Yangian Bootstrap for Massive Feynman Integrals
Florian Loebbert, Julian Miczajka, Dennis Müller, Hagen Münkler
HHU-EP-20/27
Yangian Bootstrap for
Massive Feynman Integrals
Florian Loebbert a , Julian Miczajka a ,Dennis Müller b , Hagen Münkler ca Institut für Physik, Humboldt-Universität zu Berlin,Zum Großen Windkanal 6, 12489 Berlin, Germany b Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen,Denmark c Institut für Theoretische Physik, Eidgenössische Technische Hochschule Zürich,Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland {loebbert,miczajka}@[email protected]@itp.phys.ethz.ch
Abstract
We extend the study of the recently discovered Yangian symmetryof massive Feynman integrals and its relation to massive momentumspace conformal symmetry. After proving the symmetry statementsin detail at one and two loop orders, we employ the conformaland Yangian constraints to bootstrap various one-loop examples ofmassive Feynman integrals. In particular, we explore the interplaybetween Yangian symmetry and hypergeometric expressions of theconsidered integrals. Based on these examples we conjecture singleseries representations for all dual conformal one-loop integrals in D spacetime dimensions with generic massive propagators. a r X i v : . [ h e p - t h ] O c t ontents P D +1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 m : Gauß F (Non-Dual-Conformal Example) . . . . . . . . . . . 22 : Appell F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 3 Points, m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.3 3 Points, 2 Loops, m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 m : Rational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.2 2 Points, : Rational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.3 2 Points, m : Gauß F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.4 2 Points, m m : Kampé de Fériet . . . . . . . . . . . . . . . . . . . . . . . . 337.5 2 Points, m m : Appell F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.6 3 Points, m : Generalized Lauricella . . . . . . . . . . . . . . . . . . . . . 38 m : Rational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.2 2 Points, m m : Associated Legendre P . . . . . . . . . . . . . . . . . . . . 428.3 2 Points, m m : Gauß F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.4 3 Points, m : Gauß F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458.5 3 Points, m m m : Srivastava H C , Region A . . . . . . . . . . . . . . . . . . 468.6 3 Points, m m m : Region B . . . . . . . . . . . . . . . . . . . . . . . . . . 48 n -Gon Integrals 49 n Points, m . . . m n : Conjecture in Region A . . . . . . . . . . . . . . . . . . 499.2 n Points, m . . . m n : Conjecture in Region B . . . . . . . . . . . . . . . . . . 519.3 Unit Propagator Powers for 2,3 and 4 Points . . . . . . . . . . . . . . . . . . 52
10 Outlook 54 Introduction and Summary
Conformal symmetry plays an important role in theoretical physics. On the one hand itrepresents an (approximate) symmetry of many interesting models. On the other hand, itfurnishes a powerful tool that puts strong constraints on a theory’s observables and leadsto intriguing mathematical structures. In this paper we explore an extension of conformalsymmetry into two directions: its applicability to situations with masses as well as its embed-ding into an infinite dimensional Yangian symmetry. While there is a clear phenomenologicalmotivation for going beyond the realm of massless particles, the extension to a conformalYangian brings us to the theory of integrable models where one may expect that physicalquantities of interest are fixed completely by the underlying symmetry.While conformal symmetry can be studied on different levels of a given model, here weare interested in its impact on the elementary building blocks of quantum field theory, i.e. onFeynman integrals. The focus of the present paper lies on the question of how to bootstrapmassive Feynman integrals by using conformal symmetry or its Yangian extension. In fact,we will discuss two instances of conformal symmetry, i.e. an ‘ordinary’ conformal symmetrythat is here naturally formulated in momentum space and dual conformal symmetry actingon dual region momenta. In both cases we will discuss representations of the symmetry thatalso act on the particles’ masses, which can be interpreted as extra-dimensional componentsof the coordinate vectors. The Yangian algebra employed here is then understood as theclosure of these two conformal algebras, cf. [1–3].The dual conformal symmetry of certain Feynman integrals has a long history in themassless as well as in the massive situation, see e.g. [4–6], and it strongly reduces the numberof variables a function of interest depends on. Among the dual conformal Feynman integrals,infinite classes of diagrams of fishnet structure feature an even larger Yangian symmetry aswas shown in [2, 7] for the massless case and a massive version was recently found in [3]. TheYangian algebra is well known to underly rational integrable models [8–10], and it includesthe dual conformal symmetry at the zeroth level of its infinite set of generators. In thecase of two-dimensional field theories, this nonlocal symmetry typically fixes the scatteringmatrix completely [11]. The distinguished role of fishnet-type Feynman integrals can beunderstood from the fact that their conformal Yangian symmetry is inherited from planar N = 4 super Yang–Mills (SYM) theory via a particular double scaling limit of its gamma-deformation. This double scaling limit yields a massless fishnet theory [12], whose correlatorsor scattering amplitudes are in one-to-one correspondence with individual Feynman graphsof fishnet structure. Similarly, a massive fishnet theory can be obtained from a double-scalinglimit of N = 4 SYM theory on the Coulomb branch, allowing to identify massive Feynmanintegrals with Yangian invariant scattering amplitudes [13]. Also the massive version ofthe Yangian can be understood as the closure of massive dual conformal symmetry and anovel massive extension of ordinary conformal symmetry [3]. In this paper we will study theconstraints of the Yangian and its (dual) conformal sub-algebras.The idea to bootstrap Feynman integrals using their Yangian symmetry was first dis-cussed in [14] for the examples of the massless box, hexagon and double box integrals.While the 2-variable box integral was shown to be completely fixed by its symmetries, inthis first approach it was not possible to fix the linear combination of formal Yangian invari-ant building blocks for the 9-variable hexagon and double box integrals. Here an important3tep was recently made in [15], where this linear combination was determined using a multi-variable extension of the Mellin–Barnes techniques, cf. [16]. In order to refine the algorithmicapproach towards Feynman integrals from Yangian symmetry it would be desirable to studyexamples that interpolate between the above 2- and 9-variable cases. Here the recent exten-sion of Yangian symmetry to Feynman integrals with massive propagators comes in handy,since switching on individual masses allows us to slowly increase the number of variables. Infact, initial examples of massive integrals were obtained from Yangian symmetry in [3].In the present paper we expand on the details of the massive Yangian and conformal sym-metry presented in [3]. First we prove the symmetry statements at one- and two-loop ordersin detail. We then elaborate on the relation between Yangian symmetry and the massiveextension of momentum space conformal symmetry. We explicitly discuss the implicationsof this momentum space symmetry on a few examples in Section 6. In the massless limit,the resulting constraints are precisely those that have recently been studied in the context ofthe momentum space conformal bootstrap (see e.g. [17–23]) with applications in cosmologyor condensed matter physics.We then systematically apply the bootstrap approach to massive Feynman integrals withgeneric propagator powers at one loop order. Since the treatment strongly depends on thechoice of variables, it is useful to discuss different examples in detail, even if simpler casescan (in principle) be obtained as limits of more complex cases. In particular, we will discussrepresentations of the considered integrals in terms of different hypergeometric functions.The results are summarized in the following table:Points Dual Conf. Masses Ratios
Parameters
Solution Basis Section2 no/yes
00 0 / rational/rational 7.2/—2 no/yes m / Gauß F /rational 7.3/8.12 no/yes m m / Kampé de Fériet/Legendre 7.4/8.23 no/yes
000 2 / Appell F /rational 6.1/(6.2)3 no/yes m
00 3 / Lauricella/Gauß F m m m / —/Srivastava H C —/8.5 n no/yes m . . . m n see eq. (2.30) —/conjecture —/9Based on the intuition gained with these examples, we finally conjecture two different seriesrepresentations for the most generic massive n -point integrals at one-loop order, with prop-agator powers a j that obey the dual conformal constraint, i.e. they add up to the spacetimedimension D = (cid:80) j a j : (cid:90) d D x (cid:81) nj =1 ( x j + m j ) a j = m ,a m ,a m ,a m n − ,a n − m n ,a n . . . . (1.1)The two conjectured series representations correspond to expressing the integral in terms oftwo different sets of variablesRegion A: u ij = x ij + ( m i − m j ) − m i m j , Region B: v ij = x ij + m i + m j m i m j . (1.2)4he A-series represents an n -point generalization of Gauß’ hypergeometric function F andSrivastava’s triple hypergeometric series H C for 2 and 3 points, respectively. The B-seriesclosely resembles a representation given by Aomoto [24]. Moroever we note that the v -type variables are distinguished since in the case of unit propagator powers a j = 1 , elegantpolylogarithmic expressions for this class of integrals are known up to five points [25]. In-terestingly, this family of all-mass n -gon integrals has been found to have beautiful relationsto geometry, see e.g. [25–27]. Our analysis suggests that this beauty is closely connected tothe underlying Yangian symmetry which essentially fixes these integrals completely. Notethe hint at integrability in the 1998 paper [26] by Davydychev and Delbourgo, where thesimplification of the integral representation for the constraint (cid:80) j a j = D was already calleda “generalization of the so-called uniqueness formula for massless triangle diagrams”. Inthe non-dual-conformal case of unconstrained propagator powers, hypergeometric represen-tations for these one-loop integrals were obtained in [29].Throughout this paper we use the following notation for non-dual-conformal and dualconformal one-loop n -point integrals with propagator weights a j , respectively, which is alsoreflected in the subsection titles (cf. the above table of contents): I m ...m n n [ a , . . . , a n ] , I m ...m n n • [ a , . . . , a n ] . (1.3)In the dual conformal case denoted by • , the propagator weights obey (cid:80) j a j = D . We closethe paper with an outlook in Section 10. In this section we discuss the massive dual conformal symmetry of Feynman integrals. Inte-grals with this symmetry are particularly interesting in the context of the present paper sinceonly these are invariant under the whole tower of Yangian generators and thus maximallyconstrained. The distinguished feature of these integrals is that the powers of propagatorsentering into an integration vertex obey the dual conformal constraint n (cid:88) j =1 a j = D. (2.1)After having discussed the case of one-loop integrals in large detail, we will comment ongeneralizations to higher loop orders. One-Loop Integrals and Dual Conformal Transformations.
We begin by discussingthe dual conformal symmetry of Feynman integrals with massive external legs. As a simpleexample, we consider the one-loop n -gon integral with arbitrary propagator powers, I n = (cid:90) d D x (cid:81) nj =1 ( x j + m j ) a j = a a a a n a n − . . . . (2.2) We thank Christian Vergu for bringing this to our attentention. The uniqueness formula is also called the star-triangle relation which represents a characteristic featureof many integrable models, cf. (6.2) and e.g. [28]. V n , I n = V n φ n . (2.3)Any appropriate prefactor will lead to scale invariance of the combined object under simul-taneous rescalings of the x -coordinates and the masses. If the propagator weights satisfy theconstraint (2.1), the function φ n is also invariant under the ( D + 1) -dimensional inversion I : x ˆ µ (cid:55)→ x ˆ µ ˆ x . (2.4)Here, we note that the index ˆ µ runs from 1 to D + 1 and the additional component of thevectors x j is given by x D +1 j = m j . To denote an index running from 1 to D , we employ theunhatted version µ . Moreover, we use the abbreviation ˆ x = x + m . (2.5)For the action of the inversion map, we note that ( x j + m j ) (cid:55)→ ( x j + m j ) x ( x j + m j ) , (2.6)where we have applied the ordinary D -dimensional inversion to x , which can be achievedby using an appropriate substitution under the integral. Correspondingly, the integral trans-forms as I n (cid:55)→ (cid:90) d D x ( x ) − D + (cid:80) a j (cid:81) nj =1 ( x j + m j ) a j (cid:81) nj =1 ( x j + m j ) a j . (2.7)If the conformal constraint (2.1) is satisfied, the integral thus transforms by a factor. Forthe function φ n to be an invariant, we hence need to construct a prefactor which transformsas V n (cid:55)→ V n n (cid:89) j =1 ( x j + m j ) a j . (2.8)There are several possible choices for such a prefactor, in particular since we can obtain therespective scalings from combinations of ˆ x ij or m i . A simple choice of prefactor satisfyingthe above constraint is given by V n = n (cid:89) j =1 m − a j j , (2.9)however, other choices of prefactors can be more convenient and we will also employ differentones below.The combination of the above inversion with D -dimensional translations yields the specialconformal transformations in D + 1 dimensions, x ˆ µ (cid:55)→ x ˆ µ + c ˆ µ x ˆ ν x ˆ ν c ˆ ν x ˆ ν + c ˆ ρ c ˆ ρ x ˆ ν x ˆ ν , (2.10)6lbeit with the extra-dimensional component c D +1 set to zero, since I n is not invariant underthe respective translation. These transformations are generated by the conformal generator ˜K µ = n (cid:88) j =1 ˜K µj , ˜K µj = − i (cid:0) x µj x ˆ νj − η µ ˆ ν ˆ x j (cid:1) ∂ ˆ ν . (2.11)Next we note that the invariance of φ n under the above generator can be translated to aninvariance statement for I n with an adapted generator, µ φ n = V − n K µ I n , K µ = ˜K µ + V n ˜K µ V − n = ˜K µ − i n (cid:88) j =1 a j x µj , (2.12)which is easy to see using the explicit prefactor given in (2.9) but holds for any prefactorsatisfying (2.8). The integral I n is hence invariant under the dual conformal generators P µj = − i∂ µx j , L µνj = ix µj ∂ νx j − ix νj ∂ µx j , D j = − ix j ˆ µ ∂ ˆ µx j − i∆ j , K µj = − i (cid:0) x µj x ˆ νj − η µ ˆ ν ˆ x j (cid:1) ∂ ˆ ν − i∆ j x µj , (2.13)if we set the weights ∆ j equal to the propagator powers a j . That is, for J a denoting one ofthe above generators we have J a I n = 0 . (2.14)We remind the reader that the index ˆ µ runs from 1 to D + 1 while µ runs from 1 to D . Thegenerators can hence also be understood as massless generators in D + 1 dimensions. Notehowever that in order to have invariance, we have to restrict to indices µ, ν , e.g. the integralis not invariant under translations in the mass dimension. The generators given above satisfythe conformal algebra (cid:104) D j , P ˆ µk (cid:105) = iδ jk P ˆ µk , (cid:104) D j , K ˆ µk (cid:105) = − iδ jk K ˆ µk , (cid:104) P ˆ µj , L ˆ ν ˆ ρk (cid:105) = iδ jk (cid:16) η ˆ µ ˆ ν P ˆ ρk − η ˆ µ ˆ ρ P ˆ νk (cid:17) , (cid:104) K ˆ µj , L ˆ ν ˆ ρk (cid:105) = iδ jk (cid:16) η ˆ µ ˆ ν K ˆ ρk − η ˆ µ ˆ ρ K ˆ νk (cid:17) , (cid:104) K ˆ µj , P ˆ νk (cid:105) = 2 iδ jk (cid:16) η ˆ µ ˆ ν D k − L ˆ µ ˆ νk (cid:17) , (cid:104) L ˆ µ ˆ νj , L ˆ ρ ˆ σk (cid:105) = iδ jk (cid:16) η ˆ µ ˆ σ L ˆ ρ ˆ νk + ( ) (cid:17) . (2.15)On a massless leg j , the same representation applies with m j ≡ x D +1 j = 0 .Summing up, we have found that the one-loop graph (2.2) is invariant under the dualconformal generators (2.13) provided that the propagator weights satisfy the constraint (cid:80) a j = D . Higher Loop Integrals.
