aa r X i v : . [ h e p - t h ] F e b Cuts and Isogenies
Hjalte Frellesvig , Cristian Vergu , Matthias Volk , and Matt von Hippel Niels Bohr International AcademyNiels Bohr InstituteUniversity of CopenhagenBlegdamsvej 17, 2100 København, Denmark
February 5, 2021
Abstract
We consider the genus-one curves which arise in the cuts of the sunrise and in the ellipticdouble-box Feynman integrals. We compute and compare invariants of these curves in anumber of ways, including Feynman parametrization, lightcone and Baikov (in full andloop-by-loop variants). We find that the same geometry for the genus-one curves arises inall cases, which lends support to the idea that there exists an invariant notion of genus-onegeometry, independent on the way it is computed. We further indicate how to interpretsome previous results which found that these curves are related by isogenies instead. [email protected] [email protected] [email protected] [email protected] ontents A.1 The one-loop case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16A.2 Multi-loop, the loop-by-loop approach . . . . . . . . . . . . . . . . . . . . . . . . 18A.3 Multi-loop, the standard approach . . . . . . . . . . . . . . . . . . . . . . . . . . 19A.4 The elliptic double-box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A.5 Derivation of a four-dimensional Baikov representation . . . . . . . . . . . . . . . 20
There has recently been a flurry of interest in Feynman integrals associated with elliptic curves.Many different ways to represent these integrals have been developed [1–24], culminating in basesof functions that are believed to be powerful enough to represent all such integrals [25–27]. Acommon feature of most of these representations is the characterization of each integral in termsof a single, specific elliptic curve. With the curve specified, relations can be found betweenfunctions defined on the same curve, allowing for the choice of a linearly independent basis.What these representations typically do not consider are relations between Feynman integralsassociated with different elliptic curves. This deficit is thrown into sharp relief by a pair of papers,one by Adams and Weinzierl [15], and the other by Bogner, Müller-Stach, and Weinzierl [28],investigating the two-loop sunrise integral with all equal masses and with distinct internal massesrespectively. These integrals have long been known to involve elliptic curves [1, 2, 4–6, 8, 13, 29–43]. What they found was that the sunrise integral can in fact be described by two distinct ellipticcurves in different contexts, with the curves related by a quadratic transformation, characterizedin the latter paper as an isogeny [28]. One curve appeared when analyzing the integral in terms ofits Feynman-parametric representation, while another emerged from the maximal cut expressedin the Baikov representation [44] (see also [45–50]). They refer to these as the curve from thegraph polynomial and the curve from the maximal cut, respectively.1n this work, we investigate the origin of the distinction between these two curves: whetherthey differ because one comes from the maximal cut, or due to their origin in different rep-resentations. We examine two diagrams, the sunrise with all distinct internal masses and theelliptic double-box [51, 52], in a variety of representations. In particular, we compare maximalcuts of these diagrams both in Baikov representations and in other representations (a light-conerepresentation in two dimensions, and a momentum twistor representation in four dimensions).We find that in general these representations can all give identical elliptic curves. Instead, weexplain the observations of refs. [15, 28] as a consequence of a particular choice those referencesmade when extracting an elliptic curve from the Baikov representation, involving combining twosquare roots. If we instead rationalize one of the roots, we find not an isogenous curve, but an identical curve to that found in Feynman parametrization.The paper is organized as follows: after a quick review of the relevant mathematics in section2, in section 3 we consider the sunrise integral with three distinct masses. We review the Feynman-parametric representation in subsection 3.1, and the loop-by-loop Baikov representation foundin ref. [28] in subsection 3.2. We then derive two more representations, the traditional Baikovrepresentation in subsection 3.3 and a representation in light-cone coordinates in subsection 3.4,and compare the resulting curves. In subsection 3.5 we explain the differing curves as a result ofcombining distinct square roots, and extract an alternate curve by rationalizing a quadratic rootinstead, finding consistency with other methods. We give another view on the relation betweenthe curves that avoids introducing square roots in subsection 3.6. In subsection 3.7 we close witha brief discussion of how the elliptic j -invariants of these curves shed light on the singularitiesof the diagram. In section 4 we investigate the elliptic double-box, where we compute Baikovrepresentations of the maximal cut to compare to curves extracted in prior work. Specifically,we compare a d -dimensional Baikov representation (subsection 4.1) and a Baikov representationderived in strictly four dimensions (subsection 4.2) finding agreement between the two. We thenconclude and raise some topics for future investigation in section 5.Our paper also includes an appendix, reviewing both the loop-by-loop and the standardapproach to the Baikov representation in A.2 and A.3 respectively, as well as deriving our d -dimensional Baikov representation of the elliptic double-box in A.4 and presenting more details ofour four-dimensional derivation in A.5. We also include two ancillary files: doublebox_curve.txt ,presenting the elliptic curve for the double-box, and doublebox_baikov_rep.txt , presenting theBaikov representation for the double-box. An elliptic curve is a smooth projective algebraic curve of genus one, together with a rationalpoint which serves as the origin for its group structure.There are many ways to represent such curves. One can write them as the vanishing lociof cubic polynomials in projective plane, or in terms of a quartic in a single variable with norepeated roots. One standard form is the so-called Weierstrass normal form, the equation y = 4 x − g x − g , (1)for some coefficients g and g .Two elliptic curves are called isogenous when there is a non-constant map between them givenby rational functions which sends the origin of the first to the origin of the second. To everyisogeny corresponds a dual isogeny and their composition is a homomorphism from an ellipticcurve to itself. If this homomorphism is the multiplication by two, we call the initial isogeny atwo-isogeny. If an isogeny has an inverse (that is, when the inverse map is also rational), one2 k − k pk p − k Figure 1:
