The propagator seagull: general evaluation of a two loop diagram

Transcription

1 Prepared for submission to JHEP arxiv: v [hep-th] 11 Oct 018 The propagator seagull: general evaluation of a two loop diagram Barak Kol and Ruth Shir The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel barak.kol, ruth.shir@mail.huji.ac.il Abstract: We study a two loop diagram of propagator type with general parameters through the Symmetries of Feynman Integrals SFI) method. We present the SFI group and equation system, the group invariant in parameter space and a general representation as a line integral over simpler diagrams. We present close form expressions for three sectors, each with three or four energy scales, for any spacetime dimension d. We determine the singular locus and the diagram s value on it.

2 Contents 1 Introduction 1 SFI group and equation system 3 3 Group orbits 7 4 General solution 9 5 Sector evaluation Massless m 3, m Massless m 3, m Massless m Singular locus 19 7 Summary and Discussion 1 A Collection of function definitions 1 Introduction Analytic calculation of Feynman diagrams with several mass scales is a challenge, but it is important for higher loop corrections to Standard Model/Core Theory observables involving several different particles, and it is of intrinsic interest to Quantum Field Theory. The Symmetries of Feynman Integrals method SFI) [1], see also developments in [ 7], reduces the diagram to its value at some allowed and more convenient base point in parameter space, namely the space X of masses and kinematical invariants, plus a line integral in X over simpler diagrams with one edge contracted). This method is close in spirit to the Integration By Parts IBP) method [8] and to the Differential Equations DE) method [9 1], see also the textbooks [13]. The main new feature of SFI is recognizing a continuous group G which is associated with the Feynman Integral and identifying its action in parameter space, thereby leading to the above-mentioned reduction. SFI offers a new approach to the evaluation of Feynman diagrams and it is important to demonstrate it and further develop it through application to specific diagrams. The method suggests to partially order all diagrams according to edge 1

3 V n Figure 1. Hierarchy of diagrams according to edge contraction. Each column has diagrams of fixed number of vertices so that necessary sources for each diagram are always on its left. The propagator seagull is on the third column from the left, second from bottom. contraction as shown in Fig. 1. The leftmost column consists of the tadpole which is immediate to evaluate through integration of Schwinger parameters. The next column includes the diameter and bubble diagrams which were treated through SFI in [3, 6] where novel derivations for their known values were described. The vacuum seagull is in the bottom of the third column, and its analysis through SFI enabled a a novel evaluation of a sector with three mass scales [4]. The kite diagram, which is on the fourth row from the left, second from bottom, was analyzed through SFI in [1] and it was possible to identify a locus in parameter space where it reduces to a linear combination of simpler diagrams, thereby maximally generalizing the massless case, studied in [8], and the application of the diamond rule of [14]. In this paper we analyze a diagram which we call the propagator seagull. It is second from bottom on the third column from the left in Fig. 1. It is a two-loop diagram of a propagator type which can be gotten from the vacuum seagull by cutting open one of its two parallel loops, thereby explaining its name. Being on the third column, the propagator seagull appears as a source for more complicated diagrams, such as the tetrahedron and the kite fourth column, bottom two). In addition we shall see it is special in providing a first example where the group orbits are not open in parameter space, but rather are of co-dimension 1. [15] studied all two-loops diagrams of propagator type and reduced the case of arbitrary indices powers of propagators) to the kite, the propagator seagull and the sunrise the master integrals). [16], section 4.3, expressed the general case in the small p limit in terms of a straightforward quadruple sum which is a generalized

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The propagator seagull hypergeometric series. In [17] the propagator seagull was considered with one mass scale but with arbitrary indices using the Mellin-Barnes method [18]. The propagator seagull with two mass scales was calculated using dispersion relations in [19], using dispersion relations with Mellin-Barnes method in [0], using an ɛ-expansion for the differential equation method in [1] and using Mellin-Barnes in []. Other two mass scale sectors were calculated in [3] via Integration By Parts reduction and recalculated in [4]. For three mass scales an analytic expression for any dimension involving Appell hypergeometric functions was given in [0] and in [19] and [5] in an ɛ expansion. With four mass scales the ɛ 0 term was calculated through differential equations in [6]. The paper is organized as follows. We start by deriving the SFI equation system and the associated group in section. Next, in section 3 we obtain the group orbits in parameter space and the associated group invariant. Section 4 presents a general formula for a reduction to a base point where two masses are put to zero) plus a line integral over simpler diagrams. Section 5 contains the evaluation of the line integral for several sectors with 3 and 4 energy scales. Section 6 determines the singular locus where the diagram simplifies to a linear combination of simpler diagrams rather than a line integral thereof) and presents the corresponding evaluations. We summarize our results and disucss them in section 7. Finally an appendix provides a useful collection of diagram evaluations and definitions of special functions. SFI group and equation system In this paper we will consider the two-loop propagator type diagram shown in Figure and given by the integral Ix 1, x, x 3, x 4, x 5 ) = d d l 1 d d l l 1 x 1 )l x )l 1 + l ) x 3 )l 1 + p) x 5 ).1) 3