The above invariance statement carries over to higher loopgraphs if we demand that at each vertex, the joining propagator weights sum up to thespacetime dimension, (cid:88) vertex a j + (cid:88) vertex b k = D. (2.16)Here, the variables a j denote the weights of external propagators whereas the b k correspondto internal propagators. In order to see this, consider a multi-loop integral of the form I = . . . (cid:90) d D y i ρ i σ i . . . (2.17)7ith ρ i = (cid:89) j ∈ V i (cid:2) ( x j − y i ) + m j (cid:3) a j , σ i = (cid:89) k ∈ ˜ V i ( y k − y i ) b ki , (2.18)where V i and ˜ V i denote the set of external or internal points connected to y i . The internalpropagators need to be massless in order to have dual conformal symmetry, since we are notintegrating over the ( D + 1) -component of the internal points, i.e. the mass. After carryingout the inversion given in (2.6), we pick up a factor of (cid:2) y i (cid:3) (cid:80) j ∈ Vi a i + (cid:80) k ∈ Vi b ki − D . Given the above constraint, this factor cancels at each vertex.
Conformal Variables.
Due to its invariance, the function φ n can be parametrized interms of conformally invariant variables, simplifying its functional form considerably. In themassless case, the natural variables are conformal cross ratios. In the massive case, thereare (at least) three natural kinds of massive conformal variables: u ij = m i m j ˆ x ij , v kij = m k ˆ x ij ˆ x ik ˆ x jk , w klij = ˆ x ij ˆ x kl ˆ x ik ˆ x jl . (2.19)Here we use the abbreviation ˆ x ij = x ij + ( m i − m j ) . (2.20)Sometimes it is useful to multiply these variables by overall constants, to add constants tothem, or to consider the inverse of these variables, which may lead to a more natural formof the resulting differential equations. Clearly, the v kij and the w klij are not independent ofthe u ij , since v kij = u ik u jk u ij , w klij = u ik u jl u ij u kl = v kij v klj . (2.21)Therefore, in the general n -point case with all masses non-vanishing, one would try to findan independent set among the n ( n − / different u ij . However, the other cross ratiosbecome important as soon as we consider special cases where some of the masses are set tozero. Setting k masses to zero reduces the number of degrees of freedom by k , but it leadsto the vanishing of (cid:80) ki =1 ( n − i ) of the u ij . Therefore, one may need to extend the set ofnon-vanishing u ij by an independent subset of the v kij and w klij . We note that we can andwill use the freedom to select a set of independent cross ratios in order to simplify the formof the Yangian PDEs. Independent Variables.
In the general case it can be difficult to make sure that a givenset of the above variables is indeed independent, and which combinations of values can bereached by choosing appropriate x ˆ µj . In order to answer such questions systematically, wecan employ the construction of the Dirac cone [30], which is also used in the construction of8he conformal compactification of Minkowski or Euclidean space. To this end, we map x ˆ µ toa ( D + 3) -dimensional lightlike vector with components X = 1 + ˆ x , X ˆ µ = 2 x ˆ µ , X D +2 = 1 − ˆ x . (2.22)We can map X back to x via x ˆ µ = X ˆ µ X + X D +2 . (2.23)Note that the latter mapping is invariant under a rescaling of X and hence we considerequivalence classes of lightlike vectors X or the light-cone in projective space.The main advantage of this approach is that conformal transformations of x correspondto linear mappings of X , which makes them much easier to treat. In our concrete case weact with transformations belonging to SO(1 , D + 1) , embedded in such a way that it actstrivially on the ( D + 1) -component of X , which corresponds to the mass component of thespacetime vector x .In the following we consider a configuration of 4 points x ˆ µi and successively exhaust ourfreedom to employ SO(1 , D + 1) transformations in order to reach a set of fixed configura-tions. This approach gives a different parametrization of the conformally invariant degreesof freedom of the configuration. It has the advantage that the variables we obtain are in-dependent by construction and their range is clear. We can then check if a given set of(generalized) conformal cross ratios of the form (2.19) is indeed independent by expressingthem in terms of the new variables.We employ the notation [ X ] = (cid:2) X : X D +2 : X ˆ µ (cid:3) , (2.24)such that the (massive) conformal symmetry SO(1 , D + 1) keeps the last element of theabove vector fixed. We consider the case of at least one of the external legs being massiveand assume without loss of generality that m (cid:54) = 0 . The case of all masses vanishing wasdiscussed in detail in [14]. Next, note that the vector (cid:0) X , X D +21 , X µ (cid:1) is timelike (since we are leaving out a nonvanishing spatial component of a lightlike vector),and we can hence find a transformation in SO(1 , D + 1) , such that [ X ] = [1 : 0 : . . . : 0 : 1] . (2.25)We note that this corresponds to x = (0 , . . . , , (2.26)and we have effectively used the freedom to scale our variables to set m = 1 and theremaining masses are effectively measured in units of m , which we make explicit in thefollowing by using the notation ˜ m i = m i /m .The stabilizer of the above vector X in SO(1 , D + 1) is given by the obvious
SO( D + 1) and fixing the following vectors is a straight-forward exercise leading to the configuration [ X ] = (cid:2) m : 1 − ˜ m : 0 : 0 : 0 : 0 : 2 ˜ m (cid:3) , (2.27) [ X ] = (cid:2) z + ˜ m : 1 − z − ˜ m : 2 z : 0 : 0 : 0 : 2 ˜ m (cid:3) , (2.28) [ X ] = (cid:2) z + z + ˜ m : 1 − z − z − ˜ m : 2 z : 2 z : 0 : 0 : 2 ˜ m (cid:3) . (2.29)9 D = 3 D = 4 D = 5 D = 6 Table 1: Number of degrees of freedom for n massive particles in D dimensions after exhaustingdual conformal symmetry. Clearly, after fixing n points, we have a stabilizer of SO( D + 2 − n ) , provided that n ≤ D + 1 . The number of independent, conformally invariant variables is thus given by n ( D + 1) − dim(SO(1 , D + 1)) + θ ( D + 1 − n )dim(SO( D + 2 − n ))= n ( D + 1) −
12 ( D + 1)( D + 2) + θ ( D + 1 − n )2 ( D + 1 − n )( D + 2 − n ) , (2.30)see also Table 1 for the case of few particles. The above derivation assumes that at least oneof the masses is non-vanishing. We can conclude that, as long as one non-vanishing massremains, we only need to subtract one for every constraint such as masses being equal orvanishing in order to find the corresponding number of degrees of freedom. In fact, for n ≥ ,this procedure remains valid for the case of all masses vanishing, cf. Appendix A in [14]. Example: 3 Points, m m m . As a simple example, we consider the case of threeexternal points and three distinct, non-zero masses. We take the generalized conformal crossratios to be u = − u − x − m m , v = − u − x − m m , w = − u − x − m m . (2.31)From the configurations obtained above, we note (setting D = 4 for the moment), x = (0 , , , , , x = (0 , , , , ˜ m ) , x = ( z , , , , ˜ m ) . (2.32)The cross ratios are thus given by u = − (1 − m ) m , v = − (1 − m ) + z m , w = − ( m − m ) + z m m . (2.33)These expressions can in principle be employed to find out what values the triple ( u, v, w ) can take by solving for m i . In this section, we show that one- and two-loop diagrams with massive external propagatorsare Yangian invariant. The Yangian algebra extends an underlying Lie algebra symmetry to10n infinite tower of symmetry generators, grouped into levels n . For the levels zero ( J ) andone ( (cid:98) J ), we note the commutation relations (cid:2) J a , J b (cid:3) = f abc J c , (cid:2) J a , (cid:98) J b (cid:3) = f abc (cid:98) J c . (3.1)Higher level generators can be constructed by repeated commutations of level-one generators.These commutators are constrained by the Serre relations, cf. e.g. [9].In our case, the generators of the Yangian algebra are constructed from the generatordensities of massive, dual-conformal symmetry given in (2.13). These generators combine toform level-zero and level-one generators on n -point functions J a = n (cid:88) j =1 J aj , (cid:98) J a = f abc (cid:88) j
12 1 l +2 l +1 l + r l + r − ...... b = (cid:90) d D x d D x ¯0 x b (cid:81) lj =1 ( x j + m j ) a j (cid:81) nk = l +1 ( x k + m k ) a k . (3.30)We have noted in the above discussion that at one-loop order the diagram need not be in-variant under the underlying dual conformal symmetry in order to have level-one invariance.For the two-loop discussion, however, level-zero invariance is more critical and we will set y = 0 in the following, since the Lorentz generator L D +1 µ is not a symmetry of the Feyn-man diagrams we consider, and thus the extra generators (cid:98) J extra will not generate separatesymmetries at two loops, cf. (3.24). 14 ual Conformal Case. We would like to show that there is a set of 2-loop evaluationparameters s (2 ,n ) j such that the above generator (3.9) becomes a symmetry of this diagram.Here we denote the (cid:96) -loop evaluation parameters by s ( (cid:96),n ) j . To this end, note that we cansplit up a generic level-one generator as follows: (cid:98) J a (1 ,n ) = (cid:98) J a (1 ,l ) + (cid:98) J a ( l +1 ,n ) + f abc J c (1 ,l ) J b ( l +1 ,n ) + l (cid:88) k =1 (cid:16) s (2 ,n ) k − s (1 ,l ) k (cid:17) J ak + n (cid:88) k = l +1 (cid:16) s (2 ,n ) k − s (1 ,r ) k (cid:17) J ak . (3.31)The terms in the last line are due to the differences of the evaluation parameters for thegenerators acting on the diagrams containing 1 or 2 loops, respectively. When acting on thelevel-zero invariant I (2) n , we note that f abc J c (1 ,l ) J b ( l +1 ,n ) I (2) n = f abc J c (1 ,l ) (cid:0) J b − J b (1 ,l ) (cid:1) I (2) n = − c a (1 ,l ) I (2) n , (3.32)where the dual Coxeter number c arises from the contraction f abc f cbd = 2 c δ ad . (3.33)Consequently, we have (cid:98) J a (1 ,n ) I (2) n = l (cid:88) k =1 (cid:16) s (2 ,n ) k − s (1 ,l ) k − c (cid:17) J ak I (2) n + n (cid:88) k = l +1 (cid:16) s (2 ,n ) k − s (1 ,r ) k (cid:17) J ak I (2) n . (3.34)Here, we have used that I (2) n is invariant under the partial level-one generators acting on thefirst l and last r legs, respectively, since already the integrands of the constituent one-loopgraphs are invariant.We can then take the above equation as a definition of the evaluation parameters for theYangian level-one generators at the two-loop level. Concretely, this gives s (2 ,n ) k = (cid:0) a ( k +1 ,l ) − a (1 ,k − + c (cid:1) for k ≤ l, (3.35) s (2 ,n ) k = (cid:0) a ( k +1 ,n ) − a ( l +1 ,k − (cid:1) for k ≥ l + 1 . (3.36)The dual Coxeter number can be inferred by commuting the constituents of the level-onemomentum generator (3.4). This gives i ([P µ , D] + [P ν , L µν ]) = 2(P µ + δ νν P µ − δ µν P ν ) = 2 D P µ , (3.37)and consequently c = D. (3.38) Non-Dual-Conformal Case.
