following definition Φ 9 (u...u 6 ) =: 9 Φ9 (u...u 6 ). (6) leads to a pue funtion Φ 9 (u i ) i.e. a funtion that an be witten as a linea ombination of tansendental funtions with numeial oeffiients only. Hee 9 := ( u u 2 u 3 + u 4 u u 2 + u 5 u 2 u 3 + u 6 u 3 u u u 2 u 3 u 4 u 5 u 6 ) 2 4u u 2 u 3 ( u 4 )( u 5 )( u 6 ). (7) Using this definition and D (/ 9 ) = we an ewite Eq. () as D Φ9 (u...u 6 ) = Ψ 8 (u 3 u 4 u 2 u 5 ) (8) whee D := 9 ( u 3 u 2 u 4 + u 2 u 3 u 4 u 5 ) [u u 6 (u 6 ) 6 + (u 4 ) 4 + u (u ) + u ( u 6 )u 3 3 ] = 9 ( u 3 u 2 u 4 + u 2 u 3 u 4 u 5 )(D u ) (9) and Ψ 8 (u v w) := ( u v + uvw) Ψ 8 (u v w). (2) 5

6 We find it onvenient to onvet (8) into a diffeential equation fo the symbol of Φ 9 whih eads D S( Φ 9 )(u...u 6 ) = S( Ψ 8 )(u 3 u 4 u 2 u 5 ). (2) Hee the diffeentiation of a symbol is defined by x (a... a n a n ) = x log(a n ) a... a n. (22) The following set of vaiables is useful to desibe the solution whee W i := g i 9 g i + 9 i =...6 (23) g := u u 2 + u 3 + u u 2 u 4 u 2 u 3 u 5 2u 3 u 6 + u u 3 u 6 + 2u 2 u 3 u 5 u 6 u u 2 u 3 u 4 u 5 u 6 g 4 := u u 2 u 3 + 2u u 2 u u 2 u 4 + u 2 u 3 u 5 + u u 3 u 6 2u u 2 u 3 u 5 u 6 + u u 2 u 3 u 4 u 5 u 6 and whee g 2 g 3 (g 5 g 6 ) ae obtained fom g (g 4 ) by yli mappings 2 3 ; These vaiables have a nie behavio unde the diffeential opeatos e.g. { if i = 6 D log(w i ) = and othewise D if i = 4 log(w i ) = if i = 2 o 4 (24) othewise whee D 4 is defined as the image of D unde the efletion u 4 u 6 and u 2 u 3. Given these vaiables we an wite the solution to Eq. (2) as S( Φ 9 )(u...u 6 ) = S( Ψ 8 )(u 3 u 4 u 2 u 5 ) W 6 + T (25) whee T satisfies D T =. Taking into aount the diffeential equations elated to (2) by symmety futhe estits the fom of T. The patiula solution we obtained is in geneal not an integable symbol. We theefoe poeed and add a patiula T h satisfying D i T h = (fo i =... 5) to obtain an integable symbol. Finally additional tems satisfying the homogeneous equations D i T = ae fixed by demanding that the symbol fo Φ 6 fo the massless hexagon [4 ] is epodued when u 4 = u 5 = u 6 =. Following this poedue we find that the symbol S( Φ 9 ) an then be witten as S( Φ 9 ) = 6 S(f i ) W i (26) i= whee f i ae the following degee thee funtions f := Ψ 8 (u 2 u u 6 u 4 ) + Ψ 8 (u u 2 u 5 u 4 ) + Ψ 8 (u 2 u 3 u 6 u 5 ) F(u u 2 u 3 u 4 u 5 u 6 ) f 4 := Ψ 8 (u u 3 u 5 u 6 ). (27) 6

7 Hee the quantities f 2 f 3 (f 5 f 6 ) ae obtained fom f (f 4 ) by yli mappings 2 3 ; Moeove F :=2 Ψ 8 (u u 2 u 4 ) + log u log u 5 + log u 2 log u 6 log u 3 log u 4. (28) Note that one an eaange tems in Eq. (27) beause of the identity = Ψ 8 (u 3 u 2 u 4 u 5 ) + Ψ 8 (u u 3 u 5 u 6 ) + Ψ 8 (u 2 u u 6 u 4 ) Ψ 8 (u 3 u u 4 u 6 ) Ψ 8 (u u 2 u 5 u 4 ) Ψ 8 (u 2 u 3 u 6 u 5 ). (29) 4 Twisto geomety assoiated to a thee-mass hexagon The diffeential equation tehnique allowed us to obtain the symbol of the one-loop thee-mass hexagon integal. If we want to find the analyti expession fo the integal we need to integate the symbol to a funtion. We follow