Real-Virtual contributions to the inclusive Higgs cross-section at N3LO
Charalampos Anastasiou, Claude Duhr, Falko Dulat, Franz Herzog, Bernhard Mistlberger
aa r X i v : . [ h e p - ph ] N ov Preprint typeset in JHEP style - PAPER VERSION
IPPP/13/92, DCTP/13/184, CERN-PH-TH/2013-258, MCnet-13-18
Real-Virtual contributions to the inclusive Higgscross-section at N LO Charalampos Anastasiou
ETH Zurich, 8093 Zurich, SwitzerlandE-mail: [email protected]
Claude Duhr
Institute for Particle Physics Phenomenology, University of Durham,Durham, DH1 3LE, United KingdomE-mail: [email protected]
Falko Dulat
ETH Zurich, 8093 Zurich, SwitzerlandE-mail: [email protected]
Franz Herzog
CERN Theory Division, CH-1211, Geneva 23, Switzerland.E-mail: [email protected]
Bernhard Mistlberger
ETH Zurich, 8093 Zurich, SwitzerlandE-mail: [email protected]
Abstract:
We compute the contributions to the N LO inclusive Higgs boson cross-sectionfrom the square of one-loop amplitudes with a Higgs boson and three QCD partons asexternal states. Our result is a Taylor expansion in the dimensional regulator ǫ , where thecoefficients of the expansion are analytic functions of the ratio of the Higgs boson mass andthe partonic center of mass energy and they are valid for arbitrary values of this ratio. Wealso perform a threshold expansion around the limit of soft-parton radiation in the finalstate. The expressions for the coefficients of the threshold expansion are valid for arbitraryvalues of the dimension. As a by-product of the threshold expansion calculation, we havedeveloped a soft expansion method at the integrand level by identifying the relevant softand collinear regions for the loop-momentum. Keywords:
Higgs, QCD, NNLO and N3LO. . Introduction
The theoretical prediction of the Standard Model inclusive Higgs boson cross-section is animportant reference for the experimental verification of the model at the energies probedby the Large Hadron Collider. The uncertainty which is associated with the truncation ofthe perturbative expansion at NNLO is currently estimated at the order of ±
10% (see, forexample, ref. [1]). With the accumulation of more data by ATLAS and CMS, this pertur-bative uncertainty will be ultimately one of the largest systematic uncertainties enteringthe extraction of the coupling strengths of Higgs boson interactions. A calculation of theN LO cross-section may reduce it to about ±
5% [2].A computation of the N LO cross-section is therefore very much desired. Many impor-tant steps have been taken towards this objective. The three-loop gg → h amplitude hasbeen computed in refs. [3–5]. Contributions which are associated with collinear/ultravioletcounter-terms and the partonic cross-sections at lower orders have been computed inrefs. [2, 6–8]. N LO corrections due to processes with triple real radiation were computedin ref. [9] as a threshold expansion around the limit of soft-parton emissions. An importantingredient for the threshold expansion of the Higgs boson cross-section with both real andvirtual radiation, the two-loop soft current, was presented in refs. [10, 11].In this article, we focus on a different contribution to the N LO cross-section, theintegration over phase-space of the squared one-loop amplitudes for partonic processesfor Higgs production in association with a quark or gluon in the final-state. We denotethese squared real-virtual contributions as (RV) . We have been able to perform manyindependent calculations for the (RV) cross-sections by employing different methods.First, we have two different methods that are suited for a threshold expansion. Inmethod (Ia), we reduce the one-loop amplitudes to bubble and box master integrals and findappropriate representations of the box master integrals which allow for a trivial integrationover phase-space order by order in the threshold expansion. Method (Ib) is also a thresholdexpansion method and relies on expanding the amplitudes at the integrand level in theregions where particles in the loops are soft or collinear to an external particle. We haveidentified all such regions and we have proven that the expansion by regions yields thesame result as our first method for an arbitrary order in the threshold expansion. Thecoefficients of the threshold expansion are exact in the dimensional regulator ǫ = 2 − d .Methods (II) and (III) yield expressions of the partonic cross-sections which are validfor arbitrary values of the partonic center of mass energy. Method (II) treats the combinedloop and phase-space integrations as a three-loop forward scattering amplitude, whichis reduced to master integrals with the reverse-unitarity method [12–15]. The masterintegrals themselves are evaluated by solving the differential equations they satisfy in termsof generalized hypergeometric functions. The solution of the differential equations has beenachieved by observing recursion patterns for the hypergeometric integrals, similar to theones for in multiple polylogarithms [16]. Due to the hypergeometric representation of ourmaster integrals a threshold expansion is easily performed, thus providing an independentverification of the results of Methods (Ia) and (Ib). To obtain the master integrals withfull kinematical dependence we solve the differential equations as an ǫ expansion in terms– 1 –f harmonic polylogarithms. Method (III) identifies counterterms for combined loop andphase-space integrals and proceeds with an expansion in ǫ followed by a direct integrationover phase-space variables. This is made possible by embedding the classical polylogarithmsthat arise in the ǫ expansion into a larger space of multiple polylogarithms where theintegration over phase space is trivial. This embedding is achieved by deriving the requiredfunctional equations using symbols [17–21] and the Hopf algebra structure of multiplepolylogarithms [22–24].This article is organized as follows. In Section 2 we introduce the partonic cross-sections we are computing and define our notation. In Section 3 we present our results forthe partonic cross-sections. In Section 4 we detail the methods employed for the purposesof our calculation. We present our conclusions in Section 5.