5 where the parameter space is X = {x 1, x, x 3, x 4, x 5 ) = m 1, m, m 3, p, m 5) }.) and the numbering of the x s is chosen to conform with that of the vacuum seagull which is gotten by tying together the external legs at infinity. The discrete symmetry group is a reflection exchanging propagators and 3, namely Γ = Z..3) SFI method. Before we proceed to obtain the SFI equation system let us give a short review of the SFI method. The integral.1) is invariant under transformations of the integration variables l A, and in particular under infinitesimal linear variations of the form l A l A + ɛ AB l B.4) l A l A + ɛ A p.5) with A, B = 1, and ɛ AB, ɛ A are small. 1 When applied to the integral.1) such variations induce the following equations 0 = d d l 1 d d l l B Ĩ l A 0 = d d l 1 d d l p l Ĩ A with Ĩ the integrand. Another useful equation is obtained by the variation p p + ɛ p p which does not leave the integral invariant but nevertheless gives the following useful identity p p I = d d l 1 d d l p p Ĩ..6) Specializing to the propagator seagull, one can consider in this way 7 variations of the form.4,.5,.6). While performing the variations, numerators are generated. If the numerators can be expressed as a linear combination of current squares, then one obtains a differential equation for I. However, not all current quadratics can be expressed so: the space of quadratics Q = Sp{l 1, l 1 l, l, p l 1, p l, p } is 6d while the space of squares S = Sp{l 1, l, l 1 + l ), l 1 + p), p } is 5d, and thus there is a single irreducible numerator, which can be chosen to be p l. Variations which avoid the generation of the irreducible numerator, or equivalently, those which preserve S, generate the differential equations of interest. By 1 The notation ɛ AB should not be confused with the d volume tensor. 4

6 direct computation we find that the group G of linear transformations on the space of loop and external momenta can be represented by the following variations l 1 ɛ l 1 δ l = ɛ 1 ɛ 0 l..7) p 0 0 ɛ p p SFI equation system. The variation.7) translates into operating on the integrand by either one of the following variations l 1 l 1, l l, l l 1, or p. which in p turn give a system of 4 partial differential equations in parameter space of the form: c a Ix) T x) a i x i Ix) = J a x).8) where a = 1,..., 4 enumerates the equations and i = 1,..., 5 enumerates the parameters. c a are constants which could depend on spacetime dimension d; T x is a 4 5 matrix with entries linear in the x s and J a x) are linear combinations of Feynman diagrams which arise from Ix) by contracting one propagator We choose to present the equation system in the basis defined by the following variations E 1 l 1 l 1 + l l + p p In it.8) reads E E 3 E 4 := l l l l 1 + l ) p p d 8 0 d 3 c = d 3, Jx) = 3 O O 1 )I O 3 O 1 )I 1 5 O 1 I.9).10) x 1 x x 3 x 4 x 5 0 x T x = s 1 A s 1 A x ) x 4 s 1 B This basis is adapted to the reflection symmetry of the diagram in the sense that equations 1 and 4 are invariant under reflection, while and 3 are exchanged. The notation above is defined as follows. The s variables are defined for any trivalent vertex v by s i v := x j + x k x i )/.1) 5

7 where i, j, k are the three edges which meet at v see also [6, 7]). The propagator seagull has two trivalent vertices: A := 13), B := 145), as shown in the fig. and hence s 1 A := x + x 3 x 1 )/ s 1 B := x 4 + x 5 x 1 )/..13) For i =,..., 5 propagator i belongs exactly to a single trivalent vertex either A or B) and hence the subscript denoting the vertex may be omitted. For example For later use we define the Heron / Källén invariant and for each trivalent vertex v so that in particular s 5 s 5 B := x 1 + x 4 x 5 )/..14) λ := x + y + z x y x z y z.15) λ v := λx i, x j, x k ).16) λ A := λx 1, x, x 3 ) λ B := λx 1, x 4, x 5 )..17) Motivation and geometric interpretation for the definitions of λ and s can be found in [6]. The O i operators i = 1,..., 5 which appear in J a denote the diagram gotten by omitting, or contracting, the i th propagator. Note however, that O 5 does not appear in the SFI equations. Accordingly, the possible topologies for the degenerations are [ ] { } Degen =,,.18) namely, the sunrise and the tadpole times the bubble. Finally, the derivatives are defined as i := / x i. Comments. Relation with vacuum seagull. The group G given by.7) is of the form G = G vac T,1 = 0 = 0.19) where G vac is the group for the vacuum closure of the diagram, namely the vacuum seagull [4], T,1 is a triangular block matrix describing the variations.4,.5,.6) and 6