Sunrise integral. All internal propagators are massive and we consider the most generalcase where all masses can be unequal. The momentum labeling is chosen such as to make theloop-by-loop Baikov representation easier to derive. further calls the two curves isomorphic [53]. Isomorphic curves have the same j -invariant, whichcan be specified in terms of the coefficients of the Weierstrass normal form as follows j = 1728 g ∆ , (2)where the elliptic discriminant ∆ = g − g . The elliptic curve defined by the Weierstrassmodel (1) is smooth if and only if ∆ = 0. The two-loop sunrise integral shown in fig. 1 is given by I ( p , m , m , m ) = Z d k d k ( k − m ) (( k − k ) − m ) (( p − k ) − m ) . (3)This integral is finite in two dimensions, so it is often studied in that context. In this sectionwe will extract an elliptic curve from this integral in several ways, constructing the j -invariantfor each such curve. We will find that the different methods we use provide only two distinct j -invariants, and are grouped as follows:• Feynman parametrization (subsection 3.1), solving the cut equations in light-cone coordi-nates (subsection 3.4)• Loop-by-loop Baikov representation with 4 inverse propagators (subsection 3.2), full Baikovrepresentation with 5 inverse propagators (subsection 3.3)These two j -invariants correspond to two distinct elliptic curves, which are not isomorphic.However, as described in [28], the two curves are related by a two-isogeny.In the rest of this section, we will describe how to extract an elliptic curve using each ofthese methods, and finish by reconciling the Baikov representations with the first set of methods,before briefly discussing this integral’s Landau singularities. The factor of 1728 = 2 × is required for various number theoretic reasons which will not be relevant forus. We choose to keep it in order to minimize confusion, but also because some of the formulas we will find belowactually look nicer when including this factor. .1 Feynman-parametric representation We begin by reviewing the two representations considered in ref. [28]. The first representationconsidered in that reference was for the full integral expressed in Feynman parameters. InFeynman parameters, the integral can be written as R ω F where F is the second graph polynomial, F = m x ( x + x ) + m x ( x + x ) + m x ( x + x ) + ( − p + m + m + m ) x x x (4)and ω = x d x d x − x d x d x + x d x d x . (5)The variables x , x and x are homogeneous coordinates on P and the equation F = 0 definesan elliptic curve in P . To compute the j -invariant of this curve we may first divide by p to make the expressiondimensionless, then transform to the Weierstrass normal form. For the purpose of writing the j -invariant for this curve, we define the following notation: writing µ i = m i p , we then write, ξ = µ + µ + µ , ξ = − µ + µ + µ , ξ = µ − µ + µ , ξ = µ + µ − µ . (6)With this notation, we can specify the j -invariant: j F = (cid:2) ( ξ − ξ − ξ − ξ −
1) + 16 µ µ µ (cid:3) µ µ µ ( ξ − ξ − ξ − ξ −
1) (7)where we have used a subscript F to indicate that this is computed from the Feynman parameterrepresentation. Ref. [28] presented the maximal cut of the two-loop sunrise integral in a loop-by-loop Baikovrepresentation (as distinct from the traditional, or “full” Baikov representation, see ref. [49],app. A, or the next section to clarify the difference). We review below how to derive thisrepresentation in the case of this integral.In the Baikov representation we want to change the integration variables in the integral I ( p , m , m , m ) from the loop momenta k and k to the inverse propagators. For the integralin eq. (3) the inverse propagators are D = k − m , D = ( k − k ) − m , D = ( p − k ) − m , D = k , (8)where we had to add D to be able to express all scalar products between the momenta. Inthe following we consider the integral in the Euclidean region which corresponds to p < m i > p : k = xp + k , ⊥ , k = yp + k , ⊥ . (9)The orthogonal parts satisfy p · k i, ⊥ = 0. As we are in two dimensions, k , ⊥ and k , ⊥ areproportional and we can write them as k , ⊥ = up ⊥ and k , ⊥ = vp ⊥ . Here p ⊥ is chosen so In this paper we always write P n for the complex projective space P n ( C ). p · p ⊥ = 0 and p ⊥ = p . Expressing the inverse propagators in terms of the dimensionlessquantities x , y , u and v we obtain D = p ( y + v ) − m , D = p ( x − y ) + p ( u − v ) − m ,D = p ( x − + p u − m , D = p ( x + u ) . (10)Moreover, the integration measure becomes d k d k = p d x d y d u d v .We now want to change integration variables from ( x, y, u, v ) to ( D , D , D , D ) under whichthe measure transforms as d x d y d u d v = J − d D d D d D d D . For the Jacobian factor J weget J ≡ (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( D , D , D , D ) ∂ ( x, y, u, v ) (cid:12)(cid:12)(cid:12)(cid:12) = − p u ( uy − vx ) . (11)This Jacobian now has to be expressed in terms of the new variables D i . The equations (10)are quadratic in ( x, y, u, v ) and J can therefore not be expressed rationally in terms of the D i .However, one can solve for the squares of u and uy − vx in eq. (11) rationally. While this ispossible for the full integral, here we only give the expression for the maximal cut correspondingto D = D = D = 0: Q := u = − p (cid:2) D − ( m − p ) (cid:3) (cid:2) D − ( m + p ) (cid:3) ,Q := ( uy − vx ) = − p (cid:2) D − ( m + m ) (cid:3) (cid:2) D − ( m − m ) (cid:3) . (12)Note that in the Euclidean region p is negative implying that the equation D = 0 does nothave a real solution. In order to impose the cut conditions we are thus forced to consider theanalytic continuation of the integral.Multiplying Q and Q from the previous two equations we obtain an expression for J as apolynomial of degree four in D . This approach was followed in refs. [15, 28] and is equivalentto extracting the square root of each line in eq. (12) and combining the square roots under acommon square root, i.e. to writing J = − p √ Q Q . Another, inequivalent approach is tokeep the square roots separate, i.e. to write J = − p √ Q √ Q . As Q and Q are quadratic in D , one can again change variables to rationalize either √ Q or √ Q . In subsection 3.5 we willshow that this connects the elliptic curve arising from the first approach to the curve defined bythe vanishing of the F -polynomial in subsection 3.1.Following the approach taken in ref. [28], we define an elliptic curve by the equation J =( − p ) Q Q . We can transform it to Weierstrass form and compute its j -invariant as in theprevious section, obtaining: j B = (cid:2) ( ξ − ξ − ξ − ξ −
1) + 256 µ µ µ (cid:3) µ µ µ ( ξ − ( ξ − ( ξ − ( ξ − , (13)where we have again made use of µ i = m i p and the variables ξ i defined in eq. (6). This clearlydiffers from the j -invariant computed in the previous subsection, see eq. (7). However, as observedin ref. [28], the two curves are isogenous. This has been checked in ref. [28] by computing thecomplex structure parameter τ of the elliptic curve. Here we check it by using the relations By abuse of notation we are here writing p for the absolute value of the momentum p µ . j -invariants of the two elliptic curves. The j -invariants for a pair of two-isogenouselliptic curves are related by the modular polynomial Φ ( X, Y ) (see e.g. [54, Chapter 5])Φ ( X, Y ) = X + Y − X Y + 1488 (cid:0) X Y + XY (cid:1) − (cid:0) X + Y (cid:1) + 40773375 XY + 8748000000 ( X + Y ) − . (14)See ref. [55] for details about how these modular polynomials are computed. It can be checkedthat Φ ( j F , j B ) = 0. This is an infinite precision test of two-isogeny. Ref. [28] used the approachof comparing the periods which are computed using elliptic integrals. This involves transcenden-tal functions while the approach we followed here only requires algebraic operations with rationalfunctions. For a “full” Baikov approach to an L -loop integral with E + 1 external legs one needs L ( L +1) + LE Baikov variables D a . In the present case ( L = 2, E = 1, M = L + E = 3), the variablesare D , . . . , D and the maximal cut corresponds to setting D = D = D = 0 at the end of thecomputation.We now follow [49] to derive the Baikov representation. The inverse propagators are D = k − m , D = ( k − k ) − m , D = ( p − k ) − m ,D = k , D = ( p − k ) . (15)Loosely following the notation of the paper above we set q = k , q = k and q = p and write s ij = q i · q j . The Gram determinant is G ( k , k , p ) = det s s s s s s s s p = s (cid:0) p s − s (cid:1) − s (cid:0) p s − s s (cid:1) + s ( s s − s s ) . (16)The Baikov polynomial is obtained by rewriting the Mandelstam variables s ij in terms of theinverse propagators D a in this Gram determinant, P ( D , . . . , D ) = G ( k , k , p ) (cid:12)(cid:12)(cid:12) s ij ( D a ) . (17)The cut integral ( D = D = D = 0) is of the form Z d D d D D α D α P (0 , , , D , D ) ( d − M − / . (18)Where α and α are the exponents of D and D in the original integral respectively. Since M = 3, d = 2 and α = α = 0 we get Z d D d D P (0 , , , D , D ) , (19) The astute reader may notice that this Gram determinant vanishes when in strictly two dimensions. If oneis uncomfortable with this one can instead derive a Baikov representation strictly in two dimensions. We will dosomething similar for the elliptic double-box in section 4.2. Details relevant for either case (in particular, how tohandle cases when the internal momenta are spanned by the external momenta) are presented in appendix A.5. P is a polynomial of overall degree three in D and D , P = 14 h − D D + D ( m − m )( m − p ) − ( m m − m p )( m − m + m − p ) − D ( D + ( m − m )( m − p ) − D ( m + m + m + p )) i . (20)The equation P = 0 defines an elliptic curve. We may again transform this curve to Weier-strass form. As it turns out, this curve has the same j -invariant as that from the loop-by-loopBaikov computation in the previous section. Rather than repeating it here we thus refer back toeq. (13). One convenient way to enforce on-shell conditions in two dimensions is via light-cone coordinates.We wish to enforce the conditions for the maximal cut: k − m = 0 , ( k − k ) − m = 0 , ( p − k ) − m = 0 . (21)We define the auxiliary momentum k = k − k and use that in light-cone coordinates thesquare of a momentum is given by k i = k + i k − i . Then the first two conditions in eq. (21) aresolved by k − = m k +2 , k − = m k +3 . (22)The last condition in eq. (21) becomes( p + − k +2 − k +3 )( p − − k − − k − ) − m = ( p + − k +2 − k +3 ) (cid:18) p − − m k +2 − m k +3 (cid:19) − m = 0 . (23)Introducing dimensionless quantities as k +2 = p + x , k +3 = p + y and again using µ i = m i p , theprevious equation becomes (1 − x − y ) (cid:18) − µ x − µ y (cid:19) − µ = 0 . (24)In homogeneous coordinates [ x : y : z ] and after multiplying by xyz we are left with a cubiccurve in P given by the equation P L ≡ xyz (cid:0) µ + µ − µ (cid:1) + x (cid:0) µ z − y (cid:1) + y (cid:0) µ z − x (cid:1) − z (cid:0) µ x + µ y (cid:1) = 0 . (25)This is an elliptic curve whose defining equation is closely related to the F -polynomial in (4).Specifically, their discriminants with respect to z are related bydisc z P L ( x, y, z ) = disc z F ( y, x, z ) . (26)Once again we can transform the curve to Weierstrass form, and evaluate its j -invariant. Assuggested by the relationship in eq. (26), we find it has the same j -invariant as the Feynmanparametric representation (given in eq. (7)), and a distinct (but isogenous) j -invariant to thosein the two Baikov representations. 