At the one-loop level, we noticed that all level-one gener-ators are symmetries even if the conformal constraints (2.16) are not satisfied. This findingpartially persists at the two-loop level, where we restrict ourselves to the level-one momen-tum generator, again setting y = 0 . For this generator, many aspects of the above discussion15re still valid. The only difference is that I (2) n is no longer annihilated by the dilatationgenerator D . Instead, we note that D I (2) n = − i (cid:0) D − a (1 ,l ) − b + D − a ( l +1 ,n ) − b (cid:1) I (2) n = − iαI (2) n (3.39)which only vanishes if the conformal constraints are satisfied. We can then modify (3.34) toyield (cid:98) P µ I (2) n = l (cid:88) k =1 (cid:16) s (2 ,n ) k − s (1 ,l ) k − c α (cid:17) P µk I (2) n + n (cid:88) k = l +1 (cid:16) s (2 ,n ) k − s (1 ,r ) k (cid:17) P µk I (2) n , (3.40)from which we read off the evaluation parameters s (2 ,n ) k = (cid:0) a ( k +1 ,n ) − a (1 ,k − − D + a (1 ,l ) + 2 b (cid:1) for k ≤ l, (3.41) s (2 ,n ) k = (cid:0) a ( k +1 ,n ) − a ( l +1 ,k − (cid:1) for k ≥ l + 1 . (3.42)We note that we can always adapt the evaluation parameters by employing a shift s k → s k + C , which acts as (cid:98) P µ → (cid:98) P µ + C P µ (3.43)on the level-one generator. Since P µ is a symmetry generator itself, we can omit the last termwhen discussing the respective level-one generator. In this way, we obtain the evaluationparameters s (2 ,n ) k = (cid:0) a ( k +1 ,n ) − a (1 ,k − (cid:1) for k ≤ l, (3.44) s (2 ,n ) k = (cid:0) a ( k +1 ,n ) − a (1 ,k − + D − b (cid:1) for k ≥ l + 1 . (3.45)These evaluation parameters can be stated in terms of the following rule: Pick an arbitraryleg as leg one and set s ( n )1 = a (2 ,n ) . (3.46)Then move clockwise around the diagram and update the next parameter as s ( n ) k +1 = s ( n ) k − ( a k + a k +1 ) (3.47)if legs k and k + 1 are attached to the same vertex, or as s ( n ) k +1 = s ( n ) k − ( a k + a k +1 − D ) + b (3.48)if the vertices of legs k and k + 1 are connected by an internal propagator with weight b .This rule was also stated in [3]. We note that it applies to all cases we discussed above andwe have hence omitted the explicit reference to the loop number.16oops Graphs Dual Conformal Not Dual Conformal Status1 n -gons all (cid:98) J and (cid:98) J extra all (cid:98) J and (cid:98) J extra proved2 l - r -gons all (cid:98) J (cid:98) P proved > tilings all (cid:98) J (cid:98) P conjectural Table 2: Symmetries at different loop orders.
Summary and Higher Loops.
The above findings at one- and two-loop orders are sum-marized in Table 2. Here the statements at higher loop orders reflect the following conjectureformulated in [3]: Feynman graphs cut along a closed contour from one of the three regulartilings of the plane with massless internal propagators have full Yangian symmetry in thedual conformal case, or only (cid:98) P symmetry in the non-dual-conformal case, respectively; ex-ternal propagators can be massive or massless. This conjecture is motivated by the fact thatin the massless limit, these are the classes of Feynman diagrams that are known to enjoyYangian symmetry [2]. It is further supported by numerical evidence obtained as follows.The level-one momentum generator (cid:98) P was applied to the Feynman parametrization of ex-amples of Feynman graphs in the respective categories. The resulting expression was thenintegrated numerically in Mathematica and it was compared whether the given uncertaintyneighborhood of the result includes zero. It would certainly be desirable to find an analyticproof of this conjecture on massive higher loop integrals similar to the one in the masslesscase [2], or to extend the numerical tests with more advanced numerical techniques. Let us discuss some subtleties specific to two points. Consider a two-point invariant underthe level-one symmetry which obeys (cid:98) J a I = 0 , (3.49)where at two points we have (cid:98) J a = (cid:98) J a = f abc J c J b + s (2)1 J c + s (2)2 J c . (3.50)The one-loop integrals I = (cid:90) d D x ( x + m ) a ( x + m ) a = a a , (3.51)are actually invariant under the level-zero symmetry if the dual conformal constraint a + a = D holds, i.e. they obey (2.14) (cid:0) J b + J b (cid:1) I = 0 . (3.52)This implies that we can write (cid:98) J a I = (cid:104) − f abc J c J b + s (2)1 J a − s (2)2 J a (cid:105) I . (3.53) The following statements can also be adapted to two-point integrals that are covariant under the level-zero symmetry, cf. (3.39). − f abc J c J b = − f abc (cid:2) J c , J b (cid:3) = − c J a , (3.54)with the dual Coxeter number c defined in (3.33) via f abc f cbd = 2 c δ ad . Hence, we have (cid:98) J a I = (cid:16) s (2)1 − s (2)2 − c (cid:17) J a I . (3.55)From (3.10) we can read off the one-loop evaluation parameters s (2)1 = a / and s (2)2 = − a / ,which yields (cid:98) J a I = ( a + a − c )J a I . (3.56)Hence, for the dual Coxeter number c = D as given in (3.38) this equation is trivial if thedual conformal constraint a + a = D holds and thus yields no non-trival constraints thatcould help to determine the integral.Nevertheless, there are two ways to obtain non-trivial constraints on a two-point invariantfrom the Yangian level-one generators:• Firstly, at one loop order we can employ the extra symmetry (cid:98) J extra , see (3.24) and thetwo-point examples in Section 8.• Secondly, we can give up on (parts of) the level-zero symmetry, e.g. the special confor-mal symmetry K µ , which amounts to relaxing the dual conformal condition (cid:80) j a j = D .Examples of how the resulting constraints can be used are discussed in Section 7. P D +1 As seen above, there is an extra contribution to the level-one generators containing the D + 1 -components of the generator densities, in particular the D + 1 momentum density P D +1 j = − i ∂∂m j . (3.57) Recursions from P D +1 . While these generator densities feature in the level-one symme-try of one-loop diagrams, they do not combine to form level-zero symmetry generators. It iswell known that the resulting differential equations in the mass variables are very helpful insolving Feynman integrals [31]. As an example, acting on the one-loop integral (2.2), I n = (cid:90) d D x (cid:81) nj =1 ( x j + m j ) a j , (3.58)results in P D +1 j I n [ a , ..., a n ] = 2 ia j m j I n [ a , ..., a j + 1 , ..., a n ] , (3.59)i.e. the set of all diagrams transforms covariantly under the action of the D + 1 momentumgenerator density. Hence, after the kinematic dependence of a diagram has been reducedto a linear combination of basis functions using the level-zero and level-one symmetries,these covariance equations can be used to constrain the propagator weight dependence ofthe expansion coefficients. Since there is an independent covariance equation for every mass18 j , these constraints are most powerful for integrals with all propagators massive, wherethe dependence on all propagator powers a k is fixed. Otherwise, only the dependence on thepropagator powers of massive propagators are constrained.Below, we use these identities to fix the propagator weight dependence of the non-dual-conformal all-mass two-point one-loop integral, such that the only remaining freedom is givenby numerical prefactors. This information can then be transported to all other two-pointcases by taking massless and conformal limits. From Conformal to Non-Conformal.
Another important application of the D + 1 momentum densities is to generate non-dual-conformal integrals from dual-conformal ones.As we argued in Section 2, Feynman diagrams are dual conformal if for any vertex thesum of all propagator weights gives the spacetime dimension D . Acting on the respectiveFeynman integral with a D + 1 momentum density raises the propagator weight of a legand breaks the conformal condition at the corresponding vertex. Repeated action allows toextract the integral at an arbitrary integer distance to conformality from the knowledge ofthe dual-conformal integral.As an example, consider the case of the two-point one-loop integral with one non-vanishing mass. In section 8.1 and section 7.3 we calculate both the conformal as wellas the non-conformal version of this integral respectively. They are given by I a ,a ,D = a + a = π D/ Γ a / − a / Γ a m a − a ( x + m ) a , (3.60)and I a ,a ,D = π D/ Γ a + a − D/ Γ D/ − a Γ a Γ D/ m D − a + a )1 2 F (cid:20) a ,a + a − D/ D/ ; − x m (cid:21) , (3.61)respectively. The precise relation between the two results is given by (cid:18) − a m ∂∂m (cid:19) n I a ,a ,D = a + a = I a + n,a ,D = a + a . (3.62)From the non-dual-conformal result, we find I a + n,a ,D = a + a = π D/ Γ / a − a ) Γ n +1 / a + a ) Γ a + n Γ / a + a ) m − ( a + a +2 n )1 2 F (cid:20) a ,n +1 / a + a )1 / a + a ) ; − x m (cid:21) . (3.63)To be concrete, taking n = 1 , we have I a +1 ,a ,D = a + a = π D/ Γ / a − a ) Γ a m a − a − ( a − a ) x + ( a + a ) m ( x + m ) a +1 (3.64)in agreement with the derivative of the conformal integral.Hence, knowledge of the dual-conformal case is actually enough to derive the results ofmany non-dual-conformal cases, at least the ones with integer deviations from conformality.These cases are also the ones important for phenomenological applications. Still, having theresults of arbitrarily many non-dual-conformal integrals does not necessarily allow to derivethe continuous dependence on propagator weights and spacetime-dimension.19 Yangian Bootstrap in a Nutshell
In this section we discuss the Yangian bootstrap algorithm introduced in [14] and extend itto the massive situation. After a brief explanation of how to extract the Yangian PDEs fromthe invariance equations, we illustrate the algorithm by means of a simple example. Similarsteps can be taken for the integrals discussed in the subsequent sections, but below we willrefrain from giving all details and rather highlight some key elements.
