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MPI/PhT/98–06 TTP98–04 hep-ph/9801432 January 1998 O α2 Singlet Polarization Functions at ( ) 8 s 9 9 1 K.G. Chetyrkin‡ and R. Harlander n a Institut fu¨r Theoretische Teilchenphysik, Universita¨t Karlsruhe, J D-76128 Karlsruhe, Germany 9 2 1 M. Steinhauser v Max-Planck-Institut fu¨r Physik, Werner-Heisenberg-Institut, 2 3 D-80805 Munich, Germany 4 1 0 8 Abstract 9 / h p - We consider thethree-loop singlet diagrams inducedby axial-vector, scalar p e and pseudo-scalar currents. Expansions for small and large external mo- h mentum q are presented. They are used in combination with conformal : v mapping and Padé approximations in order to arrive at results for the po- i X larization functions valid for all q2. Results are presented for the imaginary r parts which are directly related to physical quantities like the production a of top quarks or the decay of scalar or pseudo-scalar Higgs bosons. PACS numbers: 12.38.-t, 12.38.Bx, 14.65.Ha I. INTRODUCTION In the last few years a lot of effort has been devoted to developments of techniques which allow the evaluation of higher order corrections. Of special interest is thereby the evaluation of QCD corrections to two-point current correlators. Their knowledge immediately leads to a variety of important observables like the cross section σ(e+e− → hadrons) mediated by a photon or a Z boson or the decay of a scalar or pseudo-scalar ‡Permanent address: Institute for Nuclear Research, Russian Academy of Sciences, Moscow 117312, Russia. 1 Higgs boson. Whereas at the one- and two-loop level exact results are known (for a review see [1]), until recently at (α2) only expansions for large external momentum q O s respectively small quark mass m were available. In [2,3] an approach was developed which leads to semi-analytical results for the three- loop polarization functions. The essence of this procedure amounts to the combination of the low- and high-energy analytical data for a polarization function through the use of the conformal mapping and Padé approximation suggested in [4–6]. In a first step it was applied to the non-singlet diagrams induced by external vector, axial-vector, scalar and pseudo-scalar currents [3]. In this paper this procedure will be applied to the corresponding singlet diagrams. They are often also referred to as double- triangle diagrams as the external currents are not connected through the same fermion line. This completes the knowledge of the three-loop current correlators at (α2). Thus O s also the full mass dependence for the inclusive cross sections σ(e+e− tt¯) and the decays → of a scalar or pseudo-scalar Higgs boson into quarks is available at this order. The method used in [3] heavily relies on the fact that the lowest particle threshold for thenon-singletgraphsstartsatq2 = 4m2. Incontrasttothat,thesingletdiagramscontain massless cuts. The solution of this problem is described in detail for the axial-vector correlator in Section II. Also the essential ingredients are listed and the approximation procedureisbrieflyreviewed. SectionIIIdescribes thetreatment ofthescalar andpseudo- scalar diagrams and finally results are presented in Section IV. Note that for the vector correlator there are no singlet diagrams at three-loop level according to Furry’s theorem. II. SINGLET AXIAL-VECTOR CORRELATOR In this section the ingredients and the procedure for the construction of the Padé