2. The real-virtual squared cross section
We consider the partonic-processes for associated Higgs production, g ( p ) + g ( p ) → g ( p ) + H ( p ) q ( p ) + g ( p ) → q ( p ) + H ( p ) q ( p ) + ¯ q ( p ) → g ( p ) + H ( p ) (2.1)where g, q, ¯ q are symbols for gluon, quark and anti-quark partons correspondingly and H for the Higgs boson. The brackets refer to the momenta of the particles. We define thekinematic invariants, s ≡ p · p + i , t ≡ p · p − i , u ≡ p · p − i ,p = ( p + p − p ) = s − t − u = M h + i , (2.2)where we indicated explicitly the small imaginary parts carried by them. The partoniccross-sections for these processes are given by σ X = N X s Z d Φ X |A X | , (2.3)where X ∈ { gg → Hg, gq → Hq, q ¯ q → Hg } labels the different subprocesses and the sumsymbol denotes a summation over colors and polarizations of the initial and final stateparticles. We work in conventional regularization in d = 4 − ǫ for both the phase spaceand the matrix element. The d -dimensional phase-space measure is given by d Φ = (2 π ) d δ ( d ) (cid:0) p + p − p − p ) d d p (2 π ) d − δ + ( p ) d d p (2 π ) d − δ + ( p − M h ) . (2.4) N X denote the averaging over initial state spins and colors in d dimensions, N gg = 14 V (1 − ǫ ) , N gq = 14 V N (1 − ǫ ) , N q ¯ q = 14 N , (2.5) In the following we will suppress the dependence on the final state as it can always be inferred fromthe initial state for the processes we consider. – 2 –ith N and V ≡ N − s = M h z , t = s δ λ, u = s δ (1 − λ ) , (2.6)with δ = 1 − z . Note that a physical scattering process corresponds to s > < z, λ <
1, and the limit δ → d Φ = (4 π ) ǫ s − ǫ δ − ǫ π Γ(1 − ǫ ) dλ [ λ (1 − λ )] − ǫ Θ( λ ) Θ(1 − λ ) , (2.7)where Θ( x ) is the Heaviside step function. Equation (2.3) then reads σ X = s − − ǫ N X (4 π ) ǫ π Γ(1 − ǫ ) δ − ǫ Z dλ [ λ (1 − λ )] − ǫ X |A X | . (2.8)In the rest of this paper we compute in an effective theory of the Standard Modelwhere the top-quark is decoupled. The effective unrenormalized Lagrangian reads: L = L effQCD − C H G aµν G a,µν , (2.9)where the first term corresponds to an effective QCD Lagrangian with N f = 5 flavors. TheWilson coefficient C can be cast as a function of the QCD coupling, the bare heavy-quarkmasses and the Higgs field vacuum expectation value [25–29].Performing a loop-expansion of the amplitudes A X = P ∞ j =0 A ( j ) X in the effective theorywith j being the number of loops, we have: |A X | = (cid:12)(cid:12)(cid:12) A (0) X (cid:12)(cid:12)(cid:12) + 2 ℜ (cid:16) A (0) X A (1) X ∗ (cid:17) + (cid:20)(cid:12)(cid:12)(cid:12) A (1) X (cid:12)(cid:12)(cid:12) + 2 ℜ (cid:16) A (0) X A (2) X ∗ (cid:17)(cid:21) + . . . (2.10)The first two terms of the above expansion enter the already known inclusive Higgs bosoncross-section through NNLO [12, 30–35]. The third term in square brackets contributes tothe N LO coefficient. In this article, we compute the part of the partonic cross-section dueto the square of the one-loop amplitudes, namely, σ ⊗ X = s − − ǫ N X (4 π ) ǫ π Γ(1 − ǫ ) δ − ǫ Z dλ [ λ (1 − λ )] − ǫ X (cid:12)(cid:12)(cid:12) A (1) X (cid:12)(cid:12)(cid:12) . (2.11)
3. Results
The computation of the cross-sections (2.11) for the different subprocesses is the mainsubject of this paper. We evaluated the phase-space and loop integrals in different waysthat are detailed in Section 4. In this section, we summarize first our main findings.We find for the partonic cross-sections: σ ⊗ X = π ω Γ | C | s (cid:18) πs (cid:19) ǫ (cid:16) α s π (cid:17) C X Σ X ( z ; ǫ ) , (3.1)– 3 –here g s ≡ πα s , and we have C gg = NV (1 − ǫ ) , C qg = 1 N (1 − ǫ ) , C q ¯ q = VN . (3.2)In eq. (3.1) we have introduced the quantity ω Γ ≡ c Γ(1 + ǫ ) Γ(1 − ǫ ) = Γ (1 + ǫ ) Γ (1 − ǫ )Γ (1 − ǫ ) , c Γ = Γ(1 + ǫ ) Γ (1 − ǫ )Γ(1 − ǫ ) . (3.3)The function Σ gg ( z ; ǫ ) has a pole as z →
1. This pole constitutes the soft singularityof the gluon initiated cross-section and it will only be remedied by integrating the partoniccross-section with parton distribution functions. We separate this singular part manifestlyfrom the remainder and write:Σ gg ( z ; ǫ ) = Σ sing gg ( z ; ǫ ) + Σ reg gg ( z ; ǫ ) (3.4)where the z → sing gg ( z ; ǫ ) = − (1 − z ) − − ǫ N (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) (1 − ǫ ) (1 − ǫ ) ǫ − (1 − z ) − − ǫ N (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) Γ( ǫ + 1)Γ(2 ǫ + 1)Γ(1 − ǫ ) (1 − ǫ ) ǫ Γ(1 − ǫ ) Γ(1 − ǫ )Γ(4 ǫ + 1) − (1 − z ) − − ǫ N (1 − ǫ )Γ(1 − ǫ ) Γ( ǫ + 1) Γ(1 − ǫ )3 ǫ Γ(1 − ǫ ) . (3.5)The cross-sections of the q ¯ q and qg channels are regular in the limit z → qg ( z ; ǫ ) = Σ reg gq ( z ; ǫ ) , Σ q ¯ q ( z ; ǫ ) = Σ reg q ¯ q ( z ; ǫ ) . (3.6)We have calculated the regular functions Σ reg X ( x ; ǫ ) as an expansion in the dimen-sional regulator ǫ through O ( ǫ ). The expressions are composed of multiple polylogarithms(MPLs) [16] with weights up to five. Specifically, the MPLs are defined as iterated integralsvia G ( a , . . . , a n ; z ) = Z z dtt − a G ( a , . . . , a n ; t ) , (3.7)where G ( z ) = 1 and z, a i ∈ C and the weight is the number of indices a i . All our MPLs haveargument z and all a i are elements of the set {− , , } . The last property allows to relateour MPLs to a more restrictive class of functions, the so-called harmonic polylogarithms(HPLs) [36]. The relation is given explicitly by H ( a , . . . , a n ; z ) = ( − σ ( a ,...,a n ) G ( a , . . . , a n ; z ) , (3.8)where σ ( a , . . . , a n ) denotes the number of elements in the set { a , . . . , a n } that are equal to1. Software to evaluate HPLs numerically in a fast and accurate way (at least up to weightfour) is publicly available [37–41]. Due to the magnitude of the expressions obtained for thefunctions Σ reg X ( x ; ǫ ) we refrain from stating them here explicitly and make them publicly– 4 –vailable together with the arXiv submission of this article and on the web-page [42]. Inthe file results.txt we present the formulae for the Σ reg X formatted in Maple input form.Given that the triple-real contributions to the inclusive Higgs boson cross-section atN LO [9] have only been computed as an expansion around z = 1 − δ →