7 .5 Rationalizing the square roots in the Baikov representation In subsection 3.2 we derived a loop-by-loop Baikov representation of the sunrise integral andexplained how the equation J = − p √ Q Q defines an elliptic curve isogenous to the oneobtained by Feynman parameters and the light-cone computation as in ref. [28]. Combining √ Q and √ Q in this way is safe if both Q and Q are positive. However, for complex kinematics itmay lead to an incorrect phase.Instead of combining the two roots, we can rationalize one of them. Recall that Q and Q were given in eq. (12) as Q = − p (cid:2) D − ( m − p ) (cid:3) (cid:2) D − ( m + p ) (cid:3) ,Q = − p (cid:2) D − ( m + m ) (cid:3) (cid:2) D − ( m − m ) (cid:3) . (27)Choosing to rationalize √ Q , the change of variables amounts to replacing D → t (cid:20) m t − m t + 1 (cid:21) , p Q → ( t ( m − m ) + ( m + m ))( t ( m + m ) + ( m − m ))2 p ( t − . (28)It turns out that the Jacobian from the change of variables cancels against the transformed √ Q and a factor of t − √ Q . In the end we obtain I ( p , m , m , m ) (cid:12)(cid:12)(cid:12) cut = p Z d D J = − p Z d D √ Q √ Q = − p Z d t √ R , (29)where R is a polynomial of degree four in t , R ≡ p (cid:2)(cid:0) ( m − p ) − m + m ) (cid:1) t − m − m ) t − ( m − p ) (cid:3) × (cid:2)(cid:0) ( m + p ) − m + m ) (cid:1) t − m − m ) t − ( m + p ) (cid:3) . (30)The equation y = R ( t ) defines an elliptic curve as a hypersurface in a weighted projective space P . It turns out that this curve has the same j -invariant as that from the graph polynomial(given in eq. (7)). Note that this is not the same j -invariant as in the Baikov representations above,even though the loop-by-loop Baikov representation was our starting point: by rationalizinginstead of combining roots we have achieved agreement with the graph polynomial and light-cone derivations of the elliptic curve.Another way to think about how the two curves emerge is to track what happens to thebranch points of the curves under the change of variables above. In ref. [28] and subsection 3.2the elliptic curve arising from the Baikov representation is defined by J = ( − p ) Q Q . Thisis a double cover of P branched over four points. Since Q and Q are already factorized, thebranch points are easy to read off: D (1)4 , ± = ( m ± m ) , D (2)4 , ± = ( m ± p ) . (31)These are four points on a projective line parametrized by the coordinate D . They have across-ratio λ with corresponding j -invariant j = 256 ( λ − λ +1) λ (1 − λ ) . This approach gives the “Baikov” j -invariant shown in eq. (13). 8n the other hand, when rationalizing the quadric Q we write x as the image of a map froma different P with coordinate t , t x ( t ) = 2 t (cid:20) m t − m t + 1 (cid:21) . (32)Under this change of variables, Q becomes a polynomial of degree four and we again define anelliptic curve as a double cover of P , but this time the P has coordinate t . The branch pointsare the preimages of the two points D (2)4 , + and D (2)4 , − . Since the change of variables is quadraticeach point has two preimages and we indeed get four branch points as required. As we now againhave four points on a projective line, we can form a cross-ratio and the corresponding j -invariant.This is the j -invariant that comes from the F -polynomial and the light-cone approach in eq. (7).The analysis presented here applies to the loop-by-loop Baikov representation, and at firstthis may make the full Baikov result seem mysterious, as unlike the loop-by-loop representationit does not obviously involve combining square roots. However, if one derives the Baikov rep-resentation by dividing each loop momentum into perpendicular and parallel subspaces, as forexample in ref. [46], then one naturally passes through a form closely related to the loop-by-looprepresentation in which there are indeed multiple square roots. In particular, the individualequations that need to be solved to land on the cut solution will be the same. If one understandsthe Baikov representation as a result of this kind of procedure, then the elliptic curve we foundfor it earlier can be explained in the same way as the loop-by-loop curve, and a similarly morecareful treatment (especially one along the lines of the next section) will result in the same curveas was found from Feynman parameters and light-cone coordinates. In this section we study the relation between the two genus-one curves from a different point ofview. We describe the curves purely by polynomial equations and we avoid introducing squareroots.On the maximal cut we have D = D = D = 0 and these equations together with D = p ( x + u ) define a curve. We introduce a dimensionless variable d = D p . Then,the equations (10) can be simplified by solving x = d − µ + 12 , (33) u = −
14 ( d − (1 + µ ) )( d − (1 − µ ) ) , (34) v = µ − y , (35) uv = (1 − y ) d − µ + 12 + µ − µ + µ − . (36)We now obtain the equation for the curve in variables y and d , by substituting the expressionsabove in ( uv ) = u v . This equation is P ( y, d ) = − y d + 2 yd + 2 (cid:0) µ − µ − µ + 1 (cid:1) yd − (cid:0) µ + 1 (cid:1) d ++ 2 (cid:0) − µ µ + µ µ + µ − µ (cid:1) y + 2 (cid:0) µ µ + µ (cid:1) d − µ µ − µ + 2 µ µ − µ + 2 µ µ − µ = 0 , (37)9nd is a cubic equation in y and d . It is not in Weierstrass form. The expression for the Jacobiancan also be written in the variables y and d : J = p (cid:0) yd − ( d + µ − µ )( d − µ + 1) (cid:1) . (38)Note that this approach avoids introducing square roots, at the cost of working with twovariables constrained by an algebraic relation.Let us show that d d J is the holomorphic one-form on this curve. Taking the differential of P ( y, d ) = 0 we obtain (cid:18) ∂P∂y (cid:19) d y + (cid:18) ∂P∂d (cid:19) d d = − J ( y, d ) d y + K ( y, d ) d d = 0 , (39)where K ( y, d ) = − y + 4 yd + 2( µ − µ − µ + 1) y − µ + 1) d + 2 µ µ + 2 µ . (40)Since we assume that the curve described by P = 0 is nonsingular, we have that ∂P∂y = − J