8 a denotes an arbitrary entry. This is in agreement with the general result for a diagram with or 3 external legs, see [3] eq..16) and below. Degeneracy of obstructions. G has the property that its dimension is larger by 1 from the minimum dimension gotten by considering possible obstructions, namely dimobst) = dimnum) dimp rop) = 1 4 = 4.0) while dimg) = 4 dimt,1 ) dimobst) = 7 4 = 3.1) This means that not all of the possible obstructions are actually generated by the variations. We find that the actual obstructions can be characterized as those which do not include / x 1, or equivalently, those which annihilate x 1. Precisely the same comment applies to the vacuum seagull. Forbidden terms. In the presence of external legs we consider all variations except for those of the form p l. Hence, terms of the form µ p r p s.) cannot appear in the equations, where µ is any mass-squared. We note that by taking the equation system for the vacuum seagull and keeping the maximal subspace which does not include such forbidden terms one reproduces G. 3 Group orbits The matrix T x in the SFI equation system.8,.11) defines an action of the SFI group G on parameter space X. In this section we study the geometry of the G orbits, which are the characteristic surfaces for the SFI equation set. We have 4 equations in a 5d parameter space, hence the group orbits are at most 4d. According to the method of maximal minors we should compute the 4- minors M i gotten by omitting column i, taking the determinant and multiplying by an alternating sign see [5] for a full definition in terms of the ɛ tensor). We find M i = 4 λ A x4 s 4, 0, 0, x 1 s 1 B, x 1 x 4 ). 3.1) Since M i are not identically zero we conclude that generically a G-orbit is indeed 4d, or equivalently, codimg orbits) = 1. 3.) The propagator seagull is our first example for a diagram with non-zero codimension previous examples were codimension 0 and include the diameter [6], the 7

9 bubble [3], the vacuum seagull [4] and the kite [7]). It means that G has single invariant in X, which we denote by φ and which we proceed to determine. Points in parameter space with different values of φ cannot be related to each other through the SFI method. According to the method of maximal minors factorization, M i factorizes into M i = S Inv i. 3.3) Comparison with 3.1) identifies the common factor which determines the singularity locus to be S = λ A 3.4) while the invariant 1-form is Inv Inv i dx i = x 4 s 4 dx 1 + x 1 s 1 B dx 4 x 1 x 4 dx ) Note that Inv involves only the parameters x 1, x 4, x 5 associated with vertex B) and hence it can be considered as a 1-form in 3d space. This means that m, m 3 can be varied without leaving the G-orbit, and in particular every G orbit includes a point where m = m 3 = ) The invariant 1-form annihilates the tangent bundle to G-orbits, and it must be proportional to the differential of the G-invariant φ, namely Inv = f dφ 3.7) for some function f. In order to determine φ we may first solve for f through dinv = d log f Inv. 3.8) We find f = x 1 x 4 ) 3/, and substituting back into 3.7) and integrating we obtain the G-invariant φ = s5. 3.9) x1 x 4 Comments. Alternative derivation. 3.7) implies not only 3.8) but also Inv dφ = ) which can be used as an equation for φ, in fact a system of linear first order partial differential equations, to be solved by the method of characteristics, thereby providing an alternative derivation. 8

10 <latexit sha1_base64="vphmnt4cjqytm4lq9f4ky9sgcls=">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</latexit> m 1 m 5 14 p µ Figure 3. The invariant φ can be interpreted geometrically as a function of the angle α 14 in the shown triangle. Freedom of definition of invariant. Naturally any function of φ could serve equally well as the G invariant. For instance, given that λ B = s 5 ) x 1 x 4 we could choose either of the following alternative invariants φ s5 ) x 1 x 4, λ B x 1 x 4, s 5 ) λ B. 3.11) Geometric interpretation of φ. In terms of the dual on-shell diagram see [3, 7]) the invariant is related to a triangle where p µ is one of the edges and the other edges are of length m 1 and m 5. The signature of the triangle plane is determined by the data, and the triangle always exists when the data violates the triangle inequality the triangle is defined in a Lorentzian signature plane). More precisely, φ is a function of α 14, the angle between m 1 and p, see Fig. 3. For instance, if the triangle is Euclidean then φ = cos α 14. This means that α 14 is G-invariant. 4 General solution In this section we reduce the integral at a general value of the parameters to a line integral over simpler diagrams. Towards this goal we perform two intermediate steps, determining the constant-free invariants and the homogeneous solution. Constant free equations. The constant-free equation system is defined as a linear subspace of the original equation system with d independent coefficients) such that the vector of constants vanishes, c a cf = 0. The constant-free matrix of group generators associated with.8) is given by ) E + E 3 + E 4 E 1 x1 s 1 A E E 3 T x) cf = s1 A x ) 4 x 4 x 1 0 s 3 s ) This defines a two dimensional group G cf within G. It must have 5 = 3 invariants. By substituting ansatze which are either linear or quadratic in the x s we 9

11 find the following constant-free invariants p 1 = λ A, p = x 1 x 4, p 3 = s 5. 4.) Note that we could have added to p a multiple of p 3 ), so that it could be replaced by λ B for instance. As an alternative for the ansatze derivation one could have used maximal minors to compute the -minors of G cf, obtain an invariant 3-form, and attempt to factorize it into a wedge product of three differentials of the invariants. Homogeneous solution. By definition, I 0, the homogeneous solution of.8) must be a homogenous solution of the constant-free set, and hence must be a function of the constant free invariants I 0 = I 0 p 1, p, p 3 ). 4.3) Substituting this ansatz into the remaining equations we obtain the following equation system for I 0 Its general solution is d 3)I 0 p 1 I 0 = 0 p 1 4.4) I 0 + p I 0 + p 3 I 0 = 0. p p 3 4.5) I 0 = gφ) λ d 3 A. 4.6) s 5 where gφ) is an arbitrary function of the G-invariant, which we choose to set to unity. Line integral representation. In general, SFI allows to replace the integral at any point in parameter space by the integral at a conveniently chosen point on the same G-orbit plus a line integral over simpler diagrams. Here we shall employ property 3.6) and take our base point to have m = m 3 = 0. At this point the integration over the bubble the loop containing propagators and 3) is immediate, and it remains to evaluate a variant of the bubble diagram. This value appears in the literature - see 5.8). Next we should integrate the simpler source terms from m = m 3 = 0 to the point of interest. This can be done over any contour, and in particular we can take a line or a piecewise linear contour. Performing variation of the constants Ix) = cx) I 0 x) 4.7) 10