Above we have argued that certain classes of (massive) Feynman integrals are invariant underthe Yangian algebra. In order to evaluate the constraints from Yangian symmetry on a givenintegral, it is useful to translate the Yangian invariance equations into differential equationsfor the function φ n that depends only on a reduced set of variables which takes into accountPoincaré, scaling or full (dual) conformal symmetry. Doing this requires two steps: First,we bring the PDEs to a canonical form, e.g. for the level-one momentum generator we have (cid:98) P µ I n = n (cid:88) j 21 3 a a a , (4.3)with three non-vanishing masses and the kinematic variables given in (2.31), u = ˆ x − m m , v = ˆ x − m m , w = ˆ x − m m . (4.4)This integral is fully bootstrapped in Section 8.5. We set I = V φ ( u, v, w ) and choose theprefactor V that carries the scaling weight according to V = m − a m − a m − a . (4.5) Whether this is possible depends on the considered example, in particular on the number of externalpoints. (cid:98) P µ I m m m • = (cid:88) j Translating Generators. A natural question is whether the massive dual conformal Yan-gian symmetry can be understood as the closure of the massive dual conformal Lie algebrasymmetry and an ordinary (massive) conformal symmetry, similar to the situation in themassless case [1]. To address this question, we focus on the level-one momentum generatorin dual space (cid:98) P µ = i n (cid:88) i =2 i − (cid:88) j =1 (cid:0) (L µνi + η µν D i )P j,ν − ( i ↔ j ) (cid:1) + n (cid:88) i =1 s i P µi + y (cid:98) P µ extra , (5.1)where (cid:98) P µ extra = i n (cid:88) i =2 i − (cid:88) j =1 (cid:0) L µD +1 i P j,D +1 − ( i ↔ j ) (cid:1) , (5.2)and rewrite it in terms of momentum variables. The latter are related to the dual variablesin the following way: p µi = x µi − x µi +1 . (5.3)25he mass variables, on the contrary, stay untouched. Note that momentum conservationimplies the identification x µn +1 = x µ which will be implicitly assumed henceforth. Invertingthe above relation yields x µi = x µ − (cid:88) j
In order to check the consistency of equa-tion (5.6), let us verify that the right-hand side indeed annihilates the massive n -gons (5.8).For simplicity, we set y = 0 . Denoting the integrand of (5.8) as i n we find ¯K µm i n = ¯K µk, ¯ ∆ =0 i n − ∂ k ( D + 2 a + m ∂ m ) i n , (5.13)where ¯K µm = n − (cid:88) i =1 (cid:16) ¯K µi, ¯ ∆ =0 − a i + a i +1 ) ∂ µp i − ( m i ∂ m i + m i +1 ∂ m i +1 ) ∂ µp i (cid:17) , (5.14)with ¯K µk denoting the massless special conformal generator acting on the loop momentum.Due to the fact that ¯K µk is itself a total derivative, the above expression vanishes whenintegrated over. In this section we explore the idea to bootstrap Feynman integrals in momentum spacerather than in dual space. The motivation to do so is twofold. First, this idea seems naturalas the ubiquitous (dual) level-one momentum generator has been identified as the specialconformal generator in momentum space which is local and thus easier to handle. Second,such an analysis bridges the gap between the Yangian bootstrap approach and the study ofconformal constraints in momentum space pursued e.g. in [18, 20].27 .1 3 Points, : Appell F In order to start with an example that is in fact completely fixed by momentum spaceconformal symmetry, we begin by considering the non-dual-conformal massless star integralwith three external points I = (cid:90) d D x x a x a x a = 21 3 a a a . (6.1)While its dual conformal cousin with a + a + a = D is uniquely fixed by the star-trianglerelation, see e.g. [33], I • = (cid:90) d D x x a x a x a = Γ a (cid:48) Γ a (cid:48) Γ a (cid:48) Γ a Γ a Γ a π D/ x a (cid:48) x a (cid:48) x a (cid:48) , a (cid:48) i = D/ − a i , (6.2)no such statement holds for the non-dual-conformal version of the integral. We thereforeemploy the conformal momentum space symmetry from above to constrain the function. Todo so, we first express the star integral in terms of momenta by using equation (5.3) andobtain (cid:90) d D kk a ( k − p ) a ( k − p − p ) a . (6.3)Next, we employ the scaling equation ¯D ¯ ∆ =0 I = ( D − a − a − a ) I (6.4)to justify the following ansatz: I = ( p ) D/ − a − a − a φ ( u, v ) , where u = p p , v = p p . (6.5)Eliminating p from the ansatz by using momentum conservation and acting on it with ¯K µm =0 and ¯ ∆ i as specified in (5.9),i.e. ¯ ∆ i = a i + a i +1 , yields ¯K µ I = 4 ( p ) D/ − a − a − a − ( p µ PDE p + p µ PDE p ) φ , (6.6)where PDE p = (cid:0) αβ + ( α + β + 1) v∂ v + (( α + β + 1) u − γ ) ∂ u + v ∂ v + ( u − u∂ u + 2 vu∂ v ∂ u (cid:1) , PDE p = (cid:0) αβ + ( α + β + 1) u∂ u + (( α + β + 1) v − γ (cid:48) ) ∂ v + u ∂ u + ( v − v∂ v + 2 vu∂ u ∂ v (cid:1) . (6.7)Here, the Greek parameters are related to the propagator powers and dimension in thefollowing way: α = a , β = a + a + a − D , γ = 1 + a + a − D , γ (cid:48) = 1 + a + a − D . (6.8)Since p and p can be freely varied, both equations have to be fulfilled independently. Werecognize these partial differntial equations as the system defining the Appell function F , see28lso [18,20] for similar discussions of the conformal momentum space constraints. This comesas no surprise since the triangle integral has been shown to evaluate to a linear combinationof four F functions more than years ago [32]. Furthermore, the triangle can be obainedfrom the box integral by sending one of the external points to infinity [5]. The latter wasrecently computed in [14] by utilizing the Yangian bootstrap approach and we can use theexact same techniques to reproduce the result stated in [32]. Here, we only give a briefsummary of the necessary steps. For more details the reader is referred to [14].In order to solve the partial differential equations from above, we make a power seriesansatz G αβγγ (cid:48) xy ( u, v ) = (cid:88) k ∈ x + Z n ∈ y + Z f αβγγ (cid:48) kn u m v n . (6.9)Acting with the PDEs (6.6) on this ansatz yields recurrence relations for the coefficients f αβγγ (cid:48) kn which can straightforwardly be solved, for example, by using Mathematica f αβγγ (cid:48) kn = 1 Γ k +1 Γ n +1 Γ k + γ Γ n + γ (cid:48) Γ − k − n − α Γ − k − n − β . (6.10)Note that this expression leads to a (formal) solution of the PDEs for any value of x and y in (6.9). However, for generic values the series will most likely be divergent for any valueof u and v because the sum extends over a shifted Z -lattice. Only if x and y are chosenin such a way that the series terminates at a lower or upper bound for both k and n theseries has a chance of being convergent. A careful investigation of the zeros of fundamentalsolution (6.10) shows that there are choices for ( x, y ) for which the power series terminatesand converges. However, only four of them converge in a region around the origin in the u - v -plane which is the region that we want to focus on here. These are G αβγγ (cid:48) ,G αβγγ (cid:48) − γ, = u − γ G α +1 − γ,β +1 − γ, − γ,γ (cid:48) ,G αβγγ (cid:48) , − γ (cid:48) = v − γ (cid:48) G α +1 − γ (cid:48) ,β +1 − γ (cid:48) ,γ, − γ (cid:48) ,G αβγγ (cid:48) − γ, − γ (cid:48) = u − γ v − γ (cid:48) G α +2 − γ − γ (cid:48) ,β +2 − γ − γ (cid:48) , − γ, − γ (cid:48) , (6.11)where G αβγγ (cid:48) ( u, v ) = F (cid:104) α,βγ,γ (cid:48) ; u, v (cid:105) Γ − α Γ − β Γ γ Γ γ (cid:48) . (6.12)In the final step, we employ the permutation symmetries of the triangle integral to completelyfix the solution up to an overall constant N : φ ( u, v ) = N (cid:16) Γ α Γ β Γ − γ (cid:48) Γ − γ F (cid:104) α,βγ,γ (cid:48) ; u, v (cid:105) (6.13) + Γ α − γ Γ β − γ Γ γ − Γ − γ (cid:48) u − γ F (cid:104) α +1 − γ,β +1 − γ − γ,γ (cid:48) ; u, v (cid:105) + Γ α − γ (cid:48) Γ β − γ (cid:48) Γ − γ Γ γ (cid:48) − v − γ (cid:48) F (cid:104) α +1 − γ (cid:48) ,β +1 − γ (cid:48) γ, − γ (cid:48) ; u, v (cid:105) + Γ β − γ − γ (cid:48) Γ α − γ − γ (cid:48) Γ γ (cid:48) − Γ γ − u − γ v − γ (cid:48) F (cid:104) α +2 − γ − γ (cid:48) ,β +2 − γ − γ (cid:48) − γ, − γ (cid:48) ; u, v (cid:105) (cid:17) . N = π α + β − γ − γ (cid:48) Γ α Γ β − γ Γ β − γ (cid:48) Γ α − γ − γ (cid:48) . (6.14)The result (6.13) can also be shown to be in full agreement with the Feynman parametrizationof the function φ ( u, v ) reading φ ( u, v ) = π D/ Γ a Γ a Γ a ∞ (cid:90) d α d α α a − α a − Γ a + a + a − D/ (1 + α + α ) D − a − a − a ( α + α u + α α v ) a − a − a − D/ . (6.15) m Let us now consider the same integral with one massive leg: I m = (cid:90) d D x ( x + m ) a x a x a = (cid:90) d D k ( k + m ) a ( k − p ) a ( k − p − p ) a = 21 3 a a a . (6.16)Again, we utilize the scaling equation to justify an ansatz of the following form: I m = m D − a − a − a φ ( u, v, w ) , (6.17)where u = − p m , v = − p m , w = + p m . (6.18)Here, the signs have been chosen for later convenience. Eliminating p from the ansatz usingmomentum conservation and acting on the integral with ¯K µm for ¯ ∆ i = a i + a i +1 yields ¯K µm I m = − m D − a − a − a − ( p µ PDE p + p µ PDE p ) φ , (6.19)where PDE p = ( a ∂ u − a ∂ v + w∂ u ∂ w − w∂ v ∂ w + ( v − u ) ∂ u ∂ v ) , PDE p = (cid:0) (1 + a + a − D ) ∂ w − a ∂ v + w∂ w ∂ w − u∂ u ∂ v − w∂ v ∂ w (cid:1) . Since the number of independent momenta has not increased compared to the massless case,we again find two partial differential equations which need to be satisfied independently.However, as the mass introduces an additional degree of freedom, the function (6.17) nowdepends on three scale invariant variables instead of two. Hence, the number of PDEs isnot sufficient to fully constrain the function. To make matters worse, the (cid:98) P ˆ µ extra symmetryequation is trivially fulfilled and does therefore not yield any additional constraints.To be more explicit, we make the series ansatz G xyz ( u, v, w ) = (cid:88) k,l,n f kln u k k ! v l l ! w n n ! . (6.20)30he function G xyz ( u, v, w ) solves the above differential equations for f kln = c k + l + n ( a ) k + n ( a ) l + n ( a + a − D + 1) n , (6.21)with an unfixed function c k + l + n that depends on the sum of the three summation indices.This function is not fixed by momentum space conformal symmetry only.A natural course of action to generate further PDEs is to consider the full set of level-one generators in dual region momentum space instead of only the level-one momentumgenerator, cf. Table 2. In fact, since the level-zero algebra in dual space is partially broken,it is clear that considering just the level-one momentum generator is no longer sufficient.Considering the full set of dual Yangian level-one generators can of course also be done inmomentum space. However, since the level-one generators are scattered among differentlevels in momentum space, for example, the generator (cid:98) K µ in x -space corresponds to thetrilocal level-two momentum generator in p -space, we prefer to work in region momentumspace which puts all generators on an equal footing. We will come back to the above integral I m in Section 7.6 where the function c k + l + n is constrained using the remaining Yangianlevel-one generators, cf. (7.69): c k + l + n = ( a + a + a − D/ k + l + n ( D/ k + l + n . (6.22) m Consider now the two-loop integral I (2) m = (cid:90) d D x d D x ¯0 ( x + m ) a x b x a x a = (cid:90) d D q d D k ( q + m ) a ( k + q + p + p ) b ( k + p ) a k a = a a a b . (6.23)We write this integral as I (2) m = m D − b − a − a − a φ ( u, v, w ) , (6.24)where u = − x m , v = − x m , w = + x m . (6.25)Acting on the above ansatz with ¯K µm and ¯ ∆ = a + a + b − D/ , (6.26) ¯ ∆ = a + a , (6.27) ¯ ∆ = D/ − a − b , (6.28)as follows from the general rule ¯ ∆ i = ∆ i + ∆ i +1 + 2 s i − s i +1 , we find exactly the same systemof partial differential equations as in the one-loop case (6.19). In fact, as in the one-loopcase, those equations do only depend on a , a and D and not on a . The ¯K µm equations aresolved by the fundamental solution (6.21), with the yet to be determined function c k + l + n .For the one-loop integral this function is fixed by the (cid:98) K equations. Hence, the one- andtwo-loop integrals (6.16) and (6.23) only differ in the function c k + l + n .31 Changing Back: Non-Dual-Conformal Integrals While we will study dual conformal integrals in the following