and ∂P∂d = K can not vanish simultaneously. Then, we have d d J = 2 d yK . Hence, one can see thatat the zeros of J this holomorphic form does not have poles, when written with the denominator K . It can be checked that this curve is the same as the one obtained by the more traditionalBaikov approach.However, one can see that the curve we started with, in the variables x , y , u , v and d is adouble cover of the curve P ( d , y ) = 0. Given a point ( d , y ), we can uniquely find x and u , v and uv . This allows us to solve for u and v up to a sign. Hence, to a point on the curve P ( d , y ) = 0 correspond two points on the initial curve defined by D = D = D = 0 and d = x + u .To find a one-to-one projection of the curve which is easily recognizable as an elliptic curve,we proceed as follows. We can use a kind of Euclidean lightcone construction and transform theequations to y + iv = µ y − iv , (41)( x − y ) + i ( u − v ) = µ ( x − y ) − i ( u − v ) , (42)( x −
1) + iu = µ ( x − − iu . (43)Combining them, we find µ y − iv + µ ( x − y ) − i ( u − v ) − µ ( x − − iu = 1 . (44)If we introduce ζ = y − iv and ξ = x − iu , we have a curve µ ζ + µ ξ − ζ − µ ξ − , (45)which is a cubic equation in ( ζ, ξ ). Once we have ζ and ξ we obtain y = 12 (cid:18) ζ + µ ζ (cid:19) , v = 12 i (cid:18) − ζ + µ ζ (cid:19) , (46)10nd similarly for x and u . Finally, we obtain d = x + u . This time, given a point ( ζ, ξ ) wecan find a unique point on the initial curve.This second curve looks very similar to the lightcone solution of sec. 3.4 and indeed it hasthe same j -invariant. Recall that the j -invariant of an elliptic curve is j = 1728 g ∆ , (47)where ∆ is the elliptic discriminant. When ∆ vanishes, j is singular, and the elliptic curvedegenerates.For the curve arising from Feynman parametrization and the light-cone computation weobtained j F = (cid:2) ( ξ − ξ − ξ − ξ −
1) + 16 µ µ µ (cid:3) µ µ µ ( ξ − ξ − ξ − ξ −
1) (48)while for the curve arising from the Baikov representation we obtained j B = (cid:2) ( ξ − ξ − ξ − ξ −
1) + 256 µ µ µ (cid:3) µ µ µ ( ξ − ( ξ − ( ξ − ( ξ − . (49)The denominators of these expressions are distinct, but they clearly have the same zeros, justwith different multiplicities. These zeros all correspond to physical singularities of the diagram,either to thresholds, pseudo-thresholds, or vanishing internal masses. Each corresponds to aconsistent Landau diagram, for particular choices of the sign of the energies of each particle. Theeasiest to recognize are the thresholds, occurring when ( ξ ) = 1 and thus ( m + m + m ) = p ,which is the condition for energy conservation when all of the intermediate particles are travelingin the same direction. The Landau analysis also reveals that there are other singularities, arisingat pseudo-thresholds p = ( − m + m + m ) , p = ( m − m + m ) and p = ( m + m − m ) .In terms of variables ξ these are ( ξ ) = 1, ( ξ ) = 1 and ( ξ ) = 1. Finally, the singularitiesarising when one of the masses vanishes are of a different type. They arise due to the fact thatwhen one of the masses vanishes the integral becomes divergent. The elliptic double-box integral has previously been analyzed in ref. [52] from the point of viewof direct integration in a Feynman parametric representation, and in ref. [56] from the point ofview of the maximal cut in twistor space. In both papers the same elliptic curve was found usingvery different methods. In this section we derive a Baikov representation of the double-box, andshow that it also defines the same curve.
The Baikov representation is a rewriting of Feynman integrals where the integration is overLorentz-invariant quantities, such as dot products. In appendix A we derive such a representationfor the elliptic double-box integral shown in fig. 2 (see in particular appendix A.4).11 p p p p p k + p k + p k + p k k + p k − k k + p Figure 2:
Double-box integral in momentum space. Incoming momenta are assumed to be off-shell,i.e. p i = 0, and p i ··· i n ≡ p i + · · · + p i n . The internal propagators are massless. The maximal cut of the elliptic double-box can be written in a loop-by-loop Baikov repre-sentation as an integral over two Baikov parameters. The cut integrand takes the followingform:
J √G d x d x B ( x , x ) p B ( x , x ) , (50)where x and x are the two remaining Baikov variables after all propagators have been cut. Thepolynomials B and B are of degree two in x and also of degree two in x . The factors J and G only depend on the external kinematics. We include expressions for these polynomials in anancillary file, doublebox_baikov_rep.txt .To obtain an elliptic curve, we may begin by taking a residue around B ( x , x ) = 0. Withoutloss of generality, let us take this residue in x . Solving B ( x , x ) = 0 for x introduces a squareroot that contains x , and this square root can be rationalized by Euler substitution as done insubsection 3.5 for the sunrise integral. Denoting by t the variable that replaces x to rationalizethe square root we find that B ( t ) becomes a quartic polynomial in t . We can therefore define anelliptic curve by y = B ( t ) and compute its j -invariant through standard changes of variables.The problem with this approach is that the change of variables from x to t may itselfintroduce a square root in the kinematic parameters. Since the j -invariant of the elliptic curveis expected to be a rational function of the kinematics, this root is spurious and must cancel in j . The spurious kinematic root can be avoided if we view x and x as a subset of the coordinateson a P with homogeneous coordinates [ x : x : y : z ]. From the denominator in the integrandin eq. (50) we define the two quadrics Q : { [ x : x : y : z ] ∈ P | B ( x , x , z ) = 0 } ,Q : { [ x : x : y : z ] ∈ P | y − B ( x , x , z ) = 0 } . (51)The integrand in eq. (50) corresponds to a differential form on the intersection of Q and Q .For generic quadrics Q and Q this intersection is a smooth curve of genus one. Note that here we are writing B ( x , x , z ) for the homogenization of the polynomial B in eq. (50) andsimilarly for B .