12 <latexit sha1_base64="lsxymw3d+9onrcfqxhwxzo1o=">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</latexit> p M m Figure 4. The diagram for sector 1, namely IM, m, 0, p, 0). The dashed lines denote massless propagators. and substituting into equations E, E 3.9) of the SFI equation system.8) to solve for c, 3 c we obtain the following line integral representation [ x Ix) = λ d 3 A I 0 x 1 ) + d 3 0 dx λ d 1 A x3 O O 1 )I s 1 A O 3 O 1 )I ) + 3) where the evaluation point 0 denotes x = x 3 = 0 and the integration path is taken with fixed x 1, x 4, x 5. For the integrand to be well defined we assume that λ A x) 0 so that the λ A factor in the denominator starts at λ A = x 1 ) 0 and never crosses a zero along the path. In the special case where x 3 = 0 the expression simplifies to [ Ix) = x 1 x ) d 3 I x x =0 x 1 ) + dx d 3 x 1 x ) d O 3 O 1 )I) ] x. 4.9) x =0 Similarly here we assume that x x 1. 5 Sector evaluation In this section we consider certain special sectors in parameter space where the sources appearing in the general line integral 4.9) are known and we are able to evaluate the integral to obtain more explicit expressions. As a 4-scale sector we consider the sector with x 3 = 0. This is the only sector where sunrise diagrams appearing in the source terms have at least one massless propagator and hence their value is known in terms of an Appell functions. Proceeding to 3-scale sectors we consider setting in addition also x 5 or x 1 to zero. The remaining case x = x 3 = 0 is known in terms of Appell functions 5.8), and we do not consider it. ] x 4.8) 5.1 Massless m 3, m 5 Let us calculate the integral at the sector with x 3 = x 5 = 0 parameterized by B 1 = M, m, 0, p, 0) see Fig. 4), starting from A 1 = M, 0, 0, p, 0) and proceeding 11

13 along the path γ 1 = M, m t, 0, p, 0), 0 t 1. Note that A 1, B 1 and γ 1 are on the orbit with φ = M p = const. 5.1) M p The line integral 4.9) is given by IM, m, 0, p, 0) = M m ) d 3 [ 1 0 O 3 IM, m t, p ) O 1 Im t, p ) M m t) d m dt + IM, 0, 0, p, 0) ]. 5.) M ) d 3 The sources in this case are see A.5),A.1) and A.4) for definitions) where O 1 Im t, p ) = i 1 d π d/ G1, 1) J bubble d/, ; 0, m t, p ) [ = i d π d p ) d 4 a 1 d) F 1 4 d, 5 3d ; 3 d ) m p t ) d/ +a d) m p t F 1 d, 3 d, d ) 1 m ] p t 5.3) O 3 IM, m t, p ) = J bubble 1, 1; 0, M, p )J tad ; m t) = i d π d p ) d/ m t) [a d/ 3a d) 1 M ) d 3 p +a 3b d) M ) d/ 1 F p 1 1, d, d M ) ] 5.4) p a 1 d) = G1, 1)G d/, ) 5.5) a d) = a 3a d) = G1, 1) Γ d/) 5.6) a 3b d) = Γ d/)γ1 d/). 5.7) The starting point can be calculated through the Mellin-Barnes method and is given by [ IM, 0, 0, p, 0) = i d π d p ) d 4 ã 1 d) F 1 1, 5 3d, 3 d M ) p + b 1 d) M ) d 3 F p 1 1, d, d M )] 5.8) p where ã 1a d) is given in 5.14) and b 1 d) is given in 5.36). We see that the only non-trivial integrals we have to evaluate are of the form 1 t β F 1 a,b,c xt) dt, and they can be computed by expressing 0 1 yt) F α 1 by its power series definition and exchanging the order of integration and summation 1 0 t β F 1 a, b, c xt) 1 yt) α dt 1