approximants of the axial-vector polarization function are presented in detail. The scalar and pseudo-scalar singlet diagrams are discussed in Section III. Let us start with some definitions. It is convenient to introduce the dimensionless quantities q2 s 2m z = , r = , x = , v = √1 x2, (1) 4m2 4m2 √s − where √s is the center of mass energy and m is the pole mass of the produced quark. x is a convenient variable in the high energy region and v represents the velocity of the quark. The axial-vector polarization function is defined through q2g +q q Πa(q2)+q q Πa(q2) = i dxeiqx 0 Tja(x)ja(0) 0 , (2) − µν µ ν µ ν L h | µ ν | i (cid:16) (cid:17) Z with ja = ψ¯γ γ ψ. Only the transversal part Πa(q2) will be considered in the following. µ µ 5 In D = 4 2ε space-time dimensions care has to be taken concerning the treatment of γ 5 − — especially in connection with the singlet diagrams. We follow the treatment introduced in [7] and refer for more details to [8]. 2 FIG. 1. Singlet or double-triangle diagrams. In the fermion lines either the quark ψ or its isospin partner χ may be present. Inorderfortheaxialanomalytocancel, onehastotakebothmembersofaweakisospin doublet into account. It is therefore convenient to replace the current ja in Eq. (2) by µ ja = ψ¯γ γ ψ χ¯γ γ χ, where ψ and χ are isospin partners. The diagrams contributing S,µ µ 5 − µ 5 to the singlet part, Πa(q2), are depicted in Fig. 1 where in the fermion triangles either ψ S or χ may be present. Note that for a degenerate quark doublet Πa(q2) vanishes. Having S in mind the physical case (ψ,χ) = (t,b), however, we set m = m and m = 0 in the ψ χ subsequent analysis. Πa(q2) is conveniently written in the form S α 2 Πa(q2) = C T s Π(2),a(q2), (3) S F π S (cid:18) (cid:19) where C = (N2 1)/(2N ), and T is the trace normalization for an SU(N ) gauge group. F c − c c For QCD N = 3 and T = 1/2. c The imaginary part of Πa(q2), S Ra(s) = 12πImΠa(q2 = s+iǫ), (4) S S normalized in analogy to the vector case, enters, e.g., the total inclusive cross section for the production of top quarks. However, also imaginary parts arising from the massless quark χ contribute to Ra(s). The purely gluonic cut is zero according to the Landau- S Yang-Theorem [9]. Let R(2),a be the contribution of these massless cuts to Ra. Then the Sb S quantity (2),a 1 1 R (s) Πâ(q2) = dr Sb (5) S 12π2 r z Z0 − defines a function whose imaginary part for q2 4m2 coincides with that of Π(2),a(q2) ≤ S and is zero for q2 > 4m2. Its evaluation will be described below. With the help of this (2),a function we choose the overall renormalization condition for Π S lim Π(2),a(q2) Πˆ(2),a(q2) = 0. (6) q2→0 S − S (cid:16) (cid:17) In this difference the ln( z) terms cancel for z 0 which makes it possible to demand − → this QED-like renormalization condition. Alternatively one could use the MS scheme where only the 1/ε poles are subtracted. 3 An important ingredient for the Padé approximation is the expansion of Π(2),a(q2) S for small external momentum. It is obtained by applying the so-called hard mass procedure [10] which provides a consistent expansion in q2/m2: 3 Π(2),a(q2) = C(2),azn. (7) S 16π2 S,n n≥0 X The first eight coefficients have been evaluated: 37 C(2),a = Ka + l l2 S,0 6 qm − qm 471631 5215 56 (2),a C = + ζ l S,1 311040 13824 3 − 81 qm 1178417 18179 14 (2),a C = + ζ l S,2 8064000 92160 3 − 75 qm 15527720419 1811719 13312 (2),a C = + ζ l S,3 −5120962560000 14745600 3 − 165375 qm 