Two-loop self-energy master integrals on shell

Transcription

1 BI-TP 99/1 Two-loop self-energy master integrals on shell J. Fleischer 1, M. Yu. Kalmykov and A. V. Kotikov 3 Fakultät für Physik, Universität Bielefeld, D Bielefeld 1, Germany Abstract Analytic results for the complete set of two-loop self-energy master integrals on shell with one mass are calculated. Keywords: Feynman diagram, Pole mass. PACS numbers): 1.38.Bx 1 fleischer@physik.uni-bielefeld.de misha@physik.uni-bielefeld.de On leave of absence from BLTP, JINR, Dubna Moscow Region) Russia 3 kotikov@sunse.jinr.ru Particle Physics Lab., JINR, Dubna Moscow Region) Russia

2 For a wide class of low-energy observables the dominant and the next-to-leading effects originate entirely from the gauge boson propagators. These corrections are sensitive to heavy virtual particles due to quadratic divergences of the corresponding diagrams and due to non-decoupling of the heavy particles within the Higgs mechanism for the generation of masses. Only recently have the two-loop sub-leading corrections to the m W m Z interdependence of order OG F m t m Z ) been calculated in an expansion in terms of the large top mass [1]. Moreover, the two-loop Higgs boson mass dependence associated with the fermion loop to r has also become available [] 4. These results allow to reduce the theoretical uncertainties originating from higher order effects of the basic physical quantities [4]. To improve the accuracy of the calculations, corrections of order OG F m4 Z ) must be taken into account, which require in particular the calculation of two-loop selfenergy diagrams with only gauge bosons and massless particles. Keeping in mind this physical application, we present here analytical results for all two-loop on-shell self-energy master integrals with one mass. Another application of the given results is the calculation of the naive part of diagrams arising within the asymptotic expansion when the external momentum is on the mass shell of the particle with the large mass [5]. The aim of this letter is the analytical calculation of a full set of two-loop on-shell self-energy master integrals with one mass, Eqs.1)-6). The diagrams of this type are presented on Fig.1. It can be shown [6, 7], that the full set of master-integrals consists of the following ones: F11111, F00111, F10101, F10110, F01100, F00101, F10100, F00001, V1111, V1001, J111 see Fig.1) with all indices powers of propagators) equal to 1, and two integrals of the J011-type: with indices 111 and 11, respectively for the massless integrals see [8, 9]). The integration-by-parts method [9, 10] allows not only to reduce diagrams with higher indices to master-integrals, but also to calculate the master integrals themselves 5. The main idea for the latter step is the following: a diagram with the index i+1) on a massive line can be represented as the derivative of the same diagram with index i on that line w.r.t. its mass squared. This representation is used in certain equations obtained by the integration by part method, yielding a differential equation in general partial differential equations) for the initial diagram with the index i) whichcan be solved analytically. This approach [13, 14] - to construct the differential) relations between diagrams - is called the Differential Equation Method. For some particular cases where the number of the masses is small) these partial differential equations can be rewritten as ordinary differential equations 6. To obtain the finite part of two-loop physical results one needs to know the F-type integrals up to the finite part, the V- andj-type integrals up to the ε-part, and the one-loop integrals up to the ε -part. In the last years various methods have been suggested and many analytical results for one-, two- and three-loop self-energy diagrams with different values of masses and external momenta have been obtained. In particular, the ε-part of one-loop integrals is given in Ref.[16], the finite parts of two-loop self-energy diagrams with different internal masses and arbitrary external momentum are given in Refs.[1],[14], [17]-[4], the ε-part of two-loop tadpole and sunset-type integrals at threshold pseudothreshold) are calculated in Refs.[5]-[7]. Very efficient and elegant methods for numerical calculation of Feynman diagrams have been presented in Refs.[8],[9]. In 4 Recently the accuracy of these two results has been tested in [3]. 5 One of the first calculations of this type for the Standard Model and QED was performed in Refs.[11, 1]. 6 An example where no reduction to an ordinary differential equation was made, may be found in [15]. 1