14 = = k=0 k=0 a) k b) k c) k a) k b) k c) k [ = Γ1 α) 1 y)1 α Γ α) x k 1 k! 0 x k k! t β+k 1 yt) α dt β + k F 1 α, 1 + β + k, + β + k y) Γ1 + β) Γ + β α) y 1 β k=0 n=0 k=0 a) k b) k 1 + β) k x/y) k c) k α + β) k k! α + β) k+n a) k b) k 1) n x k α + β) k c) k α) n k! 1 y) n ] n! 5.9) where the fourth line is gotten by applying the hypergeometric function identity F 1 a, b, c, z) = Γc a b)γc) Γc a)γc b) F 1 a, b, a+b+1 c, 1 z)+ Γa+b c)γc) 1 z) c a b Γa)Γb) F 1 c a, c b, c+1 a b, 1 z). The first single sum can be recognized as the hypergeometric function 3 F. Once we plug in the values for a, b, c, d, α, β, it becomes F 1 while the second sum simplifies to the Appell F 1 or Appell F function. Our final result is IM, m, 0, p, 0) = i d π d [I 1 M, m, p) + I M, m, p) + I 3 M, m, p)] 5.10) where I 1 M, m, p) = ã 1a d)p ) d 4 F 1 5 3d/, 3 d, 1, 3 d/ m p, M ) p ã 1b d) p m ) d m M F d, 3 d, 1, d ) 1, 4 d m p, 1 m M I M, m, p) = 1 m ã 1c d)m M ) d m p d F 1 1, d, d M ã b d)p ) d 4 1 m m M M ) d 3 ) d 3 [ã a d) p m d m ) M M p ) d 3 F 1 I 3 M, m, p) = ã 3 d) p M ) d 1 m M with ) F 1 1, d, d M ) p 5.11) 1 M ) d 3 d F p 1 1, d, d ) m M 1, d, d M )] 5.1) p ) d 3 1 M ) d ) p ã 1a d) = Γ3 d/ 1)Γ3 d) d 4)Γ3d/ 4) π Γ3 d) ã 1b d) = d 3) tandπ/) 5.14) 5.15) 13

15 <latexit sha1_base64="kntx9c/p0q6a6xarapreujpndci=">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</latexit> m 5 p m Figure 5. The diagram for sector, namely I0, m, 0, p, m 5 ). ã 1c d) = 1) d+1 Γ 1 d/) 5.16) π ã a d) = sin dπ/)γd 1) 5.17) ã b d) = b Γ d) 1 d) = π sin d π/) 5.18) ã 3 d) = πγ3 d)γ d/ 1) sindπ/)γd ) 5.19) 5.0) We have compared this result with a Mellin-Barnes computation to find full numerical agreement at arbitrary points in parameter space. 5. Massless m 3, m 1 Now we shall calculate the propagator seagull at the sector with x 1 = x 3 = 0 parameterized by B = 0, m, 0, p, m 5), see Fig. 5. Our starting point will be A = 0, 0, 0, p, m 5) and the path will be γ = 0, m t, 0, p, m 5) with 0 t 1. Note that this trajectory is on φ =. Here our line integral 4.9) is given by I0, m, 0, p, m 5) = 1 O 1 Im t, p, m 5) O 3 Im t, p, m 5) dt 0 t d +m ) d 3 IM, 0, 0, p, m ] lim 5). 5.1) M 0 M ) d 3 Since I 0 vanishes at the starting point A, its contribution to the integral diverges and we regulate it by setting an IR cut-off x 1 = M and taking the limit M 0. The sources in this case are given by see A.1), A.4), A.6) for definitions) O 1 Im t, p, m 5) = J sunrise 1,, 1; 0, m t, m 5, p ) = i d π d m 5) d 4 [b 1 d)f 4 3 d, 4 d, d, 3 d p ) m d/ + b d) t d/ F 4 1, m 5 m 5 ), m t m 5 d d d p,, 1, m t m 5 m 5 ) ] 5.) 14

16 O 3 Im t, p, m 5) = J bubble 1, 1; 0, m 5, p )J tad ; m t) = i d π d b 3 d) m 5) d/ m ) d/ t d/ F 1 1, d, d p m 5 ) 5.3) where b 1 d) = ) πd d π csc Γ4 d) 5.4) b d) = d Γ d ) 5.5) b 3 d) = d Γ 1 d ) 5.6) The integral over O 3 It) is trivial. For the integral over O 1 It) we find that we need to perform integrals of the type 1 0 F 4 a, b, a b + 1, d x, y t) dt = t α = 1 1 α = 1 1 α = 1 1 α k=0 n=0 n=0 n=0 = 1 1 α 1 ± x) a 1 t α 0 k=0 n=0 a) k+n b) k+n 1 α) n k! n! a b + 1) k d) n α) n x k y n a) n b) n 1 α) n y n F 1 a + n, b + n, a b + 1 x) n!d) n α) n a) n b) n 1 α) n n!d) n α) n y n 1 ± x) a+n) F 1 k=0 n=0 a) k+n b) k+n k! n! a b + 1) k d) n x k y t) n dt a + n, a b + 1 a) k+n a b + 1 ) kb) n 1 α) n k! n! a b + 1) k d) n α) n ±4 ) x, a b ± x) ) k ) n ±4 x 1 ± y x) 1 ± x) where we used the quadratic transformation, F 1 a, b, a b + 1 z) = 1 ± z) a F 1 a, a b + 1, a b + 1 ± 4 z ), 1± to reach the fourth line. Again, z) once we plug in the values for a, b, c, d, α we find that this double sum simplifies to an Appell F function. IM The starting point is taken as the limit lim,0,0,p,m M 5 ) 0 where IM, 0, 0, p, m M ) 5) d 3 can be calculated via Mellin-Barnes [0] for a possible simplification from F and F 4 functions to F 1 and F 1 see [17] footnote 16) [ IM, 0, 0, p, m 5) = i d π d 1) d 4 b1 d) m d 5 m 5 ± p) d 6 F 3 d, d 1, 1, d 1, 3 d ±4p m ) 5 m 5 ± p), M m 5 ± p) 5.7) 15