1, we alsoprovide the same expansion for the (RV) cross-section functions Σ X ( z, ǫ ) of this article.We have discovered a characteristic structure for Σ X ( z, ǫ ) in the δ → δ exponentiate into factors of δ − aǫ , where a is an integer inthe interval [2 , X ( z, ǫ ) = X a =2 δ − aǫ η ( a ) X ( δ ; ǫ ) , (3.9)where the functions η ( a ) X ( δ ; ǫ ) are meromorphic functions of δ . We further decompose η ( a ) X ( δ ; ǫ ) = φ ( a ;1)Γ ( ǫ ) ˆ η ( a ;1) X ( δ ; ǫ ) , (3.10)for a = 4, while for a = 4 we write, η (4) X ( δ ; ǫ ) = X j =1 φ (4; j )Γ ( ǫ ) ˆ η (4; j ) X ( δ ; ǫ ) , (3.11)with φ (2;1)Γ ( ǫ ) = 1 ,φ (3;1)Γ ( ǫ ) = cos( πǫ )Γ(1 − ǫ ) Γ(1 − ǫ )Γ(1 − ǫ ) ,φ (4;1)Γ ( ǫ ) = Γ (1 − ǫ )Γ (1 − ǫ ) Γ (1 − ǫ ) ,φ (4;2)Γ ( ǫ ) = cos(2 πǫ )Γ(1 − ǫ ) Γ( ǫ + 1)Γ(1 − ǫ )Γ(1 − ǫ ) ,φ (4;3)Γ ( ǫ ) = Γ (1 − ǫ ) Γ (1 − ǫ )Γ (1 − ǫ ) Γ (1 − ǫ ) ,φ (5;1)Γ ( ǫ ) = cos( πǫ )Γ(1 − ǫ )Γ(1 − ǫ ) Γ( ǫ + 1)Γ(1 − ǫ )Γ(1 − ǫ ) ,φ (6;1)Γ ( ǫ ) = Γ (1 − ǫ ) Γ (1 + ǫ ) Γ (1 − ǫ )Γ (1 − ǫ ) . (3.12)The structure of Eqs. (3.9)–(3.11) originates from loop integrations over distinct kinematicconfigurations, where the loop momentum can be either soft ( s ), or collinear to the firstincoming parton ( c ), or collinear to the second incoming parton ( c ), or, otherwise, hard( h ). Every term in the squared amplitude may be thought of as associated to a product oftwo regions r and r , r i ∈ { s, c , c , h } , one from the amplitude itself and the other fromits complex conjugate. In the following we denote the contribution from such a productof regions by ( r , r ). The coefficients φ ( i ; j )Γ ( ǫ ) ˆ η ( i ; j ) X ( δ ; ǫ ) are in one-to-one correspondence– 5 –ith these regions. The correspondence is explicitly given by φ (2;1)Γ ( ǫ ) ˆ η (2;1) X ( δ ; ǫ ) ↔ ( h, h ) ,φ (3;1)Γ ( ǫ ) ˆ η (3;1) X ( δ ; ǫ ) ↔ ( c + c , h ) + ( h, c + c ) ,φ (4;1)Γ ( ǫ ) ˆ η (4;1) X ( δ ; ǫ ) ↔ ( c , c ) + ( c , c ) ,φ (4;2)Γ ( ǫ ) ˆ η (4;2) X ( δ ; ǫ ) ↔ ( s, h ) + ( h, s ) ,φ (4;3)Γ ( ǫ ) ˆ η (4;3) X ( δ ; ǫ ) ↔ ( c , c ) + ( c , c ) ,φ (5;1)Γ ( ǫ ) ˆ η (5;1) X ( δ ; ǫ ) ↔ ( c + c , s ) + ( s, c + c ) ,φ (6;1)Γ ( ǫ ) ˆ η (6;1) X ( δ ; ǫ ) ↔ ( s, s ) . (3.13)A derivation of the above decomposition will be given in Section 4. The analytic form ofthe double soft ( s, s ) terms ˆ η (6;1) X ( δ ; ǫ ) is rather simple. We find:ˆ η (6;1) q ¯ q ( δ ; ǫ ) = 1 N " δ ǫ − ǫ + 2 + δ (cid:0) − ǫ − ǫ + 1 (cid:1) ǫ − ǫ + 2 ǫ + δ (cid:0) ǫ + 6 ǫ − ǫ + 1 (cid:1) ǫ − ǫ + 6 ǫ , (3.14)ˆ η (6;1) qg ( δ ; ǫ ) = N " ǫ − ǫ + δ ǫ (6 ǫ − − δ ( ǫ + 1) (cid:0) ǫ + 12 ǫ − ǫ + 2 (cid:1) ǫ (12 ǫ − ǫ + 1)+ δ ( ǫ − ǫ (12 ǫ − ǫ + 1) − δ ( ǫ − ǫ (12 ǫ − ǫ + 1) , (3.15)ˆ η (6;1) gg ( δ ; ǫ ) = N " + 2( ǫ − δǫ − ǫ − ǫ + 2 δ (cid:0) ǫ − ǫ + 1 (cid:1) ǫ (6 ǫ − − δ (cid:0) ǫ − ǫ + 1 (cid:1) ǫ (6 ǫ − − δ (3 ǫ − (cid:0) ǫ − ǫ + 35 ǫ − ǫ + 4 (cid:1) − ǫ ) ǫ (6 ǫ − δ ( ǫ − ǫ − − ǫ ) ǫ (6 ǫ − − δ ( ǫ − ǫ − − ǫ ) ǫ (6 ǫ − . (3.16)The remaining ˆ η a ; iX terms are more complicated combinations of generalized hypergeometricfunctions, which can be readily cast as a Laurent series in δ ∞ X n = − c n ( ǫ ) δ n Due to the size and complexity of the expressions in terms of hypergeometric functions,we provide in Appendix D and in the arXiv submission file results.txt [42] the terms of theseries in the δ expansion up to O (cid:0) δ (cid:1) . With our computer programs, we have explicitlygenerated the terms of the series up to order O (cid:0) δ (cid:1) . In addition, as explained earlier, wehave computed the ˆ η ( a ; j ) X functions for arbitrary values of δ as an expansion in ǫ throughorder O (cid:0) ǫ (cid:1) . – 6 – . Methods In this section, we discuss how we evaluated the loop and phase-space integrals that con-tribute to the real-virtual squared cross-sections (3.1). We employed various methods thateach have their own strengths and weaknesses. We checked that we obtain consistentresults when comparing the different approaches. These methods are:1.
Threshold expansion of the cross section:
In this approach, we derive a repre-sentation of the cross-section as an expansion close to threshold where δ →
0. We firstexpand the loop amplitude in the limit where the final state parton is soft, and thenperform the phase-space integration order-by-order in the expansion. The thresholdexpansion of the loop amplitude is obtained in two different ways: first by finding asuitable representation in terms of convergent hypergeometric functions within theentire phase-space, and, second, by expanding around the relevant soft, collinearand hard regions of the loop momentum. We will detail our threshold expansiontechniques in Section 4.1.2.
Differential equations for master integrals:
Using a duality of loop and phase-space integrals we reduce them simultaneously to a minimal set of master integrals.The master integrals satisfy differential equations that can be solved in two ways:either order-by-order in dimensional regularization in terms of harmonic polyloga-rithms, or in terms of generalized hypergeometric functions. The boundary condi-tions for the differential equations are obtained by matching to the leading termof the threshold expansion, which we compute with one of our threshold expansionmethods mentioned above. We present our approach based on differential equationsin Section 4.2.3.
Direct integration using multiple polylogarithms:
It is possible to derive ana-lytic results of the loop integrals entering our amplitudes in terms of polylogarithmicfunctions. These expressions are singular in soft and collinear limits of the phase-space and we render all integrals convergent by constructing appropriate countert-erms. Then we perform the two-body phase-space integration by embedding thepolylogarithmic functions into a larger class of multiple polylogarithms for which theintegration is trivial. In the end, we recast the final result in terms of harmonicpolylogarithms only. This method is explained in Section 4.3.
We start by computing the one-loop amplitudes A X with X ∈ { gg → Hg, q ¯ q → Hg, qg → Hq } in dimensional regularization and for arbitrary values of the regulator ǫ . We generatethe Feynman diagrams with QGRAF [43] and perform the spin and color algebra usingFORM [44]. Using the methods of refs. [45, 46] for tensor integrals and well establishedreduction techniques for scalar integrals [47,48], the amplitudes are reduced to the one-loop– 7 –calar bubble and box master integrals,Bub( s ) = Z d d kiπ d k ( k + q + q ) , Box( s , s , s ) = Z d d kiπ d k ( k + q ) ( k + q + q ) ( k + q + q + q ) , (4.1)where the q i are considered light-like and ingoing and s ij = ( q i + q j ) .In a next step, we construct the squared one-loop amplitudes and cast them in termsof three functions, in the form: X (cid:12)(cid:12)(cid:12) A (1) gg → Hg (cid:12)(cid:12)(cid:12) = N V | C | g s (4 π ) d s t u ( | A ggg ( s, − t, − u ) | + (1 − ǫ ) h | B ggg ( s, − t, − u ) | + | B ggg ( − t, − u, s ) | + | B ggg ( − u, s, − t ) | i ) , (4.2) X (cid:12)(cid:12)(cid:12) A (1) qg → Hq (cid:12)(cid:12)(cid:12) = V | C | g s u π ) d ( (1 − ǫ ) h | A q ¯ qg ( − u, − t, s ) | + | A q ¯ qg ( − u, s, − t ) | i − ǫ ℜ (cid:2) A q ¯ qg ( − u, − t, s ) A ∗ q ¯ qg ( − u, s, − t ) (cid:3) ) , (4.3) X (cid:12)(cid:12)(cid:12) A (1) q ¯ q → Hg (cid:12)(cid:12)(cid:12) = V | C | g s s π ) d ( (1 − ǫ ) h | A q ¯ qg ( s, − t, − u ) | + | A q ¯ qg ( s, − u, − t ) | i − ǫ ℜ (cid:2) A q ¯ qg ( s, − t, − u ) A ∗ q ¯ qg ( s, − u, − t ) (cid:3) ) . (4.4)Explicit expressions for the functions A q ¯ qg , A ggg , B ggg are given in Appendix C. Threshold expansion using hypergeometric functions.