12e now review briefly how this curve may be characterized and refer to [57, Chapter 22] forfurther details. The quadrics Q and Q generate a family of quadrics { λ Q + λ Q | [ λ : λ ] ∈ P } . (52)This family is called the pencil of quadrics and the intersection C = Q ∩ Q is called the baselocus of the pencil. The members Q λ of the pencil are quadrics in P and for some choices of λ ∈ P they may be singular. If Q and Q intersect transversely, there are four such singularmembers Q λ i with i = 0 , . . . ,
3. Out of the four points λ i we can form a cross-ratio κ andsubsequently the invariant combination j = 256 ( κ − κ + 1) κ ( κ − , (53)which characterizes the pencil of quadrics up to isomorphism. One can now moreover show thatthe base locus C of the pencil is isomorphic to a genus-one curve in the plane with the same j -invariant as the pencil.An advantage of this description is that it allows us to compute the elliptic discriminant ofthe curve using only rational operations. Writing Q and Q for the 4 × λ i of the singular members of the pencil aregiven by the eigenvalues of the matrix Q − Q . The curve degenerates if two of those points in P are the same, i.e. if Q − Q has a double eigenvalue. This leads to the expression∆ = disc λ det( λ − Q − Q ) (54)for the elliptic discriminant. Moreover, a defining equation for the curve is given by y = det( x − Q − Q ) = 0. This depends rationally on the kinematic variables contained in Q and Q and aWeierstrass form and the j -invariant can subsequently be computed by rational transformations.It turns out that the elliptic curve obtained in this way has the same j -invariant as thosecomputed from twistor space in ref. [56] and from the parametric representation of ref. [52]. Aswe do not need to combine distinct square roots in this representation, this is consistent withour observations in the previous section.In the submission of this paper to the arXiv, we have attached the file doublebox_curve.txt that contains an expression for the defining equation of the curve. With minor modifications thefile should be readable with most computer programs. In this section we present a derivation of the Baikov form without using dimensional regulariza-tion. This avoids having to take the potentially somewhat tricky limit d →
4. Equivalently, onecan obtain the cut integrand as a one-form and it is not necessary to take one extra residue asin sec. 4.1.Consider the loop parametrized by k in the elliptic double-box. This loop has denominators D = k , D = ( p + k ) , D = ( p + k ) , D = ( k − k ) . (55)It has “external” momenta p , p , k + p and k . The integral measure d d k decomposes intoan integral d k k over the space spanned by the independent “external” momenta p , p and k ,13nd an orthogonal integral d d − k ⊥ . The dot products of k with the “external” momenta are k · p = 12 ( D − p − D ) , (56) k · p = 12 ( D − D − p + p ) , (57) k · k = −
12 ( D − D − k ) . (58)Using identities from appendix A.5, it follows thatd k k = d( k · p )d( k · p )d( k · k )det G ( p , p , k ) = −
18 d D d D d D + d D ( · · · )det G ( p , p , k ) , (59)d d − k ⊥ = 12 Ω d − (cid:18) det G ( k , p , p , k )det G ( p , p , k ) (cid:19) d − d D . (60)Of course, we do not need to keep the dimension d arbitrary and we can set d = 4 here. In thatcase we have Ω = 2.When computing the full d k measure the extra terms in d k k proportional to d D dropout: d k = − D d D d D d D (cid:0) det G ( p , p , k ) (cid:1) − (cid:18) det G ( k , p , p , k )det G ( p , p , k ) (cid:19) − . (61)Note that we have not canceled the factor det G ( p , p , k ) since we do not allow ourselves tocombine square roots. Note also that we have some Gram determinants whose entries contain k · p , k · p and k . We need to keep these dot products in mind when analyzing the k integral,to which we turn next.For the k integral we have new denominators D = ( k + p ) , D = ( k + p ) , D = ( k − p ) , (62)while in the Jacobian of the d d k integral we have k , k · p and k · p . We introduce two newLorentz-invariant quantities D = k and D = ( k + p ) .However, not all quantities D , . . . , D can be independent; there are five such quantities andonly four components for the vector k . The relation connecting these quantities can be obtainedby computing the Gram determinant det G ( k , p , p , p , p ) = 0. Equivalently, we canantisymmetrize in five different vectors to obtain k µ ǫ ( p , p , p , p ) − p µ ǫ ( k , p , p , p ) + p µ ǫ ( k , p , p , p ) − p µ ǫ ( k , p , p , p ) + p µ ǫ ( k , p , p , p ) = 0 . (63)When decomposed over the basis p , p , p and p , k has components k · p , etc.,with a metric given by the inverse of the Gram matrix G ( p , p , p , p ). The scalarproducts k , k · p and k · p can be computed from this decomposition. In particular, thisimplies that we can compute D = k in terms of the other D i (since here there are no transversalcomponents there is no need to introduce D at all). Let us compute the measure d k in termsof D , D , D and D . Using eq. (105), we findd k = 12 (det M ) − det M (cid:0) det M det M (cid:1) d D d D d D d D , (64)14here M = G ( p , p , p , p ) , (65) M = D ( D + D − p ) ( D + D − p ) ( D + D − p ) D ( D + D − p ) ( D + D − p ) D ( D + D − p ) D . (66)Here we have written only some of the matrix entries, the others can be determined from theseby symmetry.When taking the cuts we need to set D through D to zero, and thus we only need theexpression for det M when D = · · · = D = 0. Then det M is a quadratic polynomial in D . Taking the squares of the Jacobians obtained in this section we obtain a genus-one curveas an intersection of two quadrics. This curve has the same j -invariant as the one obtained byconsidering the curve embedded in momentum twistor space as described in ref. [56]. We have shown that the maximal cut and the Feynman parametrization of the two-loop sunriseintegral do not necessarily correspond to different elliptic curves. The observation of differentcurves for these two objects in the literature was an artifact due to combining two square roots,and a more careful treatment shows the same curve for both the Feynman-parametric and Baikovrepresentation, reinforced by the observation of the same curve in a light-cone parametrizationof the maximal cut. We have shown that similarly the Baikov