17 M ) d 3 F m 4 1, d 5, d, d p m 5, M ) ]. 5.8) m 5 b1 d) is given in 5.36). Here and in the following ± means that either sign can be chosen consistently throughout the expression and we are denoting p p. The limit where IM, 0, 0, p, m lim 5) = i d π d 1) d 4 b1 1 d) M 0 M ) d 3 m 5 p. 5.9) In this way we arrive at the complete result I0, m, 0, p, m 5) = i d π d 1) [I d 4 1 m, p, m 5 ) + I m, p, m 5 ) ] 5.30) +I 3 m, p, m 5 ) + I 4 m, p, m 5 ) 5.31) I 1 m, p, m 5 ) = b 1 d)m 5 ) d m 5 ± p) d 6 F 3 d, d 1, 3 d, d 1, 3 d ±4p m ) 5 m 5 ± p), m m 5 ± p) 5.3) with I m, p, m 5 ) = b d) m m 5) d mm 5 ± p)) F 1, d 1, 1 d, d 1, d ) 1 ±4p m 5 m 5 ± p), m m 5 ± p) 5.33) I 3 m, p, m 5 ) = b d)m m 5 ) d 4 F 1 1, d, d ) p 5.34) m 5 I 4 m, p, m 5 ) = b 1 d) m ) d ) m 5 p Γ d) b1 d) = π sin d π/) b d) = Γ 1 d ) 5.36) 5.37) Also in this case we have compared this result with a Mellin-Barnes computation to find full numerical agreement at arbitrary points in parameter space. 5.3 Massless m 3 Here we give a 4 scale result. The starting point is A 3 = M, 0, 0, p, m 5) and the final point lies on the x 3 = 0 subspace parameterized by B 3 = M, m, 0, p, m 5), see Fig. 6. The path is given by γ 3 = M, m t, 0, p, m 5) with 0 t 1. The integral to perform is 16

18 <latexit sha1_base64="jzbupkqquhqwolpc640soow54m=">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</latexit> m 5 p M m Figure 6. The diagram for sector 3, namely IM, m, 0, p, m 5 ). IM, m, 0, p, m 5) = 1 m Here [ m M M 1 0 ) d 3 ] +IM, 0, 0, p, m 5) O 3 IM, m t, p, m 5) O 1 Im t, 0, p, m 5) 1 m t ) d M dt 5.38) O 1 Im t, 0, p, m 5) = J sunrise 1,, 1; 0, m t, m 5, p ) 5.39) defined in A.6) and given explicitly in Eq. 5.3) and O 3 IM, m t, p, m 5) = J bubble 1, 1; M, m 5, p )J tad ; m t) 5.40) with J tad and J bubble defined in A.1) and A.) and given explicitly by [ J bubble 1, 1; M, m 5, p ) = i 1 d π d/ m 5) d π sinπ d/)γd/) F 4 1, d, d, d p, M ) m 5 m 5 +Γ 1 d ) ) M d/ 1 F m 4 1, d 5, d, d p, M ) ] 5.41) m 5 m 5 and J tad ; m t) = i 1 d π d/ Γ d ) m ) d/ t d/. 5.4) Integrating O 3 It) amounts to the integral 1 t d/ 0 1 m t ) dt = d d d F 1 1, d, d M The integral over O 1 It) is of the form m M ). 5.43) 1 0 t β F 4 a, b, c, d x, y t) 1 z t) α dt = k=0 n=0 a) k+n b) 1 k+n x k y n k!n!c) k d) n 0 t β+n 1 zt) α dt 17

19 = = = β β k=0 n=0 n=0 a) k+n b) k+n 1 + β) n k! n! a b + 1) k d) n + β) n x k y n F 1 α, 1 + β + n, + β + n z) a) n b) n 1 + β) n n! d) n + β) n y n F 1 a + n, b + n, a b + 1 x) F 1 α, 1 + β + n, + β + n z) 1 Γ + β)γ1 α) z 1 β 1 ± x) a 1 + β Γ α + β) a) l+n a b + 1 ) ) lb) n 1 + β) l ) n n ±4 x l! n! a b + 1) l=0 n=0 l d) n α + β) n 1 ± y x) z1 ± x) + 1 α 1 1 z)1 α 1 ± x) a a) l+n α + β) m+n a b + 1) l1) m b) n l! m! n! α) l=0 m=0 n=0 m a b + 1) l d) n α + β) n ) l ) n ±4 x 1 ± 1 z) m y x) 1 ±. 5.44) x) In the fourth line we used both the quadratic transformation F 1 a, b, a b + 1 z) = 1 ± z) a F 1 a, a b + 1, a b + 1 ± 4 ) z 1± and the identity z) F 1 a, b, c, z) = Γc a b)γc) Γc a)γc b) F 1 a, b, a + b + 1 c, 1 z) + Γa+b c)γc) 1 z) c a b Γa)Γb) F 1 c a, c b, c + 1 a b, 1 z). Here, the double sum simplifies to F 1 or an Appell F function and the triple sum to a Lauricella F K function, see A.7) for the definition of F K. The starting point in this sector, IM, 0, 0, p, m 5) is given by 5.8). Finally our result is IM, m, 0, p, m 5) = i d π d[ I 1 M, m, p, m 5 ) + I M, m, p, m 5 ) + I 3 M, m, p, m 5 ) ] where 5.45) I 1 M, m, p, m 5 ) = 1) d 4 c 1 d) m d 5 m 5 ± p) d 6 m M F K 3 d, 4 d, 1, d 1, d 1, 4 d, 3 d ±4p m ) 5 m 5 ± p), 1 m M, m m 5 ± p) + 1) d 4 c m m 5 ) d d) Mm 5 ± p)) F K 1, d, d 1, 1, d 1, 4 d, d ) 1 ±4p m 5 m 5 ± p), 1 m M, m ] m 5 ± p) I M, m, p, m 5 ) = 1) d 4 c 1 d) m5 d M d 4 m 5 ± p) M ) 1 1 m F 1 1, d 1, d 1 ±4m 5 p m 5 ± p) M I 3 M, m, p, m 5) = i d 1 π d/ c 3 d) m M ) 1 m M ) d 3 M ) d 3 d F 1 1, d, d 5.46) 5.47) m M ) 18