Loop integrals in dimen-sional regularization can be expressed, to all orders in the dimensional regulator, as (gen-eralized) hypergeometric functions. For example, the box integral defined in eq. (4.1)admits the representation (see, e.g., ref. [49]),Box( s , s , s ) = 2 c Γ ǫ s s ( ( − s ) − ǫ F (cid:18) , − ǫ ; 1 − ǫ ; − s s (cid:19) + ( − s ) − ǫ F (cid:18) , − ǫ ; 1 − ǫ ; − s s (cid:19) − (cid:0) − M h (cid:1) − ǫ F (cid:18) , − ǫ ; 1 − ǫ ; − M h s s s (cid:19) ) . (4.5)Our goal is to insert the parametrization (2.6) and then to perform the integration over λ term-by-term in the series representation of the hypergeometric functions. The result is apower series in δ , i.e., the desired expansion of the cross-section close to threshold. The bubble integral is trivial, and will not be discussed any further. – 8 –hile all the hypergeometric series in eq. (4.5) are convergent in the Euclidean re-gion where s ij <
0, the one-loop amplitude in the physical region involves the functionsBox( − t, − u, s ), Box( s, − t, − u ) and Box( s, − u, − t ). It is easy to check that the correspond-ing hypergeometric series are no longer convergent in the physical scattering region. It is,however, always possible to analytically continue the F function such that arguments lieinside the unit disc, yielding another representation in terms of F functions. While thisapproach is adequate to find a meaningful expansion around ǫ in terms of polylogarithms,it does not allow one to find (convergent) hypergeometric series expansions around δ = 0.Instead, one needs at least a double sum representation to achieve this task. It turns outthat such a representation is known in the literature [49],Box( − t, − u, s ) = 2 c Γ ǫ Γ(1 + ǫ )Γ(1 − ǫ ) e − iπǫ (cid:0) tus (cid:1) − ǫ tu − c Γ ǫ (1 + ǫ ) t − ǫ − s F (cid:16) , ǫ ; 2 + ǫ ; us (cid:17) − c Γ ǫ (1 + ǫ ) u − ǫ − s F (cid:18) , ǫ ; 2 + ǫ ; ts (cid:19) − c Γ ǫ (1 + ǫ ) e iπǫ s − − ǫ F (cid:18) ǫ ; 1 + ǫ, ǫ ; 2 + ǫ, ǫ ; us , ts (cid:19) , (4.6)Box( s, − t, − u ) = 2 c Γ ǫ t − ǫ s ( − t ) F (cid:16) , − ǫ ; 1 − ǫ ; us (cid:17) + 2 c Γ ǫ s − − ǫ e iπǫ S (cid:18) ǫ ; 1 , ǫ ; 2 , ǫ, ts , us (cid:19) . (4.7)The corresponding result for Box( s, − u, − t ) is obtained from Box( s, − t, − u ) by exchanging t and u . The generalized Kamp´e de F´eriet function S is defined as S ( a ; a , b ; a , b ; x , x ) = ∞ X n,m =0 ( a ) m + n ( a ) m + n ( b ) m ( a ) m + n ( b ) m x m x n m ! n ! , (4.8)and the Appel function F is defined as F ( a ; b , b ; c , c ; x , x ) = ∞ X n,m =0 ( a ) m + n ( b ) m ( b ) n ( c ) m ( c ) n x m x n m ! n ! . (4.9)Using these expressions for the box functions, we can easily exchange the phase-spaceintegration and the infinite summations, and all the integrals can be performed in termsof Euler’s Beta function, B ( α, β ) = Z dλ λ α − (1 − λ ) β − = Γ( α ) Γ( β )Γ( α + β ) . (4.10) Threshold expansion of hard, soft and collinear regions.
It is possible to deriverepresentations such as the ones of (4.6) and (4.7) with a more physical method, performingTaylor expansions around soft, collinear and hard regions of the integrand of loop integralsin momentum space. The method of expansions by regions [50] promises to hold in general,– 9 –lthough its generality has only been stated as a conjecture and a verification of the validityof the approach is necessary in specific cases .For the production of the Higgs boson near threshold, the partonic center of massenergy is close in value to the Higgs boson mass, and thus we have a kinematic variablewhich is small, δ = 1 − z ∼
0. From eq. (2.6) we infer that the external momenta scale as p ∼ p ∼ √ s, p ∼ √ s δ . (4.11)For a particle propagating in the loop, we find four types of non-trivial scalings of itsmomentum k : • Hard ( h ) k µ ∼ √ s , where all propagators in the loop are off-shell, • Soft ( s ) k µ ∼ √ s δ , where the loop integrand is singular at the point k µ = 0, • Collinear to p ( c ) 2 k · p s ∼ δ, k · p s ∼ , k ⊥ ∼ √ s δ , where the integrand has a singular surface as k µ ∝ p µ , • Collinear to p ( c ) 2 k · p s ∼ δ, k · p s ∼ , k ⊥ ∼ √ s δ , where the integrand has a singular surface as k µ ∝ p µ .In the above, the transverse momentum k ⊥ of the particle is defined via: k = p k · p s + p k · p s + k ⊥ . (4.12)A scaling of the loop momentum is called a region . In a given region, we can perform asystematic expansion of the integrand around δ = 0. This yields multiple new integralswhich are simpler than the unexpanded integral. For some regions, we are able to computeanalytically all (i.e., an infinite number of) terms of the expansion. For the remainingregions, we limit ourselves to a finite number of terms in the expansion and perform analgebraic reduction [47, 48] (after expansion) to master integrals. The soft and collinearregions of our loop integrals correspond to the singular surfaces which solve the Landauequations [53] while the loop momentum scalings can be identified with the scalings of the Full proofs of the validity of asymptotic expansions by regions are hard to derive or unknown. Forefforts in this direction we refer the reader to ref. [51, 52] and references therein. – 10 –oordinates which are normal to the singular surfaces [54]. In the following we discuss howwe can reproduce the hypergeometric function representations given in eq. (4.6).We start by discussing the asymptotic expansion of Box( s, − t, − u ), which we findconvenient to parametrize asBox( s, − t, − u ) = Z d d kiπ d A A A A , (4.13)with A = ( k − p ) ,A = ( k − p ) ,A = k ,A = ( k − p ) . (4.14)We find that the full integral is reconstructed by two regions:1. ( c )-region, where k is collinear to p .2. ( h ) − region, where k is hard.After Taylor expanding the loop integrand in every region in the small variable δ , wecan use integration-by-parts identities to reduce the coefficients of the Taylor expansionto a small set of master integrals. In the ( c )-region we find that all the coefficients areproportional to the one-loop bubble integral Bub ( − t ), and this region reconstructs, orderby order in δ , the first term of the hypergeometric representation for Box( s, − t, − u ) giveneq. 