23054974995287 5965963 7696 (2),a C = + ζ l S,4 −811160469504000 70778880 3 − 178605 qm 735623850793897673 16260413 6600448 (2),a C = + ζ l S,5 −23817834479222784000 264241152 3 − 252130725 qm 1301167265336208772211 425212357 3007168 (2),a C = + ζ l S,6 −46002446022727434240000 9059696640 3 − 173918745 qm 11171249029492611725205473 193004110999 132788224 (2),a C = + ζ l , S,7 −450455951454547036078080000 5218385264640 3 − 10956880935 qm (8) with ζ 1.202056903 and l = ln( q2/m2). Ka is a constant whose numerical value 3 qm ≈ − will be given below. A stringent constraint both for the real and imaginary part of Π(2),a(q2) is set by the S expansion for large external momentum. Recently, the large momentum procedure has been applied leading to the result [8]1: 3 185 385 m2 Π(2),a(q2) = Ka + ζ +21ζ S 16π2 16 − 288 3 3 q2 (cid:26) m2 2 80 320 + ζ + ζ q2 ! (cid:20)− 3 3 3 5(cid:21) m2 3 380 296 + 64ζ + 32ζ l +24l2 q2 ! (cid:20) 3 − 3 (cid:18) 3 − 3(cid:19) qm qm(cid:21) m2 4 3271 416 280 410 176 + ζ + +32ζ l + l2 l3 q2 ! (cid:20)− 243 − 9 3 (cid:18) 27 3(cid:19) qm 27 qm − 27 qm(cid:21) 1In [8] the results are listed in the MS scheme. 4 m2 5 395921 5584 + ζ q2 ! − 2916 − 27 3 (cid:20) 4111 160 1340 1660 + + ζ l + l2 l3 54 3 3 qm 9 qm − 81 qm (cid:18) (cid:19) (cid:21) m2 6 105441373 2420 + ζ q2 ! (cid:20)− 101250 − 3 3 6044237 1177331 15542 + +112ζ l + l2 l3 +... , (9) − 40500 3 qm 1350 qm − 135 qm (cid:18) (cid:19) (cid:21)(cid:27) with ζ defined above and ζ 1.036927755. 3 5 ≈ The logarithms in Eq. (8) are due to the massless cuts and do not appear in the non- singlet diagrams [3]. They spoil the procedure for constructing the Padé approximants developed in [3]. Therefore, instead of dealing with the full polarization function, one may use Eq. (5) and consider the following quantity: Π(2),a (q2) = Π(2),a(q2) Πˆ(2),a(q2). (10) S,mod S − S According tothedefinition ofΠˆ(2),a(q2)theln( z)-terms fromthelowenergyexpansion of S − Π(2),a(q2) are exactly canceled. Above z = 1, the imaginary part on the r.h.s. of Eq. (10) S is determined by Π(2),a(q2) alone. This means that for z > 1, Π(2),a (q2) contains all S S,mod possible cuts of the double-triangle diagrams and one should subtract the massless ones by using R(2),a(s) in the region s > 4m2 to get the production cross section for massive Sb quarks. An analytic formula for R(2),a(s) is available [11]2. Nevertheless it is not possible Sb to solve the dispersion integral in Eq. (10) analytically. On the other hand a purely numerical integration is excluded as the result contains ln( z) terms for z 0 which − → makes an expansion of the integrand with subsequent integration impossible. Let us therefore briefly describe the method we used for evaluation of Eq. (5). (2),a One may write R (s) as (recall Eq. (1)) Sb 3 R(2),a(s) = 3 ln(4r)+R˜(2),a(s). (11) Sb 2 Sb Then R˜(2),a(s) has a very simple limiting behaviour. For s 0 it reads Sb → 37 14 7 R˜(2),a(s) = 3 + r + r2 +... , (12) Sb − 8 27 50 (cid:20) (cid:21) where the dots represent higher orders in r. The expansion of R˜(2),a(s) around √s = 2m Sb leads to 2In [11] a different kinematical region was considered. However, we continued the result to the region under consideration by using the translation table given in the appendix of [11]. 