3 many of these cases the authors have obtained one-dimensional integrals as well for arbitrary masses and momentum. We do not know, however, of attempts to reduce these integrals to generalized polylogarithms or other analytic expressions. We consider here the following master-integrals in Euclidean space-time with dimension N =4 ε: ONS{IJ }i, j, m) K 1 J{IJ K}i, j, k, m) K d N kp i) k, Im) P j) k p, J m), p = m d N k 1 d N k P i) k 1, Im)P j) k 1 k, J m) P k) k p, Km) p, = m V{IJ KL}i, j, k, l, m) K d N k 1 d N k P i) k p, Im) P j) k 1 k, J m)p k) k 1, Km) P l) k, Lm), p = m F{ABIJ K}a, b, i, j, k, m) m K d N k 1 d N k P a) k 1, Am) where K = P b) k, Bm)P i) k 1 p, Im) P j) k p, J m) P k) k 1 k, Km) p, = m Γ1 + ε) 4π) N m ) ε, Pl) k, m) 1 k + m ) l, the normalization factor 1/π) N for each loop is assumed, and A, B, I, J, K =0,1. The finite part of most of the F-type master-integrals can be obtained from results of Ref.[4] in the limit z 1. Nevertheless, to take this limit is not an easy task in general. We have used this approach to obtain the results for F10110 and F The finite part of F10101 is given in Refs.[17, 18], F00001 and F10100 are presented in Ref.[0]. These integrals we have verified by taking the above mentioned limit z 1ofresultsin Ref.[4]. F11111 is calculated in Ref.[3] and we have checked it numerically. Instead of the usually taken F01101 integral [17, 1] we consider J111 as master integral. We recall the results of all master integrals for completeness. The last unknown master integral then remains to be F This appears to be one of the most complicated integrals and below we present details of its calculation. The finite part of the integrals of V- and J-type can be found in Refs.[0]-[1]. The calculation of the ε eε ) parts of master integrals of this type have been performed by the differential equation method [13, 14]. The results for F-type master-integrals are follows: F{ABIJ K}1, 1, 1, 1, 1,m)=a 1 ζ3) + a π 3 S + a 3 iπζ) + Oε), 1) and the coefficients {a i } are given in Table I:

4 TABLE I F11111 F00111 F10101 F10110 F01100 F00101 F10100 F00001 a a a where [11, 17] S = 4 ) π 9 3 Cl = Here we used the m iε prescription. The results for the remaining master integrals are the following ones: V{IJ KL}1, 1, 1, 1,m)= 1 5 ε +1 ε π ) b 1 ζ) 4 π S + π { 65 ln 3 + ε 3 + b ζ) b 3 ζ3) 1 π π 63S + b 4 ζ) ln 3 + 9S 3 3 ) S ln π ln 3 1 π ln 3 b 5 π ζ) 1 Ls π } Oε ), ) where the coefficients {b i } are listen in Table II: TABLE II b 1 b b 3 b 4 b 5 V V J1111, 1, 1,m)= m ε +17 4ε + 59 { } ε 16 +8ζ) { } ) 1117 ε 5ζ) + 48ζ) ln 8ζ3) + Oε 3 ), 3) 3 1 4ε 1 J0111, 1,,m)= 1 ε)1 3ε) ε + π 3 3 ζ) { +ε 8 π 3 3 ζ) 6 π ln 3 + } ) 3 3 ζ3) + 7S + Oε ), 4) 3