4.7, and we have verified this statement explicitly up to O (cid:0) δ (cid:1) . The ( h )-region yieldsthe second and last term of eq. (4.7). In this region, we have been able to calculate allterms in the expansion around δ = 0 with an analytic integration. We see that the sum ofthe ( h ) and ( c ) regions is equal to the correct expression for the one-loop box. All othersoft and collinear regions are zero, as we can readily verify.Next, we turn to the asymptotic expansion of the Box( − t, − u, s ), given byBox( − t, − u, s ) = Z d d kiπ d A A A A , (4.15)with A = ( k − p ) ,A = ( k ) ,A = ( k − p ) ,A = ( k − p + p ) . (4.16)We find that the expression for Box( − t, − u, s ) given in eq. (4.6) is reconstructed entirelyfrom the following regions:1. ( s ) − region, where the k ∼ δ , yielding the first term of eq. (4.6). It is interestingthat the ( s ) − region consists of a single term without any subleading terms in theexpansion in δ . – 11 –. ( c )-region where the momentum k is collinear to p . This region reconstructs thesecond term of eq. (4.6) as we have verified explicitly up to O (cid:0) δ (cid:1) .3. ( c )-region where the momentum k − p is collinear to p . This region reconstructsthe third term of eq. (4.6) as we have verified explicitly up to O (cid:0) δ (cid:1) .4. ( h )-region, where k is hard. This region reconstructs the last term of eq. (4.6) as wehave verified explicitly up to O (cid:0) δ (cid:1) .All other soft and collinear regions are zero.We have seen that an expansion in hard, soft and collinear regions yields series repre-sentations for the one-loop master integrals of the required amplitudes which converge inthe entire phase-space, and thus we can immediately perform the phase-space integrationin terms of Beta functions order-by-order in the expansion. While in our case the strategyof expansion by regions is only an alternative method for deriving the threshold expansionsof Section 4.1, it can be the method of choice for the phase-space integration of more com-plicated one-loop amplitudes. Here we have presented expansions by regions at the level ofmaster integrals. We would like to remark that such expansions can also be performed atthe integrand of loop-amplitudes before any reduction to master integrals has taken place.Combined with the method of reverse-unitarity [9] we have a powerful algebraic techniquefor the simultaneous threshold expansion of integrals over loop and external momenta. In this section we evaluate the real-virtual squared cross-sections using the reverse-unitarityapproach [12–15]. Reverse unitarity establishes a duality between phase-space integrals andloop integrals. Specifically, on-shell and other phase-space constraints are dual to “cut”propagators δ + ( q ) → (cid:20) q (cid:21) c = 12 πi Disc 1 q = 12 πi (cid:20) q + i − q − i (cid:21) . (4.17)A cut-propagator can be differentiated similarly to an ordinary propagator with respect toits momenta. It is therefore possible to derive integration-by-parts (IBP) identities [55, 56]for phase-space integrals in the same way as for loop integrals. The only difference is anadditional simplifying constraint that a cut-propagator raised to a negative power vanishes: (cid:20) q (cid:21) − νc = 0 , ν ≥ . (4.18)In this approach, we are not obliged to perform a strictly sequential evaluation of the loopintegrals in the amplitude followed by the nested phase-space integrals. Rather, we combinethe two types of integrals into a single multiloop-like type of integration by introducing cut-propagators and then derive and solve IBP identities for the combined integrals. We solvethe large system of IBP identities which are relevant for our calculation with the Gausselimination algorithm of Laporta [47]. We have made an independent implementation ofthe algorithm in C++ using also the
GiNaC library [57]. In comparison to AIR [48],– 12 –hich is a second reduction program used in this work, the
C++ implementation is fasterand more powerful, storing all identities in virtual memory rather than in the file system.All integrals that appear in the real-virtual squared cross section are reduced to linearcombinations of 19 master integrals, which we choose as follows: M = = Z d Φ Bub( s ) Bub ∗ ( s ) . (4.19) M =
12 12 = Z d Φ Bub( s ) Bub ∗ ( s ) . (4.20) M =
12 12 = Z d Φ Bub( s ) Bub ∗ ( s ) . (4.21) M =
12 12 = Z d Φ Bub( s ) Bub ∗ ( s ) . (4.22) M = = Z d Φ Tri( s + s ) Bub ∗ ( s ) . (4.23) M =
21 21 = Z d Φ Tri( s + s ) Bub ∗ ( s ) . (4.24) M =
12 12 = Z d Φ Tri( s + s ) Bub ∗ ( s ) . (4.25) M =
12 12 = Z d Φ Bub( s ) Box ∗ ( s , s , s ) . (4.26)– 13 – =
12 12 = Z d Φ Bub( s ) Box ∗ ( s , s , s ) . (4.27) M =
21 21 = Z d Φ Bub( s ) Box ∗ ( s , s , s ) . (4.28) M =
12 21 = Z d Φ Bub( s ) Box ∗ ( s , s , s ) . (4.29) M =
12 21 = Z d Φ Tri( s + s ) Box ∗ ( s , s , s ) . (4.30) M =
21 21 = Z d Φ Tri( s + s ) Box ∗ ( s , s , s ) . (4.31) M =
21 12 = Z d Φ Tri( s + s ) Box ∗ ( s , s , s ) . (4.32) M =
21 12 = Z d Φ Box( s , s , s ) Box ∗ ( s , s , s ) . (4.33) M =
21 21 = Z d Φ Box( s , s , s ) Box ∗ ( s , s , s ) . (4.34) M =
12 12 = Z d Φ Box( s , s , s ) Box ∗ ( s , s , s ) . (4.35)– 14 – =
21 21 = Z d Φ Box( s , s , s ) Box ∗ ( s , s , s ) . (4.36) M =
21 12 = Z d Φ Bub( M h ) Box ∗ ( s , s , s ) 1 s . (4.37)Single solid lines represent scalar massless propagators. The phase-space integrationis represented by the dashed line and the cut-propagators are the lines cut by the dashedline. The cut propagator of the Higgs boson is depicted by the double-line. Every masterintegral has a one-loop integral on the left- and a complex-conjugated one-loop integralon the right-hand side of the cut. In each side of the cut, we find scalar bubble, box ortriangle integrals, where the latter is defined byTri( s ) = Z d D ki ( π ) D/ k ( k + q ) ( k + q + q ) , Tri( p , p ) = Z d D ki ( π ) D/ k ( k + p ) ( k + p + p ) , (4.38)with q i = 0, p i = 0 and ( p + p ) = 0. The scalar bubble and box integrals have beendefined in (4.1). A comment is in order about the appearance of the triangle integrals inthis approach, which seems to be at odds with the fact that in the expression of the one-loop amplitude presented Section 4.1 only bubble and box integrals appeared. Indeed, itis well-known that eq. (4.38) can be expressed as a linear combination of bubble integrals,Tri( s ) = 1 − ǫǫ s Bub( s ) , Tri( p , p ) = 1 − ǫǫ ( p − p ) [Bub( p ) − Bub( p )] . (4.39)These relations however introduce new denominators which need to be taken into accountin the reduction of the phase-space integrals. We therefore prefer not to use eq. (4.39), butwe work directly with the triangle integrals instead.To evaluate the master integrals we employ the method of differential equations [12,58, 59]. Differentiating the corresponding cut-propagator with respect to the square of theHiggs mass, ∂∂M h (cid:18) p h − M h (cid:19) c = (cid:18) p H − M h (cid:19) c , (4.40)– 15 –esults in another phase-space integral. This new integral can again be reduced by IBPidentities to our basis of master integrals. Proceeding in this way we obtain a system oflinear first order differential equations for the master integrals, ∂∂δ M i ( δ ) = A ij ( δ ) M j ( δ ) . (4.41)The system is triangular ∂∂δ M i ( δ ) = A ii ( δ ) M i ( δ ) + y i ( δ ) , (4.42)where y i ( δ ) only depends on master integrals that can be solved for independently of M i ( δ ). In other words, the system can be solved hierarchically, starting from the differentialequations with vanishing or known functions y . Every time we solve such an equation, itssolution serves to determine the y function of a next equation. In this way, at any stage ofthis procedure the y function is a linear combination of already evaluated master integrals y i ( δ ) = X j = i A ij ( δ ) M j ( δ ) , (4.43)that can be integrated in order to determine the integral M j ( δ ). The coefficients A ij ( δ ) arerational functions in δ and ǫ and have isolated singularities in δ only at δ = 0 , ,