20 J bubble 1, 1; M, m 5, p ) 5.48) with c 1 d) = b Γ d) 1 d) = π sin d π/) Γ1 d/)γ d/) c d) = d 3 c 3 d) = d 5.49) 5.50) 5.51) We have confirmed this result by comparing numerically with a Mellin-Barnes computation. It would be interesting to test this result by restricting it to m 5 = 0 and compare with the result of Section Singular locus For any diagram, there could be special hyper-surfaces in parameter space, called the singularity locus, where the SFI equation system becomes degenerate []. On this locus the equations can be combined in such a way that the differential part cancels. In [5] it was shown how to systematically find this singular locus using maximal minors. The singularity locus of the SFI equation system.8) is defined by setting 0 = S λ A 6.1) where the singular factor S was obtained in 3.4) through the computation of 4- minors. λ A locus. At λ A = 0 there exists a group stabilizer, namely, a row vector that is a left null vector of T x. It is found to be Stb λa = 0, x 3, s 1 A, 0) 0, s 1 A, x, 0)mod λ A ) 6.) where the two forms are parallel on the λ A = 0 locus, namely mod λ A. Multiplying the equations system by Stb λa the differential part of the equation cancels and we obtain an algebraic equation for I which is solved to yield Ix) λa = 1 d 3) s x3 3 O O 1 )I s 1 A O 3 O 1 )I ) 3) 6.3) Another equivalent expression can be written involving square roots. We use s 1 A = ± x x 3 mod λ A and two other identities gotten by permutations. We must be careful about signs and recall that the λ A = 0 cone is divided into three parts, 19

21 where in each part exactly one s variable is negative, see e.g. [6] Fig. 3. In this way we get Ix) λa = 1 x ɛ x3 ) O 3 O 1 )I + ɛ 3 3 O O 1 )I. 6.4) d 3 x 1 x 1 where the signs ɛ, ɛ 3 are given by +, +) for s 1 0 ɛ, ɛ 3 ) = +, ) for s 0, +) for s ) This expression is manifestly symmetric under 3 exchange. We have checked the expressions 6.3,6.4) with FIRE [7] at the point x 1 = x 3 = m, x = 0 and general x 4, x 5. Co-dimension loci. Let us consider co-dimension hyper-surfaces in parameter space where the equation system turns algebraic. This happens when the invariant 1-form 3.5) degenerates, namely Inv = 0, and therefore so do the 4-minors. This locus is composed of two components which we now turn to consider. Component a is defined by x 1 = 0 and x 4 = x 5. The stabilizer is and the algebraic solution is Ix) a = Stb a = 1 x 3 x ), x 3, x, 1 x x 3 ) ) 6.6) x x 3 x O 3 O 1 )I 3) ) 5 O 1 I. 6.7) We have checked this result with the computer package FIRE [7]. Component b is defined by x 4 p = 0 and x 1 = x 5. Here the stabilizer is simply Stb b = 0, 0, 0, 1 ) 6.8) and accordingly the algebraic solution is Ix) b = 5 O 1 I. 6.9) This equation means that setting p = 0 is equivalent to erasing the exteral legs and further setting x 1 = x 5 leads us to the diameter diagram with a squared propagator. We note that in general, co-dimension algebraic loci occur also at the intersection of algebraic locus components, for example in the vacuum seagull. 0