5 J0111, 1, 1,m)= m 4 15ε 1 1 ε)1 3ε) 3ε) ε + 3 π 3 { 45 π +ε 9 π ln } { S + ε 1 ζ) S 867 π 3 3 ) + 07 π ln S ln 3 7 π ln 3 1 π ζ) 81 Ls π } ) +Oε 3 ), 5) ONS111, 1,m)= 1 [ 1 1 ε ε π { } π +ε 3 3 ln 3 9S +ε 9S ln 3 1 π ln 3 6 Ls ) π π ] ζ) Oε3 ), 6) where [31] Ls 3 π 3 ) = ) to our knowledge Ls π 3 3 has appeared for the first time in the calculation of the ε-part of two-loop tadpole integrals in Ref.[5] 7 ). The expansion of J1111,1,1,m) up to ε 4 terms is given in Ref.[30]. The above results were checked numerically. Padé approximants were calculated from the small momentum Taylor expansion of the diagrams [3]. The Taylor coefficients were obtained by means of the package [33] with the master integrals taken from [19]. In this manner high precision numerical results for the on-shell values of most of the diagrams were obtained [34]. Further we made use of the idea of Broadhurst [35] to apply the FORTRAN program PSLQ [36] to express the obtained numerical values in terms of a basis of irrational numbers, which were predicted by our analytical calculation, i.e. the differential equation method. Let us point out that the numbers we obtain are related to polylogarithms at the sixth root of unity and hence are in the same class of transcendentals obtained by Broadhurst [35] in his investigation of three-loop diagrams which correspond to a closure of the twoloop topologies considered here. To demonstrate our method of calculation we present in what follows some details for the F00111 integral. The basis of our approach is the differential equation method, i.e. we write a differential equation for F00111, the inhomogeneous term of which is expressed in terms of some simpler integrals. Before turning to the final step, we discuss first of all these latter integrals. All calculations are performed off shell and only at the end we take the on-shell limit. The following extra notations are needed: a) tadpoles: T 1 i, m) d N kp i) k, m) = Γi N ) ) m N i, 4π) N Γi) 7 The definition is: Ls 3 x) = x 0 ln sin θ dθ. 4

6 T i, j, k, m) d N k 1 d N k P i) k 1,m)P j) k,m)p k) k 1 k,m), b) loops with one massive line: Li, j, m) d N kp i) k, 0)P j) k p, m), c) sunset diagrams with two massive lines: Jk, i, j, m) d N k 1 d N k P k) k p, 0)P i) k 1,m)P j) k 1 k,m), 7) d) two-loop diagrams with four propagators: V 1 i, j, k, l, m) V i, j, k, l, m) d N k 1 d N k P i) k p, m)p j) k 1 k, 0)P k) k 1,m)P l) k,0), d N k 1 d N k P i) k p, 0)P j) k 1 k,m)p k) k 1,m)P l) k,m), e) two-loop diagrams with five propagators: Fa, b, i, j, k, m) d N k 1 d N k P a) k 1, 0)P b) k, 0) P i) k 1 p, m)p j) k p, m)p k) k 1 k,m). 1. One-loop diagrams with one massive line. Because it is ultraviolet finite, the integral with shifted indices, L1,,m), is more suitable for analytic calculations than the usual master integral with all indices equal to one. The recurrence relations L1, 1,m) N 3) = m L1,,m) p +m )L, 1,m), L1, 1,m) N 3) = T 1,m) p +m )L1,,m), 8) allow to express the needed integrals in terms of the diagram L1,,m), which can be written as: L1,,m) = K 1 1 ds p 0 1 s) ε 1 + as) 1+ε) = K [ ] [ log 1 + a) ε log 1 + a)+li p a) +ε 3 log3 1 + a) + log 1 + a)li a)+s 1, a) Li 3 a)] ) + Oε 3 ), where a = p /m a = -1 corresponds to the on-shell case). To obtain this result we used the representation see e.g. [4, 37]) 5