Section 8, here we considerintegrals without imposing any constraints on the propagator powers a j . In particular, thismeans that these integrals are not invariant under the dual conformal generator K µ at levelzero. Despite the absence of dual conformal symmetry, we will see that in comparisonwith the momentum space conformal symmetry discussed in the previous Section 6, theYangian level-one generators yield additional constraints for one-loop integrals as discussedin Section 3.1. In particular, we focus on the interplay between the Yangian differentialequations and their solutions via hypergeometric functions. We also discuss relations betweendifferent cases. Even if integrals with more masses and parameters can in principle be reducedto simpler examples, it is instructive to explicitly discuss different cases with increasingcomplexity. A useful variable in a more complex example with two masses may be givenby u = x /m m which in the limit m → diverges, thus obscuring the reduction of theintegral to a simpler case. m : Rational As the simplest example consider the tadpole integral I m = (cid:90) d D x ( x − m ) a = a . (7.1)Using a single spacetime point one cannot form a translationally and scaling invariant vari-able. Hence, the integral is pure weight, i.e. I m = c a m D − a . (7.2)To fix the propagator weight dependence of the constant c a , we act with P D +1 as describedin 3.4, implying c a = c Γ a − D/ Γ a . (7.3)Finally, we evaluate the integral numerically at a single point to fix the overall constant andfind I m = π D/ m D − a Γ a − D/ Γ a . (7.4) : Rational Also for two points and two vanishing masses there is no scaling-invariant variable and theone-loop integral collapses into a trivial propagator. This is also known as the group relation ,see e.g. [34]: I = (cid:90) d D x x a x a = a a = B a + a − D = B x a + a − D/ , (7.5)with the constant B = π D/ Γ a + a − D/ Γ D/ − a Γ D/ − a Γ a Γ a Γ D − a − a . (7.6)In Section 8.1 we discuss a similar situation in the dual conformal case with a + a = D ,where a massless and a massive propagator are fused into a massive propagator.32 .3 2 Points, m : Gauß F Note that the two-point integral with one mass was explicitly discussed in Section 4.3 forthe variable − m /x as an example to illustrate the bootstrap algorithm. When comparingto limiting cases of other integrals, it can be useful to consider different choices of variables,e.g. we can invert the variable used in Section 4.3 and write I m = m D − a − a φ ( u ) = a a , u = − x m . (7.7)Setting α = a , β = a + a − D , γ = D , (7.8)the Yangian PDE obtained from invariance under the level-one special conformal generator (cid:98) K reads u ( u − φ (cid:48)(cid:48) − u (1 + α + β ) − γ ] φ (cid:48) − αβφ = 0 , (7.9)and is solved by φ = c F (cid:104) α,βγ ; u (cid:105) + c u − γ F (cid:104) α − γ, β − γ − γ ; u (cid:105) . (7.10)This result can be compared to the below three-point result (7.75) in the coincidence limitof points and in Section 4.3 which yields c = π D/ Γ a + a − D/ Γ D/ − a Γ a Γ D/ , c = 0 . (7.11) m m : Kampé de Fériet Also for two non-vanishing masses different choices of variables lead to different types offunctions in which the considered two-point integral can be expressed. For a nice solutionin terms of a single Appell F series see [35]. As a first example we choose our variabels to be u = − x m , v = 1 − m m , (7.12)such that we can write I m m = m D − a − a φ ( u, v ) = a a . (7.13)For convenience we set α = a , β = a , γ = a + a − D . (7.14)Making the series ansatz G xy ( u, v ) = (cid:88) k ∈ x + Z n ∈ y + Z g kn u k v n = u x v y (cid:88) k ∈ Z n ∈ Z g k + x,n + y u k v n , (7.15) This solution requires an inspired choice of variables including square roots. 33e can solve the level-one (cid:98) P extra and (cid:98) K y =0 equations to find the fundamental solution g kn = ( α ) k ( β ) k + n ( γ ) k + n Γ k +1 Γ n +1 ( α + β ) k + n . (7.16)We can alternatively express the above series via f kn = ( − k Γ k +1 Γ n +1 Γ − k − α Γ − k − n − β Γ − k − n − γ Γ k + n + α + β , (7.17)where for a certain constant C = C ( α, β, γ, x, y ) and for integer k and n we have f k + x,n + y = C g k + x,n + y . (7.18)We find 13 doublets ( x, y ) that correspond to zeros of the fundamental solution f kn in k and n , which make the above series terminate at an upper or lower bound. Only the followingtwo of these doublets give rise to effective variables u and v : ( x, y ) = (0 , , ( x, y ) = (0 , − α − β ) . (7.19)In terms of the two basis functions G = (cid:88) k ∈ Z n ∈ Z g kn u k v n , G , − α − β = v − α − β (cid:88) k ∈ Z n ∈ Z g k,n − α − β u k v n , (7.20)we can thus make the ansatz φ = c G + c G , − α − β . (7.21)Using the covariance under the action of P D +1 , see Section 3.4, the prefactors are determinedto be c = e Γ γ Γ α + β , c = e Γ γ Γ α + β (7.22)where e and e are fixed by two random numerical configurations to e = π D/ e = 0 . (7.23)This reproduces the generalized Kampé de Fériet hypergeometric function in [29] (moduloconventions). Equal-Mass Limit. Consider the limit m → m where v → . Hence, for lim v → v n = δ n and assuming α + β < , we find lim m → m φ = c G | v → = c F (cid:104) α,β,γα/ β/ ,α/ β/ / ; u (cid:105) . (7.24)The coefficient c is fixed by the below limit c = π D/ Γ γ Γ α + β . (7.25)The given result agrees with the expression of [32] for the equal-mass two-point integral.34 ne-Point, One-Mass Limit. Consider now the combined limit m → m , x → x where u, v → . Here for α + β < we find lim x → x ,m → m φ = c G | u,v → = c . (7.26)Comparing to the tadpole result of Section 7.1 for a → a + a which reads I m | a → a + a (cid:90) d D x ( x − m ) a = π D/ m D − a − a Γ a + a − D/ Γ a + a , (7.27)we can read off (7.25). m m : Appell F Let us discuss a second choice of kinematic variables which is more symmetric than (7.12),and which will lead to a solution expressed in terms of Appell hypergeometric functions F .This result will be useful to compare to various limiting cases that were expressed in termsof Gauß’ F . In fact, this example will show that more symmetry in the choice of variablesdoes not necessarily lead to a simpler solution in the sense that the resulting expression willbe a linear combination of hypergeometric functions rather than a single hypergeometricseries as in the previous Section 7.4. We now write I m m = (cid:0) x (cid:1) D/ − a − a φ ( u, v ) = a a , (7.28)where we choose the symmetric variables u = − m x , v = − m x . (7.29)Moreover, we define the following abbreviations α = a + a + 1 − D, γ = a + 1 − D , (7.30) β = a + a − D , γ (cid:48) = a + 1 − D . (7.31)Linear combinations of the Yangian PDEs obtained from (cid:98) P and (cid:98) K invariance yield the systemof two differential equations defining the Appell hypergeometric function F , cf. (6.7): (cid:0) αβ + ( α + β + 1) u∂ u + (( α + β + 1) v − γ (cid:48) ) ∂ v + u ∂ u + ( v − v∂ v + 2 vu∂ u ∂ v (cid:1) φ ( u, v ) , (cid:0) αβ + ( α + β + 1) v∂ v + (( α + β + 1) u − γ ) ∂ u + v ∂ v + ( u − u∂ u + 2 vu∂ v ∂ u (cid:1) φ ( u, v ) . (7.32)Hence, for small u, v the solution is a linear combination, cf. (6.11), φ ( u, v ) = c αβγγ (cid:48) g + c αβγγ (cid:48) g + c αβγγ (cid:48) g + c αβγγ (cid:48) g , (7.33)of the four functions g = F (cid:104) α,βγ,γ (cid:48) ; u, v (cid:105) , (7.34) g = u − γ F (cid:104) α +1 − γ,β +1 − γ − γ,γ (cid:48) ; u, v (cid:105) , (7.35) g = v − γ (cid:48) F (cid:104) α +1 − γ (cid:48) ,β +1 − γ (cid:48) γ, − γ (cid:48) ; u, v (cid:105) , (7.36) g = u − γ v − γ (cid:48) F (cid:104) α +2 − γ − γ (cid:48) ,β +2 − γ − γ (cid:48) − γ, − γ (cid:48) ; u, v (cid:105) . (7.37)35 ermutation Symmetry. Let us now use further input to constrain the coefficients c αβγγ (cid:48) j . We note that the considered two-point integral is invariant under the permutation ( x , m , a ) ↔ ( x , m , a ) , which translates into ( u, γ ) ↔ ( v, γ (cid:48) ) . Under this map we have g ↔ g , g ↔ g , g ↔ g , (7.38)such that we conclude that permutation symmetry implies the following constraints c αβγγ (cid:48) = c αβγ (cid:48) γ , c αβγγ (cid:48) = c αβγ (cid:48) γ , c αβγγ (cid:48) = c αβγ (cid:48) γ . (7.39) Recursions from P D +1 . As a further input we can employ the shift-covariance discussedin Section 3.4. The coefficients are functions of the space-time dimension and the propagatorweights according to c αβγγ (cid:48) i = c i ( a , a , D ) . (7.40)Acting with P D +1 on the general solution (7.33) and using the linear independence of the g i leads to a set of recurrence relations for the c i in the propagator weights which are solved by c ( a , a , D ) = e Γ D/ − a Γ D/ − a Γ a + a − D/ Γ a Γ a Γ D − a − a ,c ( a , a , D ) = e ( − a Γ a − D/ Γ a ,c ( a , a , D ) = e ( − a Γ a − D/ Γ a ,c ( a , a , D ) = e ( − a + a Γ a − D/ Γ a − D/ Γ a Γ a . (7.41)Here the e i are constant complex numbers independent of a , a and D and the above con-traints (7.39) from permutation invariance imply e = e . Using random points of numericaldata we finally fix e = π D/ , e = e = ( − − D/ π D/ , e = 0 . (7.42)Hence, in total we obtain the full expression for the two-mass integral I m m = π D/ (cid:0) x (cid:1) D/ − a − a (cid:18) Γ D/ − a Γ D/ − a Γ a + a − D/ Γ a Γ a Γ D − a − a F (cid:104) α,βγ,γ (cid:48) ; u, v (cid:105) + ( − a − D/ u − γ Γ a − D/ Γ a F (cid:104) α +1 − γ,β +1 − γ − γ,γ (cid:48) ; u, v (cid:105) + ( − a − D/ v − γ (cid:48) Γ a − D/ Γ a F (cid:104) α +1 − γ (cid:48) ,β +1 − γ (cid:48) γ, − γ (cid:48) ; u, v (cid:105) (cid:19) . (7.43)Indeed, the result given in [32] agrees with the above expression obtained from bootstrap(modulo phases due to a different sign convention in the propagator).36 ne-Mass Limit. For m → we have v → , such that due to the reduction formula F (cid:104) α,βγ,γ (cid:48) ; u, (cid:105) = F (cid:104) α,βγ ; u (cid:105) , (7.44)we end up with a linear combination of two Gauß hypergeometric functions (see also Sec-tion 4.3 and Section 7.3): I m = π D/ (cid:0) x (cid:1) D/ − a − a (cid:18) Γ D/ − a Γ D/ − a Γ a + a − D/ Γ a Γ a Γ D − a − a F (cid:104) α,βγ ; u (cid:105) + ( − a − D/ u D/ − a Γ a − D/ Γ a F (cid:104) α +1 − γ,β +1 − γ − γ ; u (cid:105) (cid:19) . (7.45)Taking in addition the conformal limit D → a + a of this expression we have c → and I m • = π D/ Γ a / − a / Γ a m a − a ( m + x ) a (7.46)This agrees with the below expression (8.2). Conformal Limit. We would like to take the limit D → a + a of the above full two-mass expression in order to arrive at the dual conformal case presented in terms of associatedLegendre functions in (8.12) or in terms of hypergeometric functions in (8.31). In this limitwe have α → , γ → a − a a − , (7.47) β → a + a a + , γ (cid:48) → − a a − a − . (7.48)where we define a ± = ( a ± a ) . The four basis solutions become g = F (cid:104) ,a + a − , − a − ; u, v (cid:105) , g = u − a − F (cid:104) − a − ,a + − a − − a − , − a − ; u, v (cid:105) ,g = v a − F (cid:104) a − ,a + + a − a − , a − ; u, v (cid:105) , g = u − a − v a − F (cid:104) ,a + − a − , a − ; u, v (cid:105) , (7.49)whereas the coefficients turn into c = 0 , c = ( − a + + a − e Γ a − Γ a + + a − ,c = 0 , c = ( − a + − a − e Γ − a − Γ a + − a − . (7.50)Numerically, we find the interesting relation F (cid:104) a + + a − , a − a − , a − ; u, v (cid:105) = 1[1 + ( √− u − √− v ) ] a + + a − F (cid:104) a − +1 / ,a − + a + a − +1 ; − √ uv √− u −√− v ) (cid:105) , (7.51) Different representations of the same Feynman integrals have led to other relationships for hypergeo-metric functions [35]. I m m • = π D/ m a m a (cid:18) Γ a / − a / Γ a (cid:0) ˜ u − (cid:1) a F (cid:104) a / − a / / ,a a − a +1 ; ˜ u (cid:105) + ( a ↔ a ) (cid:19) , (7.52)with ˜ u = − m m x + ( m − m ) . (7.53)To convert this result into the expression (8.13) in terms of associated Legendre functions P and Q that we obtain from bootstrap in the subsequent Section 8, we use the identities [36] F (cid:2) ν, µ + ν ν ; u (cid:3) =4 ν +1 / e − iπ ( µ − sin( πµ )(1 − u ) − µ/ ( − u ) − − ν Γ ( − µ − ν ) √ πΓ ( − / − ν ) × (1 + cot( πµ ) tan( πν )) (cid:2) πP µν (cid:0) − u (cid:1) − iQ µν (cid:0) − u (cid:1)(cid:3) , (7.54)as well as F (cid:2) − ν,µ − ν − ν ; u (cid:3) = 4 − ν − ( − u ) − µ ( − u ) µ ( − u ) − µ + ν Γ (1 − µ + ν ) √ πΓ (1 / ν ) (1 + cot( πµ ) tan( πν )) × (cid:104) − πµ ) Q µν (cid:0) − u (cid:1) + π (cos( πµ ) + ( − µ csc( π ( µ + ν )) sin( π ( ν − µ ))) P µν (cid:0) − u (cid:1) (cid:105) , (7.55)which are valid for u < only. Remarkably, these identities imply that the expression (7.52)for the dual conformal two-point integral finally collapses to (see (8.13)): I m m • = π D/ / (1 − ˜ v ) / − a / − a / m a m a P / − a / − a / a / − a / − / (˜ v ) , ˜ v = m + m + x m m . (7.56) m : Generalized Lauricella Next we study the one-mass triangle integral I m = (cid:90) d D x ( x − m ) a ( x ) a ( x ) a = 21 3 a a a . (7.57)We write the three-point integral in terms of a scale invariant function of three arguments: I m = m D − a − a − a φ ( u, v, w ) , u = − x m , v = − x m , w = + x m . (7.58)This integral has the full level-one symmetry but no special conformal level-zero symmetrysince we do not impose the dual conformal constraint on the propagator powers. Imposinglevel-one momentum and special conformal symmetry on the above ansatz, we can readoff the coefficients of the vectors x µj . Note that in the case of the special conformal level-one generator (cid:98) K , in order to turn these coeffients into functions of u, v, w modulo overallcoefficients, we need to impose the (cid:98) P equations which makes the coefficients of x j with j = 1 , , vanish. 