5 19 3 7 1 7 R˜(2),a(s) = 3 ζ + ζ +(1 r) 3ζ + ζ Sb − 8 − 2 2 8 3 − 2 − 2 4 3 (cid:20) (cid:18) (cid:19) 3 4 + √1 r 2π πln2 +... , (13) − − 3 (cid:16) (cid:17) (cid:18) (cid:19) (cid:21) i.e., in this limit R˜(2),a(s) is a series in √1 r. Provided with this information we split Sb − the integral in Eq. (5) into three parts: 1 R(2),a(s) 1 3 3 ln(4r) δ R˜(2),a(s) 1 R˜(2),a(s) dr Sb = dr 2 + dr Sb + dr Sb (14) r z r z r z r z Z0 − Z0 − Z0 − Zδ − and replace in a second step R˜(2),a(s) in the interval [0,δ] by the expansion in (12), in the Sb interval [δ,1] by the one in (13). It turns out that the inclusion of the first 100 terms in the small energy expansion and the first 40 terms in the expansion around r = 1 leads to stable results in the range δ = 0.65...0.80 with an accuracy of 13 to 14 digits. We will not quote numbers for the full Πˆ(2),a(q2) but only for the constant Ka appearing in S Eqs. (8) and (9): Ka = 9.08040684374401... . (15) − The third kinematic region to be used for the Padé procedure is the threshold for the production of two massive quarks, z 1. In this region Π(2),a(q2) gets contributions from → S (2),a two sources: the cuts involving massive quarks and R (s). It is strongly expected that Sb the former starts at least with a term proportionaltov in analogyto the non-singlet axial- vector correlator which follows the P-wave scattering solution of the Coulomb potential. R(2),a(s) on the other hand has a smooth behaviour for s 4m2. For z < 1, Π(2),a (q2) Sb → S,mod is constructed in such a way that its imaginary part vanishes. However, the leading contribution of Π(2),a (q2) for z 1+ is given by S,mod → 3 1 19 7 Π(2),a (q2) = ln +4ln2 2ζ + ζ +... , (16) S,mod 16π2 1 z − 6 − 2 6 3 (cid:18) − (cid:19)(cid:18) (cid:19) where the ellipses represent sub-leading terms in (1 z). − The construction of the Padé approximations divides naturally into four steps (for more detail we refer to [3]): First, the threshold contribution has to be subtracted in all kinematicalregionsinordertohaveapolarizationfunctionthathasavanishing imaginary part for z 1. Then, a new polarization function, Π˜(2),a (q2), is constructed whose high → S,mod energy expansion contains no logarithmic terms any more. This must be done carefully in order not to destroy the behavior for z 0 and z 1. In a third step the conformal → → mapping [4] 4ω z = (17) (1+ω)2 is used to transform the q2 plane into the interior of the unit circle. Finally, a Padé improvement is performed in the new variable, ω. 6 In [3] only the constant and the m2/q2 corrections in the high energy expansion have been included into the analysis. For the singlet diagrams meanwhile terms up to ((m2/q2)6) are available. This makes it necessary to modify the definition of the func- O tion P(ω) for which the Padé approximation is performed. The natural extension of the definition given in [3] reads: (4ω)n−1 n−1 1 dj (1+ω)2j P (ω) = Π˜(2),a (q2) Π˜(2),a (q2) , (18) n (1+ω)2n  S,mod − j=0 j! d(1/z)j S,mod (cid:12)z=−∞! (4ω)j  X (cid:12)  (cid:12)  (cid:12) where the index n 1 indicates that the mass corrections of order (m2/q2)n are included. ≥ The Padé approximants a +a ω + +a ωi 0 1 i [i/j](ω) = ··· (19) 1+b ω + +b ωj 1 j ··· are then constructed from P ( 1) and P(k)(0),(k = 0,1,...,n+n 1), where n is the n − n 0 − 0 number of moments (see Eq. (8)) used for the construction of the Padé approximation and P(k)(0) = dk P (ω) . Taking into account all available information, i.e. n = 6 and n dωk n |ω=0 n = 7, it is possible to construct approximants like [7/6], [6/7] or [8/5]. However, it turns 0 out that the construction of