2. The firststep to solving this type of differential equation is to find a solution for the homogeneouspart. The general homogeneous solution associated to the differential equation (4.42) isgiven by M hi ( δ ) = M i (0) exp δ Z dδ ′ A ii ( δ ′ ) . (4.44)and is determined up to an integration constant M i (0). We determine this integrationconstant by calculating the soft limit of the master integral explicitly following the methodsdiscussed in Section 4.1. We find that only 7 of our 19 master integrals have non-trivialboundary conditions. Interestingly, with our choice of basis of master integrals, the non-trivial boundary conditions are in one-to-one correspondence to the leading terms of the 7regions of the soft expansion of the squared amplitude of eqs. (3.10)-(3.11). The non-trivial– 16 –oundary conditions are: M S = (4 π ) − ǫ ω Γ δ − ǫ φ (4;1)Γ ǫ (1 − ǫ ) (1 − ǫ ) , M S = (4 π ) − ǫ ω Γ δ − ǫ φ (2;1)Γ ǫ (1 − ǫ ) , M S = (4 π ) − ǫ ω Γ δ − ǫ φ (3;1)Γ ǫ (1 − ǫ ) (1 − ǫ ) , M S = (4 π ) − ǫ ω Γ δ − ǫ φ (4;3)Γ ǫ (1 − ǫ ) (1 − ǫ ) , (4.45) M S = − (4 π ) − ǫ ω Γ δ − − ǫ φ (4;2)Γ ǫ (1 − ǫ ) , M S = − (4 π ) − ǫ ω Γ δ − − ǫ φ (5;1)Γ ǫ (1 − ǫ ) , M S = − (4 π ) − ǫ ω Γ δ − − ǫ ǫ ) φ (6;1)Γ ǫ (1 + 3 ǫ ) . Only the real part of the boundary conditions is presented here, given that the imaginarypart does not contribute to the cross-section.Once the homogeneous solution is found we can compute a particular solution to theinhomogeneous equation by M pi ( δ ) = M hi ( δ ) δ Z dδ ′ y ( δ ′ ) M hi ( δ ′ ) , (4.46)The full solution for the master integral is then given by M i ( δ ) = M hi ( δ ) + M pi ( δ ) . (4.47)We perform the integration in the equation above with two different approaches. Solving differential equations as an expansion in ǫ . One well established strategyis to expand the differential equations in powers of the dimensional regulator [58,59]. Afterexpanding the integral of (4.46) a solution is naturally given by iterated integrals leadingto multiple polylogarithms [16] of the form G ( a , . . . , a n ; δ ), with a i ∈ { , , } . Expressingthe functions in terms of the variable z = 1 − δ recasts the solutions in terms of morefamiliar harmonic polylogarithms [58]. Solving differential equations in terms of hypergeometric functions.
The inte-grand of eq. (4.46) takes the form M si ( δ ) ∼ Z δ dδ ′ ( δ ′ ) c (1 − δ ′ ) c (2 − δ ′ ) c M j = i ( δ ′ ) , (4.48)– 17 –here c , c and c are linear polynomials in ǫ . This structure reminds of the Euler-typeintegral representation of hypergeometric functions. Inspired by the large variety of tech-niques available for the solution of iterated integrals in terms of multiple polylogarithms [16]we define an iterated integral with integration kernel δ a − (1 − xδ ) − b . The n th iterated in-tegral is then recursively defined by F ~a n ,...,~a ( x n , . . . , x ; δ ) = Z δ dδ ′ ( δ ′ ) a n − (1 − x n δ ′ ) − b n F ~a n − ,...,~a ( x n − , . . . , x ; δ ′ ) . (4.49)where we have abbreviated ~a i = (cid:16) a i b i (cid:17) .We find that these iterated integrals interpolate between multiple polylogarithms andhypergeometric functions. For example, in this framework the multiple polylogarithm isgiven byLi m ,...,m k ( x , . . . , x k )= k Y i =1 x k − i +1 i ! F ~ , . . . ,~ | {z } m − ,~ ,...,~ , . . . ,~ | {z } mk − ,~ (0 , . . . , | {z } m − , x , . . . , , . . . , | {z } m k − , x k . . . x ; 1) . (4.50)where ~ (cid:16) (cid:17) and ~ (cid:16) (cid:17) . The Gauss hypergeometric function is given by F ( a , a ; b ; z ) = a Γ( b )Γ( a )Γ( b − a ) F ~v~u (1 , z ; 1) , (4.51)with ~v = (cid:16) a − a − a − b (cid:17) and ~u = (cid:16) a a + 1 (cid:17) . A large variety of hypergeometric functions can beexpressed in terms of these iterated integrals. With the definition A n = n X i =1 a i , K n = n X i =1 k i , (4.52)we find an explicit sum representation for this type of iterated integrals. F ~a n ,...,~a ( x n , . . . , x ; δ ) = δ A n n Q i =1 A i ∞ X k ,...,k n =0 n Y i =1 (cid:18) ( A i ) K i ( A i + 1) K i ( b i ) k i ( x i δ ) k i k i ! (cid:19) . (4.53)Equation (4.53) is valid whenever the sums are convergent. Further properties and deriva-tions are discussed in more detail in Appendix A. The solution of differential equationsusing iterated integrals is illustrated with an example in Appendix B.The iterated integrals defined in this section are a powerful tool and enable us to solveall 19 master integrals in terms of hypergeometric functions. The results is valid to allorders in ǫ . The iterated integrals can be written as multiple sums eq. (4.53) from whichit is very convenient to extract a threshold expansion in δ .– 18 – esults for the master integrals. The master integrals that we have computed in thissection are useful for the evaluation of any cross-section for a 2 → LO. Dueto the length of their expressions, we provide them in terms of harmonic polylogarithms inthe form of the text file masters.txt enclosed with the arXiv submission of this work andon the web-page [42]. We have computed all 19 master integrals to all orders in ǫ in termsof hypergeometric functions and as an expansion in ǫ in terms of harmonic polylogarithmsas described above. We present here an alternate method to compute the (RV) Higgs boson cross-section,based on subtraction terms. The phase-space integral over the squared amplitude can bewritten schematically as, Z d Φ |A| = Z d Φ X i,j M i ( s , s , s ) M j ( s , s , s ) N i,j ( s , s , s ) . (4.54)In this expression M i denote the one-loop master integrals and N i,j are rational functions,all of which depend on the invariants s , s and s . Since the results for the requiredone-loop master integrals are known to all orders in ǫ [49], the integrals are well defined indimensional regularization. Our goal is to expand the integrals in ǫ under the integrationsign and to perform the integration order by order in ǫ . After expansion, however, theintegrals may develop soft and collinear divergencies. The strategy is to subtract thesingular limits of the integrand before expansion, and to perform the remaining (finite)integration in terms of multiple polylogarithms.The construction of the counterterms that render the integration finite proceeds in twosteps. First, we analytically continue all the hypergeometric functions that appear in theall order expressions of the one-loop master integrals such that they are convergent in thewhole phase-space. This is achieved by using the well-known identities F ( a, b ; c ; z ) = (1 − z ) − b F (cid:18) b, c − a ; c ; zz − (cid:19) , F ( a, b ; c ; z ) = Γ( b − a )Γ( c )Γ( b )Γ( c − a ) ( − z ) − a F ( a, a − c + 1; a − b + 1; z − )+ Γ( a − b )Γ( c )Γ( a )Γ( c − b ) ( − z ) − b F ( b, b − c + 1; b − a + 1; z − ) . (4.55)Second, the soft and collinear counterterms are easily constructed by expanding the in-tegrand around the collinear limits, i.e., s → s →