38he resulting PDEs can be turned into recurrence equations for the coefficients g kln inthe series ansatz G α α α γ γ xyz = (cid:88) k ∈ x + Z l ∈ y + Z n ∈ z + Z g kln u k v l w n = u x v y w z (cid:88) k ∈ Z l ∈ Z n ∈ Z g k + x,l + y,n + z u k v l w n . (7.59)For convenience we introduce the parameters α = a + a + a − D , α = a , α = a , γ = D , (7.60)as well as the depend parameter γ = 1 − γ + α + α = a + a + 1 − D/ . (7.61)Modulo an unconstrained overall constant, the recurrence equations are solved by the uniquefundamental solution g kln = ( α ) k + l + n ( α ) k + n ( α ) l + n Γ k +1 Γ l +1 Γ n +1 ( γ ) k + l + n ( γ ) n . (7.62)For a fixed constant C = C ( α , α , α , γ , γ , x, y, z ) we can alternatively express the series(7.59) in terms of f k + x,l + y,n + z = C g k + x,l + y,n + z , (7.63)where f kln = ( − n Γ k +1 Γ l +1 Γ n +1 Γ − k − l − n − α Γ − k − n − α Γ − l − n − α Γ k + l + n + γ Γ n + γ . (7.64)Hence, for any triplet ( x, y, z ) the series (7.59) furnishes a formal solution of the YangianPDEs. We find 36 zeros of the fundamental solutions, i.e. combinations of x, y, z for whichthe series terminates. However, only for 2 of these 36 possibilities, u, v and w are the effectivevariables of the series: ( x, y, z ) = (0 , , , ( x, y, z ) = (0 , , − γ ) . (7.65)We note the shift identity f α α α γ γ kl,n +1 − γ = ( − γ − f α − γ +1 ,α − γ +1 ,α − γ +1 ,γ − γ +1 , − γ kln . (7.66)which alternatively can be expressed as g α α α γ γ kl,n +1 − γ = Γ γ Γ γ Γ α − γ Γ α − γ Γ α − γ Γ α Γ α Γ α Γ − γ Γ γ − γ g α − γ +1 ,α − γ +1 ,α − γ +1 ,γ − γ +1 , − γ kln . (7.67)Hence, we may relate the second series solution specified by (7.65) to a shifted version of thefirst: G α α α γ γ − γ = w − γ Γ γ Γ γ Γ α − γ Γ α − γ Γ α − γ Γ α Γ α Γ α Γ − γ Γ γ − γ G α − γ +1 ,α − γ +1 ,α − γ +1 ,γ − γ +1 , − γ . (7.68)39aking the ansatz φ = c G α α α γ γ + ˜ c G α α α γ γ − γ = c G α α α γ γ + c w − γ G α − γ +1 ,α − γ +1 ,α − γ +1 ,γ − γ +1 , − γ , (7.69)we can fix the coefficients c and c by the two limits discussed in the following: c = π D/ Γ α Γ − γ Γ α − γ +1 Γ γ , c = π D/ Γ γ − α Γ γ − α Γ γ − Γ α Γ α Γ γ − γ +1 , (7.70)or alternatively ˜ c = π D/ Γ α Γ γ − α Γ γ − α Γ γ − Γ − γ Γ γ Γ γ Γ α − γ Γ α − γ Γ α − γ . (7.71)This result agrees with the expression given in [32]. Two-Point One-Mass Limit. Consider the coincindence limit x → x which implies v → u, w → , (7.72)and thus with lim w → w n = δ n yields lim → G α α α γ γ ( u, v, w ) = ∞ (cid:88) k,l =0 f kl u k + l = F (cid:104) α ,α + α γ ; u (cid:105) , (7.73) lim → G α α α γ γ , − γ ( u, v, w ) γ < = 0 . (7.74)We may thus conclude φ | → = c m D − a − a − a F (cid:104) α ,α + α γ ; u (cid:105) = π D m D − a − a − a Γ a + a + a − D/ Γ D/ − a − a Γ a Γ D/ F (cid:104) a + a + a − D/ ,a + a D/ ; u (cid:105) . (7.75)This can be compared with the two-point integral (7.10) for a + a → a : φ | a + a → a → = π D m D − a − a Γ a + a − D/ Γ D/ − a Γ a Γ D/ F (cid:104) a + a − D/ ,a D/ ; u (cid:105) , (7.76)which fixes c = π D/ Γ α Γ − γ Γ α − γ +1 Γ γ . (7.77) Two-Point Zero-Mass Limit. Note that in the limit of a vanishing propagator power a → , we have c = 0 and the c term yields the expected propagator type contributionwith an overall factor of Gamma functions given in (7.5): I m | a → = π D/ ( x ) D − a − a Γ D/ − a Γ D/ − a Γ a + a − D/ Γ a Γ a Γ D − a − a . (7.78)This fixes the coefficient c for a = 0 to c | a → = π D/ Γ γ − α Γ γ − α Γ γ − Γ α Γ α Γ γ − γ +1 . (7.79)40ote that using the recursions from acting with P D +1 as discussed in Section 3.4 fixes the a dependence of the two coefficients to be c = f ( a , a ) Γ α Γ α − γ +1 , c = f ( a , a ) . (7.80)This shows that in fact even for a (cid:54) = 0 we have c = π D/ Γ γ − α Γ γ − α Γ γ − Γ α Γ α Γ γ − γ +1 . (7.81) In this section we systematically apply the Yangian symmetry discussed above to constrainone-loop Feynman integrals with massless and massive propagators. In order to have thefull Yangian symmetry, we consider the case of dual conformal integrals, i.e. the Yangianconstraints on integrals for which the condition D = n (cid:88) j =1 a j (8.1)is satisfied by the propagator powers a j entering an n -point vertex in the (region momentum)Feynman graph. Again we start from the simplest examples and increase the complexitystep by step. For the non-dual-conformal examples we considered in the previous sectionwe could in principle simply take the dual conformal limit to obtain the solution. However,since the dual conformal integrals are invariant under the whole Yangian and depend onless variables, the resulting constraints allow us to bootstrap more examples than above.Moreover, in the dual conformal case it is natural to employ a different set of (constrained)variables. m : Rational For a single massive propagator, the two-point integral has no independent variable. Con-formal symmetry fixes it to take the form I m • = c m a − a ( x + m ) a = a a , (8.2)with an undetermined constant c . This combination is also invariant under the level-onegenerators. To fix the normalization we can e.g. straightforwardly compute the integral inthe incident point limit (or compare with the result given in [37]). From the conformal casewe can then read off the coefficient in (8.2) to be c = π D/ Γ a / − a / Γ a . (8.3)We can understand this solution as a fusion rule for a massless and a massive propagator inthe dual conformal case a + a = D : I m • = (cid:90) d D x ( x + m ) a ( x ) a = m , a a = A m , a = A x + m ) a . (8.4)41ere we have defined A = π D/ m a − a Γ a / − a / Γ a . (8.5)This rule is similar to the so-called group relation for two massless propagators in the non-conformal situation, cf. Section 7.2. These relations imply that chains of propagators con-nected by dual conformal vertices can be reduced according to (cid:39) . (8.6) m m : Associated Legendre P For the two-point integral with m (cid:54) = 0 , m (cid:54) = 0 there is a single conformal variable andone can consider different choices for this variable that lead to different types of well knowndifferential equations. In this subsection we consider a choice that results in two-parameterLegendre functions. This choice is natural since the number of parameters of the class offunctions matches the number of free propagator weights of the integral. We choose theconformal variable to be (cf. [3]) v = x + ( m − m ) m m + 1 = x + m + m m m . (8.7) Direct Solution of PDEs. For compactness we set α = ( a − a − , β = ( − a − a + 1) , (8.8)and make the conformal ansatz I m m • = m − a m − a (1 − v ) β/ φ ( v ) = a a . (8.9)Note that while it may seem unatural at first sight, pulling out the factor (1 − v ) β/ leads tothe canonical form of the below PDE. The systematics behind this prefactor is understoodby writing − v = det v jk , (8.10)for the matrix v with elements v jk = ( x jk + m j + m k ) / m j m k , see also Section 8.5. Acting onthis function with the level-one momentum generator (cid:98) P extra leads to the following associatedLegendre differential equation: (cid:16) α ( α + 1) + β v − (cid:17) φ − vφ (cid:48) + (1 − v ) φ (cid:48)(cid:48) = 0 . (8.11)This equation is solved by φ ( v ) = c P βα ( v ) + c Q βα ( v ) , (8.12)with P βα and Q βα being the associated Legendre function of the first and second kind, respec-tively. The coefficients can be fixed using numerical data points to find φ ( v ) = 2 β π − β P βα ( v ) . (8.13)42 olution of Recurrence Relations. As an alternative to the above solution, we canapply the steps of the formal boostrap outlined in Section 4.2. We define a function ¯ φ by I m m • = m − a m − a ¯ φ ( v ) , (8.14)and we make the series ansatz ¯ φ ( v ) = (cid:88) k f k v k , (8.15)such that the level-one momentum constraints turn into the recurrence relation ( k + a )( k + a ) f k − ( k + 1)( k + 2) f k +2 = 0 . (8.16)This relation is straightforwardly solved and yields the two fundamental solutions f Pk = ( − k (ˆ a ) ˆ k (ˆ a ) ˆ k Γ k +1 , f Qk = 2 k (ˆ a ) ˆ k (ˆ a ) ˆ k Γ k +1 , (8.17)where we abbreviate ˆ a = a/ . Indeed, numerically we find the interesting identity (1 − v ) β/ φ ( v ) = π ( D +2) / − a − a Γ ˆ a +1 / Γ ˆ a +1 / ∞ (cid:88) k =0 v k ( − k (ˆ a ) ˆ k (ˆ a ) ˆ k Γ k +1 = π D/ ∞ (cid:88) k =0 v k ( − k Γ a / k/ Γ a / k/ Γ k +1 Γ a Γ a . (8.18) From Legendre to Gauß. We note the relation for v > , P βα ( v ) = 1 Γ − β (1 + v ) β/ (1 − v ) − β/ F (cid:104) − α,α +11 − β ; − v (cid:105) , (8.19)which implies the alternative representation in terms the Gauß hypergeometric function F : I m m • = 2 β π − β Γ − β (1 + v ) β m − a m − a F (cid:104) − α,α +11 − β ; − v (cid:105) . (8.20)This suggests to introduce the variable u = x + ( m − m ) − m m = 1 − v . (8.21)Numerically we find for v > that β π − β Γ − β (1 + v ) β F (cid:104) − α,α +11 − β ; − v (cid:105) = π / − β Γ / − β Γ − β F (cid:104) − α − β,α +1 − β − β ; − v (cid:105) , (8.22)which implies the representation I m m • = π / − β Γ / − β Γ − β m a m a F (cid:104) − α − β,α +1 − β − β ; u (cid:105) = π D/ Γ D/ Γ D m a m a F (cid:104) a ,a ( D +1) / ; u (cid:105) . (8.23)In Section 9 we conjecture a generalization of this expression in u -type variables to higherpoint integrals. In the subsequent Section 8.3 we will bootstrap the same integral in thevariable w = 1 /u . 