7 1 0 ds d N kp i) k, m 1 )P j) Γi + j N/) k p, m )= 4π) N Γi)Γj) 1 i+j N/) P p, m 1 s i+1 N/) 1 s) j+1 N/) s + m ). 9) 1 s. The sunset diagram with two massive lines. The set of master-integrals of this type consists of two integrals, e.g. J0111,1,1,m) and J0111,1,,m). To avoid ultraviolet singularities, however, we choose again integrals with shifted indices: J1,,,m)andJ1, 1, 3,m). Appropriate recurrence relations connecting these integrals need to be derived see also Refs.[6, 38]). Applying the integration by parts relation with the massless line as distinctive one 8,weget J1, 1,,m) N ) = d N k 1 d N k P 1) k p, 0)P ) k 1,m)P ) k 1 k,m) k k 1 ) µ k p) µ = d N k 1 d N k P 1) k p, 0)P ) k 1,m)P ) k 1 k,m) k k µ p) µ, 10) where for the last step the symmetry of the integral is taken into account. Expanding the scalar product k,k p)=k +p k ) p,wehave J1, 1,,m) N ) = 1 p T 1,m) J1,,,m) + 1 d N k 1 d N k P 1) k p, 0)P ) k 1,m)P ) k 1 k,m) k. 11) Applying now the integration-by-parts relation with the massive line with index as distinctive one, we obtain J1, 1,,m)N 6) = 4m J1, 1, 3,m) m J1,,,m) d N k 1 d N k P 1) k p, 0)P ) k 1,m)P ) k 1 k,m) k. 1) Combining Eq.11) and Eq.1) we get the final relation: J1, 1,,m)1 3ε) =T 1,m) p +m )J1,,,m) 4m J1, 1, 3,m). 13) The second recurrence relation, J1, 1, 1,m)1 ε) 1 3 ) [ ] ε = ε p 5ε) m J1, 1,,m) m 4 J1,,,m) m p +m )J1, 1, 3,m), 14) 8 In the integration-by-parts procedure one line is considered distinctive, i.e. its propagator contains nothing but the momentum k µ which is used for the partial integration see, e.g. the discussion in Ref.[4]). 6

8 can be obtained by applying the operator / p µ ) to the sunset integral J1, 1, 1,m), where the replacement k k +p is performed in 7). Further use is made of the relation / pµ ) P i) p, m) =4ii+1 N/)P i+1) p, m) 4ii +1)m P i+) p, m), and the dimension property of the integral see e.g. in Ref.[39]): p ) p + m Ji, j, k, m) =N i j k)ji, j, k, m). m is taken into account. The third recurrence relation, which we need, has the following form: J, 1, 1,m)=J1, 1,,m) m J1, 1, 3,m). 15) ε Eq.15) is obtained in a similar manner as Eq.14): the operator / p µ ) is applied to the sunset J1, 1, 1,m) integral with the initial distribution of momenta 7), and to the same diagram with the replacement k k + p. Their difference yields the recurrence relation 15). We would like to note that Eq.15) shows that the sunset diagram J1, 1, 3,m) should be known up to terms of order ε. After tedious calculation using representation 9), we obtain J1,,,m)= K 1 log y +ε[ p log3 y ζ) log y +log1 y)log y + log y 3Li y)+li y) ) 3ζ3) 1Li 3 y) 6Li 3 y)] ) + Oε ), [ { ) J1, 1, 3,m)= K 1 1 m 1 ε) ε + r y log y + ε logylog 1 + y) 3 1 y) } { + ε ζ) 6Li y) Li y) 8Li 3 y) 4Li 3 y) 11ζ3) ζ) log [ ] +4 Li y)+3li y) 6S 1, y ) 8S 1, y) + 4S 1, y) y 1 + y) 3 1 y) log 1 + y) 3 1 y) ) + log y log 1 y)log y1 y) ) log3 y + 3 ) ) log y log y y }] log 1 + y) 1 + y) 6 1 y) 4 [ { 3a +4 ε log y ε 1Li 3 y) + 4Li 3 y)+6ζ3) a + log y ζ) 4Li y) 6Li y) ) }]) log ylog 1 y) log 3 y + Oε 3 ), ) where r =4+a)/a and y =r 1)/r + 1). Putting a = 1, we get 7