0. The counterterms can betrivially integrated to all orders in the dimensional regulator in terms of Γ functions.At the end of this procedure we are left with finite one-dimensional integrals. Weexpand the hypergeometric functions appearing in the integrand in ǫ using HypExp [60],resulting in a representation for the integrand in terms of classical polylogarithms up toweight four. More specifically, we are left with integrals of the form Z dλ X i P i ( λ, z ) λ (1 − λ ) Li n ( R i, ( λ, z )) Li m ( R i, ( λ, z )) , (4.56)– 19 –ith n + m ≤ P i is a polynomial and R i,k are rational functions. Notethat, while individual terms in the sum are singular for λ → ,
1, the sum is finite byconstruction, and so the integral is well defined. In order to perform the integration over λ , we rewrite the classical polylogarithms in terms of multiple polylogarithms [16] of theform G ( a ( z ) , . . . , a n ( z ); λ ), where a i ( z ) are rational functions of z using symbols [17–21]and the Hopf algebra structure of multiple polylogarithms [9, 22–24]. All the integrals canthen easily be performed using the recursive definition of the multiple polylogarithms, Z dλλ − a G ( a , . . . , a n ; λ ) = G ( a , . . . , a n ; 1) . (4.57)Finally, we observe that the results of the integration can also be expressed in terms ofharmonic polylogarithms [36] of weight up to five, and we checked that the results are inagreement with the differential equation approach.
5. Conclusions
In this article, we have taken one more step towards the evaluation of the Higgs bosoncross-section at N LO in perturbative QCD. Specifically, we have analytically evaluatedthe one-loop contributions to the partonic cross-sections from 2 → -corrections are simpler than other, yet unknown, contributionsto the N LO Higgs boson cross-section. However, they share many of the complexities thatappear in corrections with mixed real and virtual radiation. Therefore, our calculations inthis paper serve as a perfect testing ground for our computational techniques.In this publication, we have achieved two goals. We have obtained analytic expressionsfor the (RV) partonic cross-sections which are valid for arbitrary values of the Higgs massand energy as an expansion in ǫ through the finite part. Given that a calculation of thefull hadronic Higgs boson cross-section at N LO will most likely be achieved first as anexpansion around threshold, we have obtained here a threshold series expansion for the(RV) partonic cross-sections. The coefficients of the threshold expansion are valid forarbitrary values of ǫ . We performed our calculations with various methods and techniques.Our first method is based on reverse unitarity and integration by part identities in orderto reduce the integrals of the partonic cross-sections to 19 master integrals. For these masterintegrals we derived a system of differential equations which we could solve to all ordersin the dimensional regulator in terms of iterative functions of the kind of hypergeometricfunctions which we defined explicitly for this purpose. Their series representations yielda threshold expansion for the master integrals and the partonic cross-sections. The newlyintroduced hypergeometric functions share many properties with multiple polylogarithms,for example they satisfy a shuffle algebra. They also appear to cover a wide range ofknown multi-dimensional hypergeometric functions. It will be an exciting new direction– 20 –or further research to establish further properties of these functions and their usefulnessin higher order perturbative corrections. We have also solved the differential equationsfor the master integrals order by order in ǫ in terms of harmonic polylogarithms. Thisyields the main result of this publication, which is the (RV) partonic cross-sections forarbitrary values of the Higgs mass and the partonic center of mass energy as an expansionin ǫ through O (cid:0) ǫ (cid:1) .We have also followed a direct integration approach to obtain the same results. Forthis task, we introduced counter-terms at the level of the squared matrix element in orderto subtract its collinear and soft divergences. Having rendered the phase-space integrandfinite, we then expand in the dimensional regulator and perform a direct integration interms of harmonic polylogarithms. The integration is made possible by embedding theclassical polylogarithms that result form the expansion of the hypergeometric function ina larger space of multiple polylogarithms and exploiting their Hopf algebra structure.We have also followed a different strategy, with a more restricted objective to obtaina threshold expansion of the cross-sections. Experience from NNLO has shown that verygood approximations to the inclusive Higgs boson cross-section can be obtained with thefirst few terms of this expansion. We obtain the series around threshold by expandingthe integrand of the loop amplitudes in soft, collinear and hard regions and perform thephase-space integration term by term. We have successfully applied this method to com-pute a sufficiently large number of terms in the threshold expansion to all orders in thedimensional regulator. Unlike some of the other methods we have used, the strategy ofregions is generic, which makes it a particularly attractive option for computing the (RV)and (RV) contributions in more complicated processes, where a direct evaluation in termsof polylogarithms may be unfeasible or just extremely difficult.Our results are rather lengthy and we have only typeset parts of them in this document.Instead, we provide in the source of the arXiv submission two text files results.txt and masters.txt with the expressions for the partonic cross-sections and the master integralsrespectively. The two files can also be downloaded from [42].We believe that with our calculation and the methods that we have developed in thispaper to be closer in our objective to compute the Higgs boson cross-section at N LO. Welook forward to applying the techniques presented here towards this objective.