and twistor representations of theelliptic double-box also describe the same elliptic curve.In some ways, the appearance of the same curve in different representations of these integralsshould not be surprising. If one thinks of the maximal cut as a variety in loop momentum space,that variety should already define an elliptic curve. Whether we parametrize it with Baikov,light-cone, or twistor coordinates, we are performing changes of variables which should preserveinvariant features of the geometry, such as the j -invariant. From this perspective, the surpriseis actually that this curve is preserved in Feynman parameters. Feynman parameters do notcorrespond straightforwardly to a change of variables from the initial loop momenta, so the factthat they apparently preserve the geometry deserves further explanation.One of the implications of our work is that analytic continuation of the Baikov representationaway from from the Baikov integration domain has to be done with some care. Inside this domainthe Jacobians involved in changing coordinates are positive and one can pick the positive solutionof any square roots that appear. However, while this is possible for Euclidean kinematics, thereis no canonical choice of square roots outside this region.In ref. [58], an extension of the notion of leading singularity was put forward which appliesto integrals containing genus-one curves as well. The construction in that reference implicitlyassumes a fixed geometry for the genus-one curve. If there were a genuine ambiguity in theunderlying genus-one curve it is not clear how one should modify their construction. Fortunately,the results of this paper imply that such a modification may not be necessary.In previous investigations of the elliptic double-box, conformal symmetry served as an impor-tant constraint that allowed for particularly clean representations. The Baikov representation isby its nature not conformal, as it uses momentum invariants as variables. It would be interestingto find a variant of Baikov that preserves conformal symmetry, to make better use of this kindof representation in the context of, e.g., N = 4 super Yang-Mills.15inally, there is a broader concern raised by the observations of refs. [15, 28] that we do notfully address. While we do find the same curve for both the cut and Feynman parametrizationof the sunrise integral, this by no means shows that isogenies are never relevant to the ellipticintegrals that occur in physics. In particular, while our work suggests that each elliptic Feynmanintegral has a preferred curve, it may be that there exist distinct diagrams whose correspondingcurves are isogenous. If such an example were to be found, it would suggest the need for aformalism that relates not merely iterated integrals on the same elliptic curve, but iteratedintegrals on isogenous curves as well. Acknowledgements
We thank Stefan Weinzierl for helpful discussions. This work was supported in part by theDanish Independent Research Fund under grant number DFF-4002-00037 (MV), the DanishNational Research Foundation (DNRF91), the research grant 00015369 from Villum Fonden, aStarting Grant (No. 757978) from the European Research Council (MV, MvH, CV) and the Euro-pean Union’s Horizon 2020 research and innovation program under grant agreement No. 793151(MvH). This project has received funding from the European Union’s Horizon 2020 researchand innovation program under the Marie Skłodowska-Curie grant agreement No. 847523 ‘IN-TERACTIONS’ (HF). The work of HF has been partially supported by a Carlsberg FoundationReintegration Fellowship.
A Baikov representations with derivations
In this appendix we carefully derive the Baikov representation in its loop-by-loop and its standardforms. This derivation mostly follows ref. [46] and the loop-by-loop part additionally ref. [49].
A.1 The one-loop case
As both the loop-by-loop and standard Baikov representations build off of the Baikov represen-tation at one loop, we will start by reviewing the situation there. Writing a generic one-loopintegral, I = Z d d kiπ d/ N ( k ) P ( k ) a · · · P P ( k ) a P (67)we then split the integral up in parts parallel and perpendicular to the space spanned by the E independent external momenta:d d k = d E k k d d − E k ⊥ (68)= d E k k | k ⊥ | d − E − d | k ⊥ | d d − E − Ω . (69)Using Z d n − Ω = Ω n = 2 π n/ Γ( n/
2) (70)we get I = 2Γ(( d − E ) / iπ E/ Z N ( k ) d E k k | k ⊥ | d − E − d | k ⊥ | P ( k ) a · · · P P ( k ) a P . (71)16e may write the parallel component as k k = E X i =1 z i p i , (72)which implies that k k · p j = k · p j = E X i =1 z i p i · p j . (73)We introduce the Gram matrix G with entries G ij = p i · p j . This allows us to write, z i = E X j =1 G − ij ( k · p j ) . (74)We further have that k k = E X i,j =1 z i z j G ij = E X i,j =1 ( k · p i )( G − ) ij ( k · p j ) . (75)We may pick a basis in which the quantities ς i := k · p i . (76)are the components of the vector k k . In that case, the metric is nontrivial and is given by theinverse of the Gram matrix. The integration measure is thend E k k = (det G − ) E Y i =1 d ς i . (77)The orthogonal part has norm k ⊥ = k − k k . Including the expression for k k we have k ⊥ = k − E X i,j =1 ( k · p i )( G − ) ij ( k · p j ) . (78)Let us form the ( E + 1) × ( E + 1) Gram matrix,ˆ G = (cid:18) k k · p i k · p j G ji (cid:19) . (79)Using the expression for the determinant of a matrix written in terms of blocks, we have thatdet ˆ G = k − E X i,j =1 ( k · p i )( G − ) ij ( k · p j ) det G. (80)Hence, ( k ⊥ ) = det ˆ G det G .Using the expression of k ⊥ from eq. (80), we find that | k ⊥ | d | k ⊥ | = | k | d | k | + . . . , where themissing terms contain components d ς i which vanish when wedged into d E k k . This means thatwe get the relation d | k ⊥ | d E k k = | k ⊥ | − d ς d E k k (81)17here we have used the notation ς = k .Inserting eqs. (81, 78, 77) into eq. (71) we get I = G ( E − d +1) / Γ(( d − E ) / iπ E/ Z N ( ς ) B ( ς ) ( d − E − / d E +1 ςP ( ς ) a · · · P P ( ς ) a P , (82)where we have defined B := det ˆ G = det G ( k, p , . . . , p E ) , G := det G = det G ( p , . . . , p E ) , (83)with G denoting the Gram matrix.Now the only step left is to change to the Baikov variables x i , which equal the propagators.If there are too few propagators (P < E + 1) one will need to introduce additional variables, butthis is mostly relevant at higher loops. The Jacobian J for the change of variables will dependon the exact expressions used for the propagators, but for most conventions it equals, J = ± − E . (84)Thus the final result for a one-loop Baikov representation is I = J G ( E − d +1) / Γ(( d − E ) / iπ E/ Z N ( x ) B ( x ) ( d − E − / d E +1 xx a · · · x a P P . (85) A.2 Multi-loop, the loop-by-loop approach