9 J0111,,,m)= ) ζ) εζ3) + Oε ), 16) 3m and 1 1 J0111, 1, 3,m)= 1 ε)m ε π { + ε 3 π ln 3 ζ) } S { ) π ζ3) +ε 3 7Ls π ζ) 9 π ln } ) S ln 3 + Oε 3 ). 17) Note that the integrals J0111,1,3,m) and J0111,,,m) are much simpler than the master integrals 4) and 5). 3. The F00111 diagram. The diagram F00111 demands special consideration. The investigations are similar to those done in [4]. Applying the integration-by-parts relation three times to one of the triangles in F1, 1, 1, 1, 1,m) every time with different distinctive line) and using the identity F1, 1,, 1, 1,m)+F1, 1, 1, 1,,m)= d F1, 1, 1, 1, 1,m), dm we obtain the differential equation: [ N 4)p 4 +p m +3m 4 ) m p 4 +p m +m 4 ) d ] F1, 1, 1, 1, 1,m)=f, 18) dm where the inhomogeneous term f contains only diagrams of V i, J and L-type: L1, ) f =p 4 m)[ 4,m)+L, 1,m) L1, 1,m) ] V 1, 1, 1, 1,m) V, 1, 1, 1,m) +3m p + m )V 1,, 1, 1,m) m p m )V 1 1,, 1, 1,m). 19) Expressing all L s in terms of L1,,m) see 8)) and using the recurrence relations for V- type integrals obtained via integration by parts, we derive a new, simpler representation for the inhomogeneous term f: f = p m [ ) 1 4ε 1 ε) 1 ε T 1,m) p 3 8ε)+m 1 4ε) T 1,m)L1,,m) 1 ε) ] +p + m )J1,,,m)+4m J1, 1, 3,m)+m V 1 1, 1,, 1,m) +6m [ 1+p m ) d ] m V dm 1, 1,, 1,m), 0) where for the V-type diagrams the following differential equations hold: 8

10 [ N 4)p + m ) m m p + m ) d ] V dm 1 1, 1,, 1,m) [ ] =m J1,,,m)+p m ) T 1 3,m)L1, 1,m) J1, 1, 3,m), 1) [ N 6)p +3N 4)m m p + m ) d ] V dm 1, 1,, 1,m) = p J1,,,m)+3m T 1,,,m), ) and see e.g. Eq.5.5) in Ref.[19]) T 1,,,m)= K m 3S +Oε). We integrate Eqs.18), 1), ) with the boundary condition that all diagrams tend to zero as m. Finally for the on-shell case we obtain 1 F001111, 1, 1, 1, 1,m)=8 0 { 3 1 x dx x4 x) arctan x 4 x ) π x 1 arctan ln 1 x + x )}. 6 3 This integral is evaluated numerically with a precision of 40 decimals. Then the program PSLQ see above) is applied with 15 basis elements occurring in 1) to 6) with the result given in Table I. Acknowledgments We are grateful to A. Davydychev, D. Kreimer and O. Veretin for useful discussions and careful reading of the manuscript and to O. Veretin for his help in numerical calculation. Two of us J.F. and M.K.) are very indebted to O. V. Tarasov for useful comments. M.K. and A.K. s research has been supported by the DFG project FL41/4-1 and in part by RFBR # References [1] G. Degrassi, P. Gambino and A. Vicini, Phys. Lett. B ) 19; G. Degrassi, P. Gambino and A. Sirlin, Phys. Lett. B ) 188. [] S. Bauberger and G. Weiglein, Phys. Lett. B ) 333. [3] P. Gambino, A. Sirlin and G. Weiglein, JHEP ) 05. [4] F. Jegerlehner, Z. Phys. C ) 45; M. Consoli, W. Hollik and F. Jegerlehner, Phys. Lett. B ) 167; G. Degrassi, S. Fanchiotti and A. Sirlin, Nucl. Phys. B ) 49; G. Degrassi and A. Sirlin, Nucl. Phys. B ) 34; F. Halzen and B. A. Kniehl, Nucl. Phys. B ) 567; G.Degrassi,P.Gambino,M.PasseraandA.Sirlin,Phys.Lett.B ) 09. 9

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14 F F01111 F11110 F00111 F10101 F10110 F01101 F01100 F00101 F10100 F00110 F00100 F00001 F00000 V1111 V0111 V1011 V1110 V1010 V0110 V1001 V0011 V0010 V1000 V0001 V0000 J111 J011 J001 J000 Figure 1: The full set of two-loop self-energies diagrams with one mass. Bold and thin lines correspond to the mass and massless propagators, respectively. 13