hee the appoah of Ref. [8] whih afte making a suitable hoie fo the funtions that should appea in the answe allows us to edue the poblem of integating the symbol to a poblem of linea algeba. The algoithm of Ref. [8] howeve equies the aguments of the symbol to be ational funtions (of some paametes). Fom Eq. (26) it is lea that in ou ase this equiement is not immediately fulfilled beause the vaiables W i ae algebai funtions of the oss atios u i. In ode to bypass this poblem we have to paametize the six oss atios suh that 9 beomes a pefet squae. A onvenient way to find a paametization that tuns 9 into a pefet squae is to wite the six oss atios as atios of twisto bakets. Indeed even though we wok in D = 6 dimensions whee the link to twisto spae is not immediately obvious we an nevetheless onside the oss atios as being paametized by oss atios in twisto spae CP 3 beause the funtional dependene of Φ 9 is only though the six onfomally invaiant quantities u i whih do not make efeene to the six-dimensional spae. In othe wods we an onside the extenal momenta to lie in a fou-dimensional subspae even as we integate ove six omponents of loop momentum. Futhemoe in Ref. [5] it was noted that in tems of momentum twisto vaiables the equivalent of 9 in the massless ase beomes a pefet squae. Hene momentum twistos seem to povide a natual famewok to seah fo a suitable paametization. We theefoe biefly eview the geomety of a thee-mass hexagon onfiguation in momentum twisto spae. In ode to desibe this geomety we assume that the dual oodinates x i ae elements of fou-dimensional Minkowski spae M 4. As the dependene of Φ 9 is solely though oss atios we an assume that this ondition is satisfied as long as the pojetion to the fou-dimensional spae leaves the oss atios invaiant. The twisto oespondene then assoiates to eah point x i in M 4 a pojetive line X i in momentum twisto spae and two points x i and x j in M 4 ae lightlike sepaated if and only if the oesponding lines X i and X j inteset. In ou ase this implies that the six lines must inteset paiwise (See Fig. 2). Denoting the intesetion points by Z Z 4 and Z 7 we an define six moe twistos by X i = Z i Z i i { }. (3) 7

8 x x 2 x 4 Z 9 Z Z 2 Z 3 x 8 Z 8 x 5 Z 5 Z 4 x 7 Z 7 Z 6 Figue 2: The one-loop thee-mass hexagon integal (left) and its geometi onfiguation in momentum twisto spae CP 3 (ight). Only the intesetion points Z Z 4 and Z 7 have an intinsi geometial meaning wheeas all othe twistos an be moved feely along the lines. Note that the only points in twisto spae that have an intinsi geometi meaning ae Z Z 4 and Z 7 wheeas the othe six points ae defined though Eq. (3) whih is left invaiant by the edefinitions Z 2 Z 2 + α 2 Z Z 5 Z 5 + α 5 Z 4 Z 8 Z 8 + α 8 Z 7 Z 9 Z 9 + α 9 Z Z 3 Z 3 + α 3 Z 4 Z 6 Z 6 + α 6 Z 7 (3) whee α i ae non-zeo omplex numbes. These shifts simply expess the fat that we an move the points along the line without alteing the geometi onfiguation. Futhemoe the intesetion of two lines X i and X j an be expessed though the ondition X i X