43 nit Propagator Powers. In order to compare with the discussion in [25] we can takethe limit a j → or α, β → − / in D = 2 which implies φ ( v ) = 2 π (1 − v ) − / arcsin (cid:16)(cid:112) (1 − v ) / (cid:17) . (8.24)Alternatively, we can solve the above PDE (8.11) directly for a j = 1 which yields φ ( v ) = (1 − v ) − / (cid:16) c + c log (cid:104) √ v − v (cid:105)(cid:17) . (8.25)Comparing to (8.24) for v > , the constants are fixed to c = 0 , c = πi. (8.26) m m : Gauß F As another alternative variable to the above we set w = − m m x + ( m − m ) = 21 − v = 1 u , (8.27)and we define the conformal function φ via I m m • = m − a m − a u a φ ( w ) . (8.28)Here it is convenient to introduce three shorthands α = ( a − a + 1) , β = a , γ = 2 α. (8.29)Then acting on the above function with the level-one momentum generator produces theGauß hypergeometric differential equation w (1 − w ) φ (cid:48)(cid:48) + [ γ − ( α + β + 1) w ] φ (cid:48) − αβφ = 0 , (8.30)which is solved by φ ( w ) = c F (cid:104) α,βγ ; w (cid:105) + c w − γ F (cid:104) − α, − α + β − γ ; w (cid:105) . (8.31)The coefficients for this dual conformal integral can be fixed from a limit of the non-dual-conformal two-point integral (see (7.52)) to find I m m • = π D/ m − a m − a (cid:20) Γ a / − a / Γ a (cid:0) w − (cid:1) a F (cid:104) α,βγ ; w (cid:105) + ( a ↔ a ) (cid:21) . (8.32) Unit Propagator Powers. It is interesting to evaluate the limit a , a → for D = 2 ,which corresponds to α → / and β → and yields I m m • | a =1 ,a =1 = πw arccsc ( √ w ) m m √ w − . (8.33)44 ne-mass Limit. In the limit where m → we have w → such that F → . We aretherefore left with lim m → I m m • = lim m → (cid:18) c m a − a ( x + ( m − m ) ) a + c m a − a ( x + ( m − m ) ) a (cid:19) = c m a − a ( x + m ) a , (8.34)where we assumed a > a . This matches the result from Section 8.1 for c = π D/ Γ a / − a / Γ a . (8.35) m : Gauß F In the case of the three-point integral with two vanishing masses and evaluated at the dualconformal point D = a + a + a , the integral depends on a single variable which we chooseas u = m x ( x + m )( x + m ) . (8.36)Taking the scaling weight of the integral into account, we write I m • = ( m ) γ − ( x + m ) α ( x + m ) β φ ( u ) = 21 3 a a a , (8.37)where we abbreviate α = a , β = a , γ = 1 + ( − a + a + a ) . (8.38)Then level-one momentum invariance of the integral directly implies Gauß’ hypergeometricdifferential equation u (1 − u ) φ (cid:48)(cid:48) + [ γ − ( α + β + 1) u ] φ (cid:48) − αβφ = 0 , (8.39)which is solved by φ ( u ) = c F (cid:104) α,βγ ; u (cid:105) + c u − γ F (cid:104) α − γ, β − γ − γ ; u (cid:105) . (8.40)The coefficients are fixed by the below limits to read c = π D/ Γ a / − a / − a / Γ a , c = π D/ Γ D/ − a Γ D/ − a Γ D/ − a Γ a Γ a Γ a . (8.41) Coefficients from Star-Triangle Relation. To determine the coefficients c and c , wetake the limit m → which implies u → and thus F → . Hence, for γ > we have lim m → I m • = c x α − γ )12 x β − γ )13 x γ − = c x D − a x D − a x D − a . (8.42)45his should be compared with the star-triangle relation for the massless three-point integralgiven in (6.2): I • = (cid:90) d D x x a x a x a = Γ a (cid:48) Γ a (cid:48) Γ a (cid:48) Γ a Γ a Γ a π D/ x a (cid:48) x a (cid:48) x a (cid:48) , a (cid:48) i = D/ − a i . (8.43)We conclude that c = π D/ Γ D/ − a Γ D/ − a Γ D/ − a Γ a Γ a Γ a . (8.44) Two-point limit Another way to relate this integral to a simpler object is the two-pointlimit in which x → x . In this case, the integral should be given by the conformal one-masstwo-point integral from Section 8.1, i.e. we expect lim x → x I m • ! = π D/ Γ a / − a / − a / Γ a m a + a − a ( x + m ) a + a . (8.45)On the other hand, performing this limit using the general solution (8.40), we find for γ < that lim x → x I m • = c m a + a − a ( x + m ) a + a (8.46)This allows us to immediately fix c = π D/ Γ a / − a / − a / Γ a . (8.47) m m m : Srivastava H C , Region A We introduce the conformal variables u = x + ( m − m ) − m m , v = x + ( m − m ) − m m , w = x + ( m − m ) − m m , (8.48)and write the integral as I m m m • = φ ( u, v, w ) m a m a m a = 21 3 a a a . (8.49)In Section 4.1 this integral was discussed as an example for how to extract the explicitPDEs in the conformal variables. The (cid:98) P -equations split into two contributions coming from (cid:98) P y =0 and (cid:98) P extra that annihilate the integral separately. We will show that we can find thefundamental solution for the integral by working only with the constraints arising from (cid:98) P y =0 . Reading off the coefficients of x µjk /m j m k with j, k = 1 , , yields the following threedifferential operators that annihilate φ ( u, v, w ) : PDE (cid:98) P y =0 = 2 a ∂ u − ∂ v ∂ w + (2 w − ∂ u ∂ w + (2 v − ∂ u ∂ v , (8.50) PDE (cid:98) P y =0 = 2 a ∂ v − ∂ u ∂ w + (2 w − ∂ v ∂ w + (2 u − ∂ u ∂ v , (8.51) PDE (cid:98) P y =0 = 2 a ∂ w − ∂ u ∂ v + (2 v − ∂ v ∂ w + (2 u − ∂ u ∂ w . (8.52)46e make the series ansatz φ ( u, v, w ) = (cid:88) k,l,n g kln u k v l w n , (8.53)such that (8.50,8.51,8.52) translate into three recurrence equations for the coefficients g kln ,e.g. from (8.50) we obtain a ( k + 1) g k +1 ,l,n + 2( k + 1) lg k +1 ,ln + 2( k + 1) ng k +1 ,ln − ( l + 1)( n + 1) g k,l +1 ,n +1 − ( k + 1)( n + 1) g k +1 ,l,n +1 − ( k + 1)( l + 1) g k +1 ,l +1 ,n . (8.54)These reccurence equations can be solved in Mathematica, which yields the following fun-damental solution that is unique up to overall constants: g kln = ( a ) k + l ( a ) k + n ( a ) l + n Γ k +1 Γ l +1 Γ n +1 ( γ ) k + l + n , γ = D + 12 . (8.55)Remember that here we have D = a + a + a . For a fixed constant C and for integer k, l, n the fundamental solution g kln can be written as f k + x,l + y,n + z = C ( a , a , a , x, y, z ) g k + x,l + y,n + z , (8.56)where f kln = 1 Γ k Γ l Γ n Γ k + l + n + D/ / Γ − k − l − a Γ − k − n − a Γ − l − n − a . (8.57)The transpositions of the three external legs of the integral translate into ( a ↔ a , l ↔ n ) , ( a ↔ a , k ↔ l ) , and ( a ↔ a , k ↔ n ) , which are manifest symmetries of the abovefundamental solution. The fundamental solution f kln has 29 zeros which corresponds to 29possible choices for the set ( x, y, z ) such that the following series terminates at an upper orlower bound G xyz ( u, v, w ) = (cid:88) k ∈ x + Z l ∈ y + Z n ∈ z + Z f kln u k v l w n . (8.58)Numerical analysis shows that only the choice ( x, y, z ) = (0 , , of the 29 possible zeros ( x, y, z ) of the fundamental solution leads to a series G xyz with effective variables u, v, w .This leads us to an ansatz φ = c G ( u, v, w ) , (8.59)in terms of Srivastava’s triple hypergeometric function H C , cf. e.g. [38]: G ( u, v, w ) = H C (cid:104) a ,a ,a γ ; u, v, w (cid:105) = ∞ (cid:88) k,l,n =0 ( a ) k + l ( a ) k + n ( a ) l + n ( γ ) k + l + n u k k ! v l l ! w n n ! . (8.60)This series is known to converge for | u | + | v | + | w | − (cid:112) (1 − | u | )(1 − | v | )(1 − | w | ) < . (8.61)Comparing the above ansatz to numerical data from the Feynman parametrization of theintegral I m m m • we can fix the overall coefficient to find (for D = a + a + a ): I m m m • = π D/ Γ D/ Γ D m a m a m a H C (cid:20) a ,a ,a D +12 ; u, v, w (cid:21) . (8.62)In Section 9.3 we compare this series to a result for unit propagator powers in D = 3 and inSection 9.1 we conjecture an n -point generalization of this representation.47 .6 3 Points, m m m : Region B Consider now the alternative conformal variables (these generalize the single variable for thetwo-point Legendre solution (8.7)) u = x + m + m m m , v = x + m + m m m , w = x + m + m m m . (8.63)We refer to the kinematic region covered by a series representation in these variabels asregion B. We note that for Euclidean x jk and m j these variables are never small (as opposedto (8.48)), while at the same time we expect the corresponding series solution to convergeonly for small u, v, w .We define the function φ according to I m m m • = φ ( u, v, w ) m a m a m a = 21 3 a a a . (8.64)For the series ansatz G xyz ( u, v, w ) = (cid:88) k ∈ x + Z l ∈ y + Z n ∈ z + Z g kln u k v l w n , (8.65)the recurrence equations arising from (cid:98) P ( y =0) read l + 1)( n + 1) g k,l +1 ,n +1 + ( k + 1)( l + n + a ) g k +1 ,l,n , (8.66) k + 1)( n + 1) g k +1 ,l,n +1 + ( l + 1)( k + n + a ) g k,l +1 ,n , (8.67) k + 1)( l + 1) g k +1 ,l +1 ,n + ( n + 1)( k + l + a ) g kl,n +1 . (8.68)On the support of these equations, the recurrences following from the invariance under (cid:98) P extra take the form k + l + a )( k + n + a ) g kln − ( k + 1)( k + 2) g k +2 ,l,n , (8.69) k + l + a )( l + n + a ) g kln − ( l + 1)( l + 2) g k,l +2 ,n , (8.70) k + n + a )( l + b + a ) g kln − ( n + 1)( n + 2) g kl,n +2 . (8.71)While we have not determined the general solution to these equations, we note that they aresolved by g kln = ( − k + l + n (ˆ a ) ˆ k +ˆ l (ˆ a ) ˆ k +ˆ n (ˆ a ) ˆ l +ˆ n Γ k +1 Γ l +1 Γ n +1 , (8.72)with the shorthand ˆ k = k/ . To motivate this expression we note that in (8.18) we havefound that the two-point two-mass integral can be expressed as I m m • = π ( D +2) / m a m a − D Γ ˆ a +1 / Γ ˆ a +1 / ∞ (cid:88) k =0 v k ( − k (ˆ a ) ˆ k (ˆ a ) ˆ k Γ k +1 , (8.73)which shows that (8.72) is the natural three-point generalization of the two-point summandin (8.73). 48ased on (8.72) we can now investigate the possible basis series for our ansatz. For aconstant C = C ( a , a , a , x, y, z ) we can write f k + x,l + y,n + z = C g k + x,l + y,n + z . (8.74)where f kln = 2 k + l + n Γ − ˆ a − ˆ k − ˆ l Γ − ˆ a − ˆ k − ˆ n Γ − ˆ a − ˆ l − ˆ n Γ k +1 Γ l +1 Γ n +1 (8.75)This function has 17 zeros ( x, y, z ) in k, l, n but only the triplet ( x, y, z ) = (0 , , leads toa series in the effective variables u, v and w . We thus conclude that for small u, v, w thecorrect Yangian invariant is proportional to this series: φ ( u, v, w ) = c G . (8.76)The coefficient can be fixed using numerical data points such that we find I m m m • = π ( D +3) / m a m a m a − D Γ ˆ a +1 / Γ ˆ a +1 / Γ ˆ a +1 / ∞ (cid:88) k,l,n =0 ( − k + l + n (ˆ a ) ˆ k +ˆ l (ˆ a ) ˆ k +ˆ n (ˆ a ) ˆ l +ˆ n u k k ! v l l ! w n n ! . (8.77)In the following section we will conjecture an n -point generalization of this expression. n -Gon Integrals Based on the evidence from the previous Section 8, we propose the below conjectural seriesrepresentations for the dual conformal, i.e. Yangian invariant n -point one-loop integrals withall propagators massive: I m ...m n n • = (cid:90) d D x (cid:81) nj =1 ( x j + m j ) a j = a a a a n a n − . . . , n (cid:88) j =1 a j = D. (9.1)Here the masses m j take generic non-zero values. n Points, m . . . m n : Conjecture in Region A We first choose the kinematic variables u α according to u ij = x ij + ( m i − m j ) − m i m j . (9.2)Note that these variables can become small for real Euclidean x ij and m j , e.g. for largemasses. This is relevant since we believe that the below series merely converges for small u ij . We refer to this series representation as the series in region A as opposed to the B-seriespresented in the following subsection. 49hen expressed in the above variables we conjecture the dual conformal n -point all-massintegral to be given by the expression I m ...m n n • = π D/ Γ D/ Γ D (cid:81) nj =1 m a j j ∞ (cid:88) k ,k ,...,k n − ,n =0 (cid:81) nj =1 ( a j ) (cid:80) α ∈ Bn | j k α ( D +12 ) (cid:80) α ∈ Bn k α (cid:89) α ∈ B n u k α α k α ! , (9.3)where B n = { , , , ..., ( n − , n ) } is the set of all ordered pairs of distinct numbersbetween and n , whereas B n | j is the subset of B n which is comprised of pairs containing j .We note that the Feynman parametrization of the corresponding dual conformal all-mass n -point integrals in terms of the variables (9.2) is given by I m ...m n n • = π D Γ D/ n (cid:89) i =2 ∞ (cid:90) d α i (cid:32) n (cid:89) i =1 α a i − i Γ a i m a i i (cid:33) (cid:32) n (cid:88) i In this section we compare the above A- and B-series to some lower point expressions forthe Yangian invariant all-mass integrals with unit propagator powers. A-Series for 2 Points. The unit propagator power limit of the result in Section 8.2 in D = 2 reads I m m • = 2 πm − a m − a (1 − v ) − / arcsin (cid:16)(cid:112) (1 − v ) / (cid:17) . (9.11)On the other hand, we can evaluate the A-series for a = a = 1 with D = 2 and using v = − u + 1 . This yields the relation F (cid:104) , / ; − v (cid:105) = 2(1 − v ) − / arcsin (cid:16)(cid:112) (1 − v ) / (cid:17) . (9.12) B-Series for 2 Points. For the B-series the relation in (8.18) implies that ∞ (cid:88) k =0 ( − k Γ k/ / v k k ! = 2(1 − v ) − / arcsin (cid:16)(cid:112) (1 − v ) / (cid:17) . (9.13) A-Series for 3 Points: Comparison with Nickel. In [39] Nickel has computed theall-mass three-point integral in the v -type variables for propagator powers a j = 1 in D = 3 dimensions and found I m m m • a j =1 = π φ N ( v , v , v ) m a m a m a , (9.14)where φ N ( v , v , v ) = 1 √ det G arctan (cid:18) √ det G v + v + v (cid:19) . (9.15)Here the matrix G is defined to have matrix elements G jk = v jk . We can compare this resultto the above 3-point result in u -type variables (9.7). Here we note that v jk = − u jk + 1 . andfor the prefactor of the series expression in D = 3 have Γ / /Γ = √ π/ . We thus concludethat in the region of convergence (8.61) for H C we have φ N ( v , v , v ) = H C (cid:2) , , ; u , u , u (cid:3) . (9.16)Expanding the left hand side in Mathematica assuming < u jk < indeed shows that atleast up to and including order 8 in the variables u , u , u , the expansions of both sidesof this equation coincide. For general D and propagator powers 1, the integral can be written in terms of Appell functions