P(k)(0) for large values of k suffers from huge cancellations. n It is therefore necessary to evaluate the expressions in Eq. (14) with highest possible accuracy in order to arrive at reliable results for high-order Padé approximations. After all, the above mentioned 13 to 14 digits are enough to get stable results. Of course, also lower order Padé’s have been evaluated both for consistency checks and to examine the convergence properties. We should mention that some of the Padé approximants develop poles for ω < 1 which result in poles in the physical z plane. Since this is not | | acceptable, only Padé approximants free from poles in the physical region are considered in the discussion of Section IV. We refrain from listing explicit formulae for Π(2),a(q2) at S (2),a this point and instead present in Section IV results for the imaginary part, R (s). S III. SCALAR AND PSEUDO-SCALAR CASE It is now straightforward to extend the procedure described above to the scalar and pseudo-scalar case. Here, in contrast to the singlet axial-vector contribution only the diagram with two massive triangles contributes. However, the cut through the two gluons does not vanish as it was the case for the axial-vector coupling, so that again there is a cut starting at z = 0. The polarization functions are defined through (κ stands for s and p, denoting the scalar and pseudo-scalar case, respectively): q2Πκ(q2) = i dxeiqx 0 Tjκ(x)jκ(0) 0 , (20) h | | i Z Rκ(s) = 8πImΠκ(q2 = s+iǫ), (21) 7 where the currents are given by js = ψ¯ψ and jp = iψ¯γ ψ. As for jp one again has to deal 5 with γ in D = 4 dimensions, we again adopt the definition of [7], referring for details 5 6 to [12]. In analogy to (5) we define: 1 1 R(2),κ(s) Πˆκ(q2) = dr gg . (22) S 8π2 r z Z0 − R(2),κ(s) corresponds to the two gluon cut actually describing the Born decay of a scalar gg or pseudo-scalar Higgs boson to gluons [13,14]: 3 r 1 2 3 R(2),s = 1+ − f(r) , R(2),p = (f(r))2, (23) gg 2r r gg 2r (cid:18) (cid:19) with arcsin2(√r), r 1 ≤ f(r) =  1 log 1+√1−1/r iπ 2 , r > 1. (24)  −4 1−√1−1/r − (cid:20) (cid:21) Although these functions are quite simple an analytic integration is hard to perform. So we adopt the same procedure as for the axial-vector case and expand R(2),κ(s) for gg s 0, where we take 200 terms into account, and for s 4m2, where 50 terms are → → enough to get a precision of 17 to 18 digits in the interval δ [0.65,0.80]. ∈ In analogy to Eq. (6) the overall renormalization condition reads: lim Π(2),κ(q2) Πˆ(2),κ(q2) = 0. (25) q2→0 S − S (cid:16) (cid:17) In this scheme, the low-energy expansion of Π(2),κ(q2) looks as follows: S C(2),s = Ks S,0 4609 721 4 (2),s C = + ζ l S,1 2880 1152 3 − 9 qm 2719121 10871 28 (2),s C = + ζ l S,2 5806080 36864 3 − 135 qm 519513881 1330021 1543 (2),s C = + ζ l S,3 3483648000 7372800 3 − 14175 qm 2460910303 50939 904 (2),s C = + ζ l S,4 57480192000 409600 3 − 14175 qm 8958934229477 2526649 221416 (2),s C = + ζ l S,5 2929190584320000 27525120 3 − 5457375 qm 37498822356303853 5991294557 5844896 (2),s C = + ζ l S,6 −2999491158343680000 84557168640 3 − 212837625 qm 112110439141686419569 245566743541 20750416 (2),s C = + ζ l S,7 −6118961963021107200000 4348654387200 3 − 1064188125 qm 8 14445289941190001679673 1604104532801 140071424 (2),s C = + ζ l , S,8 −723397280961606451200000 34789235097600 3 − 9741414375 qm (26) C(2),p = Kp S,0 55 175 (2),p C = + ζ l S,1 16 96 3 − qm 20143 5047 2 (2),p C = + ζ l S,2 11520 