j := (i ) i (j ) j = Z i Z i Z j Z j = ǫ IJKL Z I i ZJ i Z K j ZL j =. (32) Using the twisto bakets the oss atios u i an be paametized as u = X 2X 5 X X 7 X X 5 X 2 X 7 u 2 = X 5X 8 X 4 X X 4 X 8 X X 5 u 3 = X 8X 2 X 7 X 4 X 2 X 7 X 4 X 8 u 4 = X 2X 4 X X 5 X X 4 X 2 X 5 u 5 = X 5X 7 X 4 X 8 X 4 X 7 X 5 X 8 u 6 = X 8X X 7 X 2 X 8 X 2 X X 7. (33) It is lea that the dihedal symmety of the integal is efleted at the level of the twistos by Z i Z i+3 and Z i Z 8 i (34) whee again all indies ae undestood modulo 9. This ation on the twistos indues an ation on the lines X i and the planes Z i = Z i Z i Z i+ via X i Z i X i+3 and X i X 9 i Z i+3 and Z i Z 8 i. (35) 8

9 We now hoose a patiula epesentation fo the twistos. Sine the points Z Z 4 and Z 7 play a speial ole we hoose thei homogeneous oodinates as Z = Z 4 = Z 7 =. (36) As the othe six points do not ay any intinsi geometi meaning we pefe not to fix them but hoose thei homogeneous oodinates to be Z i = x i y i fo i { }. (37) z i (The x i and y i defined hee should not be onfused with the pevious definitions whee they wee dual oodinates.) In this paametization the oss atios then take the fom u = (y 9 y 6 ) (z 2 z 5 ) (y 2 y 6 ) (z 9 z 5 ) u 2 = (x 5 x 8 ) (z 3 z 9 ) (x 3 x 8 ) (z 5 z 9 ) u 3 = (x 6 x 3 ) (y 8 y 2 ) (x 8 x 3 ) (y 6 y 2 ) u 4 = (z 2 z 3 )(z 9 z 5 ) (z 2 z 5 )(z 9 z 3 ) u 5 = (x 5 x 6 )(x 3 x 8 ) (x 3 x 6 )(x 5 x 8 ) u 6 = (y 6 y 2 ) (y 8 y 9 ) (y 8 y 2 ) (y 6 y 9 ). (38) Note that the oss atios only depend on 2 out of the 8 homogeneous oodinates defined in Eq. (37) whih is a onsequene of the shift invaiane (3). The ation of the dihedal symmety that pemutes the oss atios is implemented in this paametization via x i y i+3 z i+6 x i and x i z 8 i and y i y 8 i. (39) This ation seems to be inonsistent with Eq. (34). Howeve we have boken the symmety by feezing Z Z 4 and Z 7 to onstant values and the symmety is now efleted at the level of the oss atios via Eq. (39). Finally we note that 9 beomes a pefet squae in these vaiables 9 = ((x 6 x 8 ) (y 9 y 2 )(z 3 z 5 ) + (x 5 x 3 )(y 8 y 6 ) (z 2 z 9 )) 2 (x 3 x 8 ) 2 (y 6 y 2 ) 2 (z 9 z 5 ) 2 (4) and Eq. (4) is manifestly invaiant unde the tansfomations (39). Having obtained a paametization that makes 9 into a pefet squae we an wite the symbol in a fom in whih all the enties ae ational funtions of the vaiables we just defined and hene the symbol now takes a fom whih allows it to be integated using the algoithm of Ref. [8]. Futhemoe using this paametization it is tivial to hek that the symbol of Φ 9 obtained in the pevious setion has the oet dihedal symmety. In patiula we find that [S( Φ 9 )] = S( Φ 9 ) and [S( Φ 9 )] = S( Φ 9 ). (4) 9