F andarctan’s [40]. -Series for 3 Points: Comparison with Nickel. Similarly, we can compare the aboveresult (9.15) by Nickel with the B-series (9.10), which is actually formulated in the same v -type variables. Also here we find agreement at leading orders when expanding the twoexpressions, i.e. φ N ( v , v , v ) = 1 √ π ∞ (cid:88) k,l,n =0 ( − k + l + n Γ k/ l/ / Γ k/ n/ / Γ l/ n/ / v k k ! v l l ! v n n ! . (9.17) B-Series at 4 Points: Comparison with Murakami–Yano. In [25] the Murakami–Yano formula [41], which gives a compact expression for the volume of a hyperbolic/sphericaltetrahedron, has been leveraged to give a concise dilogarithmic expression for the all-massbox integral in four dimensions with unit propagator powers. This result provides a valuablecross check of our series representation (9.9), which we deem worth detailing.The volume of a spherical tetrahedron is most elegantly phrased in terms of its dihedralangles. We therefore begin by making explicit the relation between these angles and ourvariables v ij . Let G be the matrix whose elements are given by the variables v ij , i.e. G ij = v ij . (9.18)The matrix G encodes the distances between all pairs of points forming the tetrahedron. Thedihedral angles are readily obtained by employing the formula θ ij = arccos (cid:32) − ( G − ) ij (cid:112) ( G − ) ii (cid:112) ( G − ) jj (cid:33) , (9.19)see [25] for a detailed discussion of the underlying geometry. Given these angular variables,we define c = e iθ , c = e iθ , c = e iθ , c = e iθ , c = e iθ , c = e iθ . (9.20)The expression for the volume of a spherical tetrahedron makes use of the positive root z + = − q + (cid:112) q − q q q , (9.21)of the quadratic equation q z + q z + q = 0 , where q = c c c c c c + c c c + c c c + c c c + c c c + (cid:80) i =1 c i c i +3 , (9.22) q = − (cid:80) i =1 (cid:0) c i − c − i (cid:1) (cid:0) c i +3 − c − i +3 (cid:1) ,q = ( c c c c c c ) − + ( c c c ) − + ( c c c ) − + ( c c c ) − + ( c c c ) − + (cid:80) i =1 ( c i c i +3 ) − . Furthermore, we require the function L ( z ) = 12 (cid:20) Li ( z ) + Li (cid:16) zc c c c (cid:17) + Li (cid:16) zc c c c (cid:17) + Li (cid:16) zc c c c (cid:17) − Li (cid:16) − zc c c (cid:17) − Li (cid:16) − zc c c (cid:17) − Li (cid:16) − zc c c (cid:17) − Li (cid:16) − zc c c (cid:17) + (cid:88) i =1 log( c i ) log( c i +3 ) (cid:21) . (9.23)53n terms of the function L ( z + ) , the volume of the tetrahedron described by the matrix G isgiven by V ( G ) = − Re( L ( z + )) + π (cid:16) arg( − q ) + (cid:88) i 10 Outlook The results presented in this paper suggest plenty of different directions for further investi-gation. Let us detail a few.In the case of one-loop Feynman integrals we have demonstrated that Yangian symmetryis in fact highly constraining. For the simple examples studied in Section 7 and Section 8,this results in a small basis of hypergeometric series whose linear combination yields theintegral under study. Here the number of basis elements depends on the chosen variables ascan for instance be seen in Section 7.4 and Section 7.5, where the same integral is studiedfor two different choices of variables. In particular, the solution basis is generically expectedto grow with the number of variables, as becomes apparent in the case of the 9-variablemassless double box and hexagon integrals considered in [14]. Here the close connection tothe Mellin–Barnes approach deserves further study. In particular, it would be desirable tofind a symmetry principle that selects the specific subsets of formal Yangian invariants thatspan the solution, see [15]. Eventually it seems plausible that to fix these linear combinationsusing only conformal Yangian symmetry, kinematical configurations with on-shell externalparticles have to be taken into account. These may result in deformations of the symmetrygenerators similar to the case of scattering amplitudes in N = 4 SYM and ABJM theoryand thus in additional relations, cf. [42].In certain cases and for particular choices of the conformal variables, the Yangian boot-strap selects a single series solution to the symmetry constraints. In these cases merely anoverall coefficient remains to be fixed in order to determine a representation of the integral.In particular, this is the case for the all-mass n -gon integrals subject of Section 9 and allowedus to conjecture two different single series representations (9.3) and (9.9) for generic one-loop integrals in D spacetime dimensions. While the outstanding properties of these integralshave been studied in the past for unit propagator powers (cf. e.g. [25]), the novel Yangiansymmetry sheds new light on their distinguished role. It would be very interesting to furtherexplore the space of Yangian invariant integrals in order to identify families with similarlybeautiful properties. Here the next step is to proceed to two loops. While the analysis ofthe massless double box in [14] shows the increase of complexity for a higher number of ex-ternal points, the present paper suggests that it may be beneficial to first consider situationswith massive propagators. With regard to the one-loop integrals, it would be interesting to54etter understand the mathematical properties of the two series representations (9.3) and(9.9). While we have not found a name for the B-series that closely resembles that givenin [24], at least for n = 2 and n = 3 points the A-series coincides with Gauß’ hypergeometricfunction F and Srivastava’s triple hypergeometric function H C , respectively, and thus canbe assumed to represent a useful generalization.When proceeding to more complicated examples, we note that at loop orders beyond two,the statements about level-one Yangian symmetry are still conjectural, cf. Table 2 and [3].It would be important to make progress on understanding these cases in detail. Ideally onecould find an analytic proof similar to the one in the massless case using the Lax operatorformalism [2]. Alternatively, it would be interesting to systematically map out the space ofhigher loop integrals using advanced numerical integration techniques. Into this direction itwould also be interesting to study massive Feynman integrals with particles different fromscalars which appear in non-scalar fishnet theories [43]. In fact, in the massless case certainbrick wall Feynman graphs including fermionic lines were found to be Yangian invariant [7]and thus represent a natural starting point.Physical application of our results asks for an extension into two further directions.Firstly, while Feynman integrals with generic propagator powers are clearly of interest, itwould be desirable to better understand the considered Yangian approach for integer (inparticular for unit) propagator powers, which at one loop order results in polylogarithmicexpressions. Here the situation of the massless box integral was explicitly discussed in [14],which underlines the importance of the choice of kinematic variables. Secondly, while wehave focussed on the case of Euclidean spacetime signature, there is an obvious demand toexplicitly discuss the results in Minkowski signature. Though this step should not modify theYangian constraints (and thus the solution basis), identifying the correct linear combinationneeds more care. For the case of the massless box integral, the results of [44] indeed showthat also in Minkowski spacetime the integral is spanned by the Yangian invariant buildingblocks given in [14].The four-point Basso–Dixon diagrams of [45] represent another example of Feynmanintegrals with an intriguing connection to integrability, see also [46]. Using sophisticatedintegrability techniques from AdS/CFT and the Steinmann relations, a conjecture for thepolylogarithmic result for these integrals was given at generic loop order. While a Yangiansymmetry has not been formulated in this case, these integrals can be understood as coin-cident point limits of the more generic Yangian invariant fishnet integrals discussed in [2, 7].If one assumes a connection between this Yangian symmetry and the simplicity of the ex-pressions given by Basso and Dixon, one may wonder whether massive propagators can beintroduced into their formula.As mentioned in the introduction, the conformal Yangian and its massive generalizationstudied here can be considered as the closure of two distinct conformal algebras. As such,it would be very interesting to identify its place within the large landscape of results onthe conformal and momentum space conformal bootstrap. Clearly, the Yangian constraintscan be studied independently of Feynman integrals and it would be interesting to search forfurther applications. These might correspond to lifting a conformal setup to an integrableone. Even if one does not consider the full Yangian, it seems natural to generalize the variousapplications of the momentum space conformal bootstrap (see e.g. [22] and references therein)to the massive extension of the algebra given in (5.10).55 cknowledgements We are grateful to Cristian Vergu for illuminating discussions and for providing us with animplementation of the Murakami–Yano formula. The work of FL is funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foundation)–Projektnummer 363895012.JM is supported by the International Max Planck Research School for Mathematical andPhysical Aspects of Gravitation, Cosmology and Quantum Field Theory. DM was supportedby DFF-FNU through grant number DFF-FNU 4002-00037. The work of HM was supportedby the grant no. 615203 of the European Research Council under the FP7 and by the SwissNational Science Foundation by the NCCR SwissMAP. A Yangian Level-One Generators We provide the expressions for all Yangian level-one generators over the (dual) conformalalgebra: (cid:98) P µ = i (cid:88) j Here we give some details on the all-mass Yangian constraints for four points. Region A. We write the conformal four-point integral as I m m m m • = φ ( u , u , u , u , u , u ) m a m a m a m a , (B.1)56ith u jk as defined in (9.2). From the (cid:98) P invariance equation we can read off the coefficientsof the vectors x µjk /m j m k to find the annihilators of the function φ , e.g. PDE (cid:98) P ( y =0) = ∂ u ∂ u + ∂ u ∂ u − a + a ) ∂ u + (2 − u ) ∂ u ∂ u + (1 − u ) ∂ u ∂ u + (1 − u ) ∂ u ∂ u + (1 − u ) ∂ u ∂ u + (1 − u ) ∂ u ∂ u . (B.2)When applied to the series ansatz φ = (cid:88) { k ij } f k k k k k k u k u k u k u k u k u k , (B.3)the partial differential equations PDE jk φ = 0 translate into recurrence equations, e.g. for PDE (cid:98) P ( y =0) − k + 1)( a + a + k + k + k + k ) f k +1 ,k k k k ,k (B.4) − k + 1)( k + 1)[2 f k ,k +1 ,k ,k ,k +1 ,k − − f k ,k +1 ,k ,k ,k +1 ,k ]+ ( k + 1)( k + 1) f k ,k +1 ,k +1 ,k k ,k + ( k + 1)( k + 1) f k ,k ,k ,k +1 ,k +1 ,k + ( k + 1)( k + 1) f k +1 ,k k k ,k +1 ,k + ( k + 1)( k + 1) f k +1 ,k k ,k +1 ,k ,k + ( k + 1)( k + 1) f k +1 ,k ,k +1 ,k k k + ( k + 1)( k + 1) f k +1 ,k +1 ,k k k k . It seems not straightforward to solve these recurrences directly, but based on the previousexperience we find that they are solved by the fundmantal solution corresponding to ourconjectural A-series for n = 4 : f k k k k k k = ( a ) k + k + k ( a ) ˆ k + k + k ( a ) k + k + k ( a ) k + k + k Γ k +1 Γ k +1 Γ k +1 Γ k +1 Γ k +1 Γ k +1 ( γ ) Σ k . (B.5)Here we abbreviate γ = ( D + 1) / and Σ k = k + k + k + k + k . Region B. We write I m m m m • = φ ( v , v , v , v , v , v ) m a m a m a m a , (B.6)with the v jk as given in (9.8). Acting with the level-one momentum generator, we can readoff the annihilators PDE jk of the function φ as the coefficients of the vectors x µjk /m j m k tofind for instance PDE (cid:98) P ( y =0) =( a + a ) ∂ u + ∂ u ∂ u + ∂ u ∂ u + 2 u ∂ u ∂ u + u ∂ u ∂ u + u ∂ u ∂ u + u ∂ u ∂ u + u ∂ u ∂ u . (B.7)Making the series ansatz φ = (cid:88) { k ij } f k k k k k k u k u k u k u k u k u k , (B.8)57he invariance equation PDE (cid:98) P ( y =0) φ = 0 for instance is transformed into the recurrenceequation k + 1)( a + a + k + k + k + k ) f k +1 ,k k k k k + ( k + 1)( k + 1) f k k k ,k +1 ,k +1 ,k + 2( k + 1)( k + 1) f k ,k +1 ,k k ,k +1 ,k − + ( k + 1)( k + 1) f k ,k +1 ,k +1 ,k k k . 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