4608 3 − 3 qm 2468869 3969 7 (2),p C = + ζ l S,3 2419200 5120 3 − 15 qm 920009009 974281 328 (2),p C = + ζ l S,4 1393459200 1638400 3 − 945 qm 5898858645227 753259 3832 (2),p C = + ζ l S,5 12875563008000 1572864 3 − 14175 qm 5523023003231 11702895 3776 (2),p C = + ζ l S,6 16531587072000 29360128 3 − 17325 qm 63171728144529503 205215857 4266896 (2),p C = + ζ l S,7 249957596528640000 603979776 3 − 23648625 qm 481264894165689829721 513043585411 32459264 (2),p C = + ζ l , (27) S,8 2447584785208442880000 1739461754880 3 − 212837625 qm where Π(2),κ(q2) and C(2),κ are defined in analogy to Eqs. (3) and (7), and S S,n Ks = 0.62280338337755... ,Kp = 1.81359971877046... . (28) The high energy expansion terms up to (1/z4) are already listed in [12] in the O MS scheme. We have added the (m2/s)5 and (m2/s)6 mass correction terms. In the renormalization scheme defined in Eq. (25) the result reads (l = ln( q2/µ2), with µ qµ − being the renormalization scale): 3 5 49 m2 Π(2),s = Ks ζ + 68+2ζ 20ζ 24l S 16π2 − 8 − 16 3 q2 3− 5− qµ (cid:26) (cid:20) (cid:21) m2 2 + 84+8ζ +160ζ +( 36+72ζ )l q2 ! − 3 5 − 3 qm (cid:20) (cid:21) m2 3 37 3 + 62ζ +320ζ + 36ζ l +33l2 +12l3 q2 ! (cid:20) 8 − 3 5 (cid:18)−4 − 3(cid:19) qm qm qm(cid:21) m2 4 178423 4472 22289 + ζ + +16ζ l q2 ! 243 − 9 3 81 3 qm (cid:20) (cid:18) (cid:19) 28 26l2 l3 − qm − 3 qm (cid:21) m2 5 12256783 8551 1594853 + ζ + +46ζ l q2 ! (cid:20) 62208 − 18 3 (cid:18) 5184 3(cid:19) qm 9 1697 236 + l2 + l3 72 qm 9 qm (cid:21) m2 6 474209987 58672 56656079 576 + ζ + + ζ l q2 ! (cid:20) 1620000 − 225 3 (cid:18) 202500 5 3(cid:19) qm 50407 5758 l2 + l3 +... , (29) − 90 qm 27 qm (cid:21)(cid:27) 3 21 m2 Π(2),p = Kp ζ + 16ζ 20ζ S 16π2 − 4 3 q2 − 3 − 5 (cid:26) (cid:20) (cid:21) m2 2 + 44+24ζ +( 12 72ζ )l q2 ! − 3 − − 3 qm (cid:20) (cid:21) m2 3 221 363 + +114ζ + 36ζ l 63l2 12l3 q2 ! (cid:20) 8 3 (cid:18)− 4 − 3(cid:19) qm − qm − qm(cid:21) m2 4 68146 1288 7727 + + ζ + 80ζ l 86l2 44l3 q2 ! (cid:20) 243 9 3 (cid:18) 81 − 3(cid:19) qm − qm− qm(cid:21) m2 5 12754021 787 1164055 + + ζ + 210ζ l q2 ! (cid:20) 20736 6 3 (cid:18) 1728 − 3(cid:19) qm 7 l2 172l3 − 8 qm − qm (cid:21) m2 6 11857111 15108 40729829 3024 + ζ + ζ l q2 ! (cid:20) 22500 − 25 3 (cid:18) 15000 − 5 3(cid:19) qm 11417 + l2 738l3 +... . (30) 10 qm − qm (cid:21)(cid:27) It should be noted that in the scalar case the singlet polarization function explicitly depends on the renormalization scale µ. For the approximation procedure the choice µ2 = m2 will be adopted. The imaginary part, however, is independent of µ. The analogue of Eq. (10) reads Π(2),κ (q2) = Π(2),κ(q2) Πˆ(2),κ(q2). (31) S,mod S − S The leading threshold term of this function both for κ = s and p will be determined in complete analogy to the axial-vector case (see Eq. (16)), despite the fact that for the non-singlet part in the pseudo-scalar case it originates from the S-wave solution of the Coulomb potential. If by this procedure an essential contribution to the threshold part is missing, different Padé results should develop a large spread close to v = 0. One obtains 3 1 3 1 π4 Π(2),s (q2) = ln +... , Π(2),p (q2) = ln +... . (32) S,mod 16π2 1 z S,mod 16π2 1 z 16 (cid:18) − (cid:19) (cid:18) − (cid:19) The approximation procedure is applied in complete analogy to the axial-vector cor- (2),s/p relator. As for the scalar and pseudo-scalar case also the eighth moment, C , is S,8 available, Padé’s like [7/7], [8/6] or [6/8] may be evaluated. 10