10 The paametization (38) also makes it vey easy to hek the vaious soft limits of H 9. Indeed we have u 4 z 3 z 2 u 5 x 6 x 5 u 6 y 9 y 8. (42) We heked that in taking these limits S( Φ 9 ) edues to the symbols fo the massless and one-mass hexagon integals [4 2]. 5 Integating the symbol: the one-loop thee-mass hexagon integal As the paametization of the oss atios in tems of momentum twistos intodued in the pevious setion tuns 9 into a pefet squae we an now integate the symbol using the algoithm of Ref. [8]. Howeve even though the paametization (38) makes all the symmeties manifest it uses a edundant set of paametes. We theefoe hoose a minimal set of paametes by beaking the S 3 symmety down to its altenating subgoup A 3 Z 3 by fixing six of the twelve paametes x 6 = y 9 = z 3 = and x 3 = y 6 = z 9 =. (43) The oss atios then take the fom u = z 2 z 5 ( y 2 ) ( z 5 ) u 2 = x 5 x 8 ( x 8 ) ( z 5 ) u 3 = y 8 y 2 ( x 8 ) ( y 2 ) u 4 = z 2 ( z 5 ) z 2 z 5 u 5 = x 5 ( x 8 ) x 5 x 8 u 6 = y 8 ( y 2 ) y 8 y 2 (44) and 9 an now be witten as 9 = (x 8y 2 z 5 + ( x 5 ) ( y 8 )( z 2 )) 2 ( x 8 ) 2 ( y 2 ) 2 ( z 5 ) 2. (45) We note in passing that the Jaobian of the paametization (44) is non-zeo fo genei values of the paametes. In a nutshell the algoithm of Ref. [8] poeeds in two steps:. Given the symbol of Φ 9 omputed in Setion 3 it onstuts a set of ational funtions {R i (x 5 x 8 y 2 y 8 z 2 z 5 )} suh that e.g. symbols of the fom S(Li n (R i )) span the veto spae of whih S( Φ 9 ) is an element. 2. One a suitable set of ational funtions has been obtained it makes an ansatz ϕ = i i Li 3 (R i ) + ij ij Li 2 (R i ) log R j + ijk ijk log R i log R j log R k (46)

11 whee the i ij and ijk ae ational numbes to be detemined suh that S( ϕ) = S( Φ 9 ). (47) As the objets appeaing in this last equation ae tensos (i.e. elements of a veto spae) the oeffiients i ij and ijk an equally well be seen as oodinates in a veto spae and the poblem of finding the oeffiients edues to a poblem of linea algeba. We have implemented the algoithm of Ref. [8] into a Mathematia ode whih we have applied to the funtion Φ 9 (x 5 x 8 y 2 y 8 z 2 z 5 ). The esult we obtain takes a stikingly simple fom Φ 9 (u...u 6 ) = 4 σ(g) L 3 (x + ig x ig ) (48) 9 i= g S 3 whee σ(g) denotes the signatue of the pemutation (+ fo { 2 } fo { 2 }) and whee we defined L 3 (x + x ) := ( l (x + ) l (x ) )3 + L 3 (x + x ) (49) 8 and 2 L 3 (x + x ( ) k ) := (2k)!! logk (x + x ) ( l 3 k (x + ) l 3 k (x ) ) (5) with k= l n (x) := 2 (Li n(x) Li n (/x)). (5) The aguments appeaing in the polylogaithms an be witten in the fom x ± ig := g(x± i g S 3 with x + := χ( 4 7) x+ 2 := χ(2 5 7) x+ 3 := χ(2 4 8) x+ 4 := χ( 5 8) x := χ( 4 7) x 2 := χ(2 5 7) x 3 := χ(2 4 8) x 4 := χ( 5 8) ) fo (52) whee we defined χ(i j k) := 47 X ix k X j 7 7 X j X k X i 47 (53) with i j = i(j )j(j + ). The funtion χ is elated to χ by Poinaé duality χ(i j k) := 47 X ix k X j 7 7 X j X k X i 4 7. (54) The funtion Φ 9 manifestly has the yli symmety. The efletion symmety howeve needs some explanation beause Φ 9 is odd unde efletion. In twisto vaiables 9 beomes a pefet squae and so we an emove the squae oot and ewite 9 as a ational funtion of twisto bakets. This poedue howeve intodues an ambiguity fo the sign of the squae oot. In patiula the ational funtion we obtained is now odd unde the efletion (34) so that Φ 9 is again even.

12 We stess that Eq. (48) is only valid in the egion whee 9 <. In this egion sine χ and χ ae elated by Poinaé duality the funtion Eq. (48) is manifestly eal and we heked numeially that Eq. (48) agees with the paameti integal epesentation fo Φ 9 given in Eq. (5). Note that as multiple zeta values ae in the kenel of the symbol map we ould a pioi add to Eq. (48) tems popotional to ζ 2 without alteing its symbol 2. The numeial ageement with the integal epesentation (5) howeve shows that suh tems ae absent in the pesent ase. 6 Conlusion Using a diffeential equation method to detemine the symbol of a funtion and an algoithm to eonstut the funtion fom its symbol we have omputed analytially the one-loop nonadjaent thee-mass hexagon integal in D = 6 dimensions. Just as fo the massless and one-mass hexagon integals the esult is given in tems of lassial polylogaithms of unifom tansendental weight thee whih ae funtions of six dual onfomally invaiant oss-atios. Beause of the high degee of symmety of the integal the esult is extemely ompat: it an be expessed as a sum of 24 tems involving only one basi funtion whih is a simple linea ombination of logaithms dilogaithms and tilogaithms. Given the elation between one-loop hexagon integals in D = 6 dimensions and highe-loop amplitudes in D = 4 dimensions we expet that ou esult will help to undestand the stutue of N = 4 SYM amplitudes and Wilson loops patiulaly at two loops. Aknowledgements VDD CD and JMH ae gateful to the KITP Santa Babaa fo the hospitality while this wok was aied out. CD is gateful to Hebet Gangl fo valuable disussions on the symbol tehnique. This wok was patly suppoted by the Reseah Exeutive Ageny (REA) of the Euopean Union though the Initial Taining Netwok LHCPhenoNet unde ontat PITN-GA by the Russian Foundation fo Basi Reseah though gant by the National Siene Foundation unde Gant No. NSF PHY5-564 and by the US Depatment of Enegy unde ontat DE AC2 76SF55. Note added: Afte this alulation was ompleted we wee infomed of an independent omputation of the symbols of hexagon integals using a diffeent method [9]. 2 Note that a onstant tem popotional to ζ 3 is exluded beause of the eality ondition on the funtion. 2

13 A Speial ases Fo u 4 = u 5 = u 6 = the diffeential equations simplify onsideably. We have [u + u (u ) ] Φ 9 (u u 2 u 3 ) = Ψ 8 (u 2 u 3 ) (55) whee Ψ 8 (u v ) = log u log v/(u )/(v ) and the two ylially elated equations. The solution is simply 3 log u i Φ 9 (u u 2 u 3 ) = u i. (56) The ase u 5 = u 6 = is also vey simple i= Φ 9 (u u 2 u 3 u 4 ) = log u 3 u 3 Ψ 8(u u 2 u 4 ). (57) B Aguments in tems of spae-time oss atios In this appendix we pesent the expessions of the funtions x + i the spae-time oss atios u i x + = 2u 3 ( u 6 )[ u 3 u 6 u 2 ( u 3 u 5 u 6 )] ( u 3 u 6 ) ( g 9 ) 2u 3 ( u 6 )[ u 2 u 3 ( u 2 u 5 )u 6 ] x + 2 = 2u u 3 ( u 6 )[ u 2 u 4 u 3 ( u 2 u 4 u 5 )] ( u 3 ) ( g 6 9 ) 2u ( u 6 )[ u 2 u 4 u 3 ( u 2 u 4 u 5 )] defined in Eq. (52) in tems of x + 3 = 2u 3 ( u 6 )[( u 2 u 5 )( u 3 u 5 ) u ( u 5 )] ( u 3 u 5 ) ( g 9 ) 2u u 3 u 5 ( u 6 ) [ u 2 u 4 u 3 ( u 2 u 4 u 5 )] x + 4 = u 6 2u 3 ( u 6 )[ u 5 u ( u 2 u 4 u 5 )( u 3 u 5 u 6 )] + ( u 3 u 5 u 6 ) ( g 6 9 ) 2 ( u 6 )[ u 2 u 3 ( u 2 u 5 ) u 6 ] The vaiables x i ae obtained fom x + i by eplaing 9 by 9. Also in Eq. 48 we define the ation of the odd pemutations g to inlude the eplaement (58) 3

14 The twisto vaiables x i y i and z i ationalize the x ± i so that they take the fom x + = x 8 y 8 x + 2 = x 8(y 2 y 8 ) ( x 8 )( y 8 ) x + 3 = x 8( y 2 ) x 5 ( y 8 ) x + 4 = x 8 y 8 ( y 8 )(x 5 x 8 ) x = ( x 5)[ x 8 ( y 2 ) y 8 z 2 ( x 8 y 8 )] y 2 [( x 5 )( y 8 ) z 5 ( x 8 y 8 )] x 2 = ( x 5)(y 2 y 8 )[ x 8 ( y 2 ) y 8 z 2 ( x 8 y 8 )] y 2 ( x 8 )[(z 2 ( x 5 ) z 5 )( y 8 ) + z 5 x 8 ( y 2 )] x 3 = ( y 2)( x 5 )[(x 5 ( y 8 ) x 8 )( z 2 ) + x 8 y 2 ( z 5 )] y 2 x 5 [(z 2 ( x 5 ) z 5 )( y 8 ) + z 5 x 8 ( y 2 )] x 4 = y 8( x 5 )[(x 5 ( y 8 ) x 8 )( z 2 ) + x 8 y 2 ( z 5 )]. y 2 (x 5 x 8 )[( x 5 )( y 8 ) z 5 ( x 8 y 8 )] (59) Note that these expessions oespond to a patiula hoie fo the sign of squae oot. Refeenes [] Z. Ben L. J. Dixon D. C. Dunba and D. A. Kosowe One-loop self-dual and N = 4 supe-yang-mills Phys. Lett. B (997) [axiv:hep-th/9627]. [2] Z. Ben L. J. Dixon and D. A. Kosowe Dimensionally egulated pentagon integals Nul. Phys. B (994) [axiv:hep-ph/93624]. [3] N. Akani-Hamed J. L. Boujaily F. Cahazo and J. Tnka Loal integals fo plana satteing amplitudes [axiv:2.632]. [4] L. J. Dixon J. M. Dummond and J. M. Henn The one-loop six-dimensional hexagon integal and its elation to MHV amplitudes in N = 4 SYM [axiv:4.2787]. [5] J. M. Dummond J. M. Henn and J. Tnka New diffeential equations fo on-shell loop integals JHEP 4 83 (2) [axiv:.3679]. [6] D. J. Boadhust Summation of an infinite seies of ladde diagams Phys. Lett. B (993). [7] J. M. Dummond J. Henn V. A. Sminov and E. Sokathev Magi identities fo onfomal fou-point integals JHEP 7 64 (27) [axiv:hep-th/676]. 4

15 [8] Z. Ben L. J. Dixon D. A. Kosowe R. Roiban M. Spadlin C. Vegu and A. Volovih The two-loop six-gluon MHV amplitude in maximally supesymmeti Yang-Mills theoy Phys. Rev. D (28) [axiv:83.465]. [9] J. M. Dummond J. Henn G. P. Kohemsky and E. Sokathev Hexagon Wilson loop = six-gluon MHV amplitude Nul. Phys. B (29) [axiv:83.466]. [] J. M. Dummond J. Henn G. P. Kohemsky and E. Sokathev Dual supeonfomal symmety of satteing amplitudes in N = 4 supe-yang-mills theoy Nul. Phys. B (2) [axiv:87.95]. [] V. Del Dua C. Duh and V. A. Sminov The massless hexagon integal in D = 6 dimensions [axiv:4.278]. [2] V. Del Dua C. Duh and V. A. Sminov The one-loop one-mass hexagon integal in D = 6 dimensions [axiv:5.333]. [3] V. Del Dua C. Duh and V. A. Sminov An analyti esult fo the two-loop hexagon Wilson loop in N = 4 SYM JHEP 3 99 (2) [axiv:9.5332]. [4] V. Del Dua C. Duh and V. A. Sminov The two-loop hexagon Wilson loop in N = 4 SYM JHEP 5 84 (2) [axiv:3.72]. [5] A. B. Gonhaov M. Spadlin C. Vegu and A. Volovih Classial polylogaithms fo amplitudes and Wilson loops Phys. Rev. Lett (2) [axiv:6.573]. [6] A. B. Gonhaov A simple onstution of Gassmannian polylogaithms [axiv: ]. [7] J. M. Dummond J. Henn G. P. Kohemsky and E. Sokathev Confomal Wad identities fo Wilson loops and a test of the duality with gluon amplitudes Nul. Phys. B (2) [axiv:72.223]. [8] C. Duh H. Gangl and J. Rhodes Symbol alulus fo polylogaithms and Feynman integals in pepaation. [9] M. Spadlin and A. Volovih to appea. 5