Acknowledgements
We thank Thomas Becher, Aneesh Manohar, Eric Laanen, George Sterman and LeonardoVernazza for useful discussions. This research is supported by the ERC Starting Grant forthe project “IterQCD”, the Swiss National Foundation under contract SNF 200021 143781,by the Research Executive Agency (REA) of the European Union under the Grant Agree-ment number PITN-GA-2010-264564 (LHCPhenoNet) and the FP7 Marie Curie InitialTraining Network MCnetITN under contract PITN-GA-2012-315877. The work of F.H.was supported by ERC grant 291377. – 21 – . Hypergeometric functions through iterated integrals
In this appendix we define a class of iterated integrals as also introduced in Section 4.2.First let us define the integral F ~a ( c ; δ ) = δ Z dt t a − (1 − ct ) − b = δ a a F ( a, b ; a + 1; c ¯ z )= δ a a ∞ X n =0 ( a ) n ( b ) n ( a + 1) n ( c ¯ z ) n n ! , (A.1)where we have abbreviated for later convenience ~a = (cid:16) ab (cid:17) . We have made use of Gauss’hypergeometric function with the third argument being the first argument increased byone. Next, we define recursively the n th iterated integral by F ~a n ,...,~a ( x n , . . . , x ; δ ) = Z δ dt t a n − (1 − x n t ) − b n F ~a n − ,...,~a ( x n − , . . . , x ; t ) . (A.2)The integration kernel t a − (1 − ct ) − b has the same form for every iteration step with indices a, b and argument c changing. Next, we derive a hypergeometric series representation forthese iterated integrals. To simplify the expressions we rewrite eq. (A.1) and introduce afunction f that is implicitly given by F ~a ( c ; δ ) = ∞ X n =0 f ( a, b, c, n ) δ a + n . (A.3)In the next step we integrate over the integration kernel and the F ~a ( c ; t ) F ~a ,~a ( c , c ; δ ) = δ Z dt t a − (1 − c t ) − b F ~a ( c ; t )= ∞ X n =0 δ Z dt t a + a + n − (1 − c t ) − b f ( a , b , c , n )= ∞ X n,m =0 δ a + a + n + m a + a + n ( a + a + n ) m ( b ) m ( a + a + n + 1) m c m m ! f ( a , b , c , n ) . Using the identity ( a + n ) m = ( a ) n + m Γ( a )Γ( a + n ) , (A.4)we can write F ~a ,~a ( c , c ; δ ) = δ a + a ( a + a ) a ∞ X n,m =0 ( a + a ) m + n ( a + a + 1) m + n ( a ) n ( a + 1) n ( b ) m ( b ) n ( c δ ) m m ! ( c δ ) n n ! . – 22 –e now proceed iteratively, and find the following series representation for the iteratedintegrals F ~a n ,~a n − ,..., ~a ( c n , . . . , c ; δ ) = δ A n n Q i =1 A i ∞ X k ,...,k n =0 n Y i =1 ( A i ) K i ( A i + 1) K i ( b i ) k i ( c i δ ) k i k i ! , (A.5)with the abbreviations A i = i X n =1 a n and K i = i X n =1 k n . (A.6)Following the the same procedure as for the sum representation we can derive a gen-eral Mellin barnes representation for our iterated integrals by utilizing the Mellin-Barnesrepresentation of the Gauss Hypergeometric function F ( a, b ; c ; δ ) = 12 πi Γ( c )Γ( a )Γ( b ) Z i ∞− i ∞ ds Γ( a + s )Γ( b + s )Γ( c + s ) Γ( − s )( − δ ) s . (A.7)This leads to F ~a n ,~a n − ,..., ~a ( c n , . . . , c ; δ ) = δ A n Z i ∞− i ∞ n Y i =1 dk i πi Γ( A i + K i )Γ( A i + K i + 1) Γ( b i + k i )Γ( b i ) Γ( − k i )( − c i δ ) k i . (A.8)These iterated integrals interpolate between multiple polylogarithms [16] and hypergeo-metric functions. In the framework of the above definitions the multiple polylogarithm isgiven byLi m ,...,m k ( x , . . . , x k )= F ~ , . . . ,~ | {z } m − ,~ ,...,~ , . . . ,~ | {z } mk − ,~ (0 , . . . , | {z } m − , x , . . . , , . . . , | {z } m k − ; x k . . . x ) k Y i =1 x k − i +1 i . (A.9)Here the indices of the iterated integrals only take the form ~ (cid:16) (cid:17) and ~ (cid:16) (cid:17) . Evenfor general indices we discover further similarities of these iterated integrals with multiplepolylogarithms. As in the case of polylogarithms this class of hypergeometric functionsmay be written as multiple nested sums F ~a n ,~a n − ,..., ~a ( c n , . . . , c ; δ ) = ∞ X k n ≥···≥ k =0 δ A n + k n n Y i =1 A i + k i ( b i ) k i − k i − ( c i ) k i − k i − ( k i − k i − )! , (A.10)where k = 0. The representation (A.10) may be useful to expand the iterated integrals interms of the dimensional regulator (see, e.g., ref. [61]).The definition of these function as iterated integrals implies that they form a shufflealgebra, F ~a i ,...,~a ( c i , . . . , c ; δ ) F ~a n ,...,~a i +1 ( c n , . . . , c i +1 ; δ ) = X σ ∈ Σ( i,n − i ) F ~a σ ( n ) ,...,~a σ (1) ( c σ ( n ) , . . . , c σ (1) ; δ ) , (A.11)– 23 –here Σ( i, n − i ) denotes the set of all shuffles of n elements, i.e., the subset of the symmetricgroup S n defined byΣ( i, n − i ) = { σ ∈ S n | σ − (1) < · · · < σ − ( i ) and σ − ( i + 1) < · · · < σ − ( n ) } . (A.12)To illustrate an application of the shuffle-product for generalized iterated integrals, letus look at the following example. We would like to integrate an iterated integral over anon-standard integration kernel. I = Z δ dt t a − (1 − c t ) − b (1 − c t ) − b F ~a ( c ; t ) . (A.13)To simplify the integral we make use of F (cid:16) aa + 1 (cid:17) (1; δ ) = δ a a (1 − cδ ) − a (A.14)and find I = b Z δ dt t a − b − (1 − c t ) − b F (cid:16) b b + 1 (cid:17) ( c ; t ) F (cid:16) a b (cid:17) ( c ; t ) (A.15)Next, we apply the shuffle product and find I = b Z δ dt t a − b − (1 − c t ) − b (cid:20) F (cid:16) b a b + 1 b (cid:17) ( c , c ; t ) + F (cid:16) a b b b + 1 (cid:17) ( c , c ; t ) (cid:21) = b F (cid:16) a − b b a b b + 1 b (cid:17) ( c , c , c ; δ ) + b F (cid:16) a − b a b b b b + 1 (cid:17) ( c , c , c ; δ ) . (A.16)Further identities among iterated integrals can be derived using integration-by-parts orby partial fractioning products of integration kernels. Further properties and parallels ofgeneralized iterated integrals and generalized poly-logarithms are under investigation. B. NNLO RV Master Integrals as hypergeometric functions
In this appendix we demonstrate how certain differential equations for master integralsappearing in physical cross-sections can be solved using iterated integrals as introducedin Appendix A. We consider the example of the master integrals contributing to the RV Higgs boson cross-section at
N N LO . The master integrals were introduced and evaluatedas an expansion in the dimensional regulator ǫ in ref. [12] and evaluated to even higherorder in ref. [7]. Here we solve them to all orders in ǫ in terms of hypergeometric functions.The master integrals and the corresponding differential equations are given by Y = = Z d Φ Bub ∗ ( s ) . (B.1) ∂ δ Y = (1 − ǫ ) δ Y . (B.2)– 24 – = = Z d Φ Bub ∗ ( s ) . (B.3) ∂ δ Y = (1 − ǫ ) δ Y . (B.4) Y = = Z d Φ Tri ∗ ( s + s ) . (B.5) ∂ δ Y = 2 ǫδ − δ Y − (1 − ǫ )(1 − ǫ )(1 − δ ) δǫ Y + (1 − ǫ ) (1 − δ ) − − ǫ δǫ Y . (B.6) Y = = Z d Φ Box ∗ ( s , s , s ) 1 s . (B.7) ∂ δ Y = − (1 + 2 ǫ ) δ Y − (2 δ − − ǫ )(1 − ǫ )(1 − δ ) δ ǫ Y + 2(1 − ǫ ) (1 − δ ) ǫ δ ǫ Y − ǫ (1 − δ ) δ Y . (B.8) Y = = Z d Φ Box ∗ ( s , s , s ) . (B.9) ∂ δ Y = − (1 + 4 ǫ ) δ Y + 2(1 − ǫ )(1 − ǫ )(1 − δ ) δ ǫ Y − ǫ (1 − δ ) δ Y . (B.10)We have abbreviated ∂ δ = ∂∂δ . Solid lines represent scalar propagators. The phase-spaceintegral is represented by the dashed line and the cut-propagators are the lines cut by thedashed line. The cut propagator of the Higgs boson is depicted by the double-line. Thecomplex conjugated one-loop integral is on the right-hand side of the phase-space cut. Thedifferential equations were obtained using the methods described in