22 7 Summary and Discussion In this paper we have analyzed the propagator seagull diagram for arbitrary values of its parameters: the masses and p, and for general spacetime dimension d, obtaining the following results The SFI equation system of differential equations was determined.8). The associated group G GL3, R) was found to be 4d and non-simple, and is shown in.7). The G-orbits were shown to be defined by a single invariant, φ, given by 3.9) or functions thereof) and interpreted geometrically as an angle in the on-shell triangle associated with the trivalent vertex B. A general reduction was given in 4.8) in terms of a base point where m = m 3 = 0 together with a line integral over simpler diagrams. Explicit expressions in terms of special functions, including the hypergeometric, Appell and Lauricella functions, were obtained for a pair of 3-scale sectors in 5.10,5.30) and for a 4-scale sector in 5.45). The three expressions were confirmed against an independent computation using the Mellin-Barnes transform of massive propagators. The singular locus was determined, both at co-dimension 1 and at co-dimension. In co-dimension 1 the locus has a single component given by λ A = 0, and the integral is represented by simpler diagrams in 6.3) or 6.4). The codimension locus was found to be composed of two components. The first is at x 1 = p x 5 = 0 where the solution is given by 6.7), and the second is at p = x 1 x 5 = 0 and the solution is 6.9). Certain confirmations for these results were achieved through FIRE [7]. The results are interesting both with respect to the diagram and with respect to the SFI method. Regarding the diagram, sector results withstood several tests and we believe that they are novel and naturally interesting. Regarding the SFI method, this case is our first example where we find a group invariant, and it provides valuable experience with performing the reduction to a line integral and with analysis of the singular locus. Discussion. It is interesting to compare the propagator seagull PS) with the vacuum seagull VS). Thanks to the adopted numbering convention for propagators, the SFI equation system for PS is identical with 4 of the 5 equations of VS. The fifth equation is forbidden in PS anyway in the sense of the last comment in section. The singular locus of PS is S P S = λ A which is one of the components of the singular locus of VS, namely S V S = x 1 λ A λ B. 1

23 Open questions. Value at base point. As a base manifold we chose setting m = m 3 = 0. At this value the integral reduces to a sort of bubble integral which is given by 5.8). It would be interesting to analyze this integral through SFI. Sectors expressions: simplification and analysis. The sector expressions include several special functions. It would be interesting to analyze the expressions, for example, to study their analytic structure. Another interesting question is whether a simplification of the expressions might be possible. Allowing for numerator. The propagator seagull has a potential irreducible numerator. We leave it for future work to analyze the integral with this numerator. Relation with Landau equations. The singular locus of the SFI equation system appears to be closely related to the Landau equations. It would be very interesting to establish this relation. Acknowledgments We would like to thank Philipp Burda, Subhajit Mazumdar and Amit Schiller for discussions. RS is very grateful to Philipp Burda for participation in the early stages of this work. This research was supported by the Quantum Universe I-CORE program of the Israeli Planning and Budgeting Committee. A Collection of function definitions Here we collect the functions used in the definition of O 1 I, O 3 I in Sec. 5. As shown in.18) the possible topologies for the source diagrams are that of tadpole bubble and sunrise. Where tadpole, bubble and sunrise. The diagram with tadpole topology with general powers on the loop propagator is given by J tad n; m ) = i 1 d d/ Γn d/) π m ) d/ n. A.1) Γn) The bubble topology with general masses m 1, m and powers n 1, n on the propagators is given by see e.g. [18]) [ J bubble n 1, n ; m 1, m, p ) = π d/ i 1 d m ) d/ n 1 n Γd/ n1 )Γn 1 + n d/) Γd/)Γn ) F 4 n 1, n 1 + n d, d, n d ) p, m 1 m m ) m d/ n1 + 1 Γn 1 d/) F 4 n, d Γn 1 ), d, d + 1 n 1 m p m ) ], m 1. m

24 When one of the propagators is massless this expression simplifies to J bubble n 1, n ; 0, m, p ) = i 1 d π d/ m ) d/ n 1 n Γd/ n 1 )Γn 1 + n d/) Γd/)Γn ) F 1 n 1, n + n d, d ) p A.3) m A.) and by analytic continuation is given also by the following expression, which we find easier to use in Sec. 5.1 [ J bubble n 1, n ; 0, m, p ) = i 1 d π d/ Gn 1, n ) F 1 n 1 + n d, n 1 + n + 1 d, n + 1 d ) m p ) + m d/ n Γn d) F p 1 n 1, n d Γn ), d )] + 1 n m p ) d/ n 1 n A.4) p where Gn 1, n ) = Γn 1 + n d/)γd/ n 1 )Γd/ n ). A.5) Γn 1 )Γn )Γd n 1 n ) The sunrise diagram with one massless propagator and general powers n 1, n, n 3 on the propagators is given by see [1]) J sunrise n 3, n, n 1 ; 0, m, M, p ) = i d π d M ) d n 3 n n 1 Γd/ n 3 ) d Γ n F 4 ) Γn 1 + n + n 3 d)γ n 3 + n 3 d ) n + n 3 d, n 1 + n + n 3 d, d, 1 + n d ) p M, m + M ) m d/ n Γ n M d ) Γn 3 )Γ n 1 + n 3 d ) F 4 n 3 + n 1 d, n 3, d, 1 + d ) n p m, m M where F 4 is the fourth Appell hypergeometric function. The Lauricella F K function is defined by F K a, b, c, d, e, f x, y, z) = l=0 m=0 n=0 a) l+n b) m+n c) l d) m d) l e) m f) n with a) n = aa + 1) a + n 1) is the Pochhammer symbol. References Γn 3 )Γn 1 )Γn )Γd/) x l y m z n l! m! n! [1] B. Kol, Symmetries of Feynman integrals and the Integration By Parts method, arxiv: [hep-th]. A.7) ) A.6) 3

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