With this representation in hand, we now want to apply it to multi-loop cases. A multi-loopFeynman integral is given by I = Z d d k iπ d/ · · · d d k L iπ d/ N ( { k } ) P ( { k } ) a · · · P P ( { k } ) a P (86)Our strategy will be to go through the steps from the previous section one loop at a time, startingwith loop number L and then going down towards 1. We call E l the number of momenta externalto loop number l . This may include the loop momenta of lower-numbered loops. We will denotewith G l the Gram-matrix of the momenta external to loop l , while B l is the same but with theloop-momentum k l included. If we follow the steps of the previous section with this notation, wearrive at the correspondenced d k l iπ d/ → G ( E l − d +1) / l B l ( ς l ) ( d − E l − / Γ(( d − E l ) / iπ E l / d E l +1 ς l (87)where ς l corresponds to the set of dot-products between k l and itself along with the momentaexternal to the l th loop. Putting this together for each loop gives I = ( − i ) L π − ( P i E i ) / Q Ll Γ(( d − E l ) / Z N ( ς ) (cid:16)Q Ll G ( E l − d +1) / l B ( d − E l − / l (cid:17) d ( P i E i )+ L ςP ( ς ) a · · · P P ( ς ) a P (88)and changing to the Baikov variables gives the final expression for the loop-by-loop Baikovrepresentation: I = J ( − i ) L π − ( P i E i ) / Q Ll Γ(( d − E l ) / Z N ( x ) (cid:16)Q Ll G ( E l − d +1) / l B ( d − E l − / l (cid:17) d ( P i E i )+ L xx a · · · x a P P (89)18here the Jacobian for the final variable change still depends on the specific expressions used forthe propagators, but is usually given as J = ± − ( P i E i ) . (90)The expression of eq. (89) may also be found in ref. [59]. A.3 Multi-loop, the standard approach
The standard approach to multi-loop Baikov parametrization can be thought of as a version of theloop-by-loop approach, but with the assumption that all loops depend on all lower loop-momentaand all external momenta. This means E l = E + l − G l = B l − since their definitions will be the same. We also have that thepower of G l which appears in the expression, ( E l − d + 1) /
2, is equal to minus the power withwhich B l − appears, making the two contributions cancel. This will happen pairwise for eachloop, leaving only B L and G . Renaming these to B and G means we have B = det G ( p , . . . , p E , k , . . . , k L ) and G = det G ( p , . . . , p E ) . (92)Then eq. (89) becomes I = J ( − i ) L π L − n G ( E − d +1) / Q Ll =1 Γ(( d +1 − E − l ) / Z N ( x ) B ( d − E − L − / d n xx a · · · x a P P (93)where we have used and defined n ≡ L + X i E i = EL + L ( L + 1) / J = ± L − n . We see that for L = 1 eq. (93) reduces nicely toeq. (85). A.4 The elliptic double-box
Let us look at the example of the elliptic double-box shown in fig. 2. We have the propagatingmomenta q = k , q = k + p , q = k + p ,q = k + p , q = k + p , q = k + p , (95) q = k − k , q = k , q = k + p . The last two q and q do not actually appear in the diagram, but they are needed to express allscalar products in terms of the Baikov variables.We have E = 3, counting the three momenta k , p , p that are external to the k -loop,while E = 4 since this is the maximum number of independent momenta in four space-timedimensions. The four Gram determinants appearing are B = det G ( k , k , p , p ) , G = det G ( k , p , p ) , B = det G ( k , p , p , p , p ) , G = det G ( p , p , p , p ) . (96)19e have J = ± − (cid:16) det G ( p ,p ,p ,p )det G ( p ,p ,p ,p ) (cid:17) . Putting this together in eq. (89) we obtain the expression I = J π − / G (5 − d ) / Γ(( d − /
2) Γ(( d − / Z N ( x ) G (4 − d ) / B ( d − / B ( d − / d xx a · · · x a . (97) A.5 Derivation of a four-dimensional Baikov representation
In this section we consider the case when there is no orthogonal component k ⊥ = 0, which willbe needed for our derivation of a Baikov representation in four dimensions. We also introducethe vectors v i which are defined from the denominators D i = ( k − v i ) , corresponding to masslesspropagators. We take all of these vectors to be nonvanishing. In other words, we will use as newcoordinates the quantities D i , but k will not be one of these coordinates.Then, we have d d k = (det G − ) d Y i =1 d( k · v i ) . (98)We want to express this in terms of D i = ( k − v i ) instead of k · v i . We have d Y i =1 d D i = d Y i =1 k − v i ) · d v = 2 d d Y i =1 ( k · d k − d( k · v i ))= ( − d d Y i =1 d( k · v i ) − d X j =1 ( − j − ( k · d k ) Y i = j d( k · v i ) . (99)Plugging in k · d k = P k,l ( k · v k )( G − ) kl d( k · v l ), we obtain d Y i =1 d D i = ( − d − d X j,k =1 ( k · v k )( G − ) kj d Y i =1 d( k · v i ) . (100)Let us rewrite the Jacobian in a simpler way − d X j,k =1 ( k · v k )( G − ) kj det G = det (cid:18) k · v j G ji (cid:19) = det ij (( v i − k ) · v j ) . (101)To compute this last determinant, consider the decomposition of k − v i and k − v j on the basisof vectors v k . Upon taking the dot product in this basis we obtain( k − v i ) · ( k − v j ) = d X k,l =1 (( k − v i ) · v k )( G − ) kl (( k − v j ) · v l ) , (102)whence det ij (( k − v i ) · ( k − v j )) = (cid:0) det ik (( k − v i ) · v k ) (cid:1) (det G ) − . (103)Since( k − v i ) · ( k − v j ) = 12 (cid:2) ( k − v i ) + ( k − v j ) − ( v i − v j ) (cid:3) = 12 (cid:2) D i + D j − ( v i − v j ) (cid:3) , (104)20he determinant det ij (( k − v i ) · ( k − v j )) can be written in terms of D variables and constants( v i − v j ) . This determinant is the Cayley-Menger determinant which arises when computing thevolume of a simplex in Euclidean space.In the end, we findd d k = ( − d d (det G ) − det G (cid:0) det ij (( k − v i ) · ( k − v j )) det G (cid:1) d Y i =1 d D i . (105) References [1] D. J. Broadhurst, J. Fleischer, and O. V. Tarasov, “Two loop two point functions withmasses: Asymptotic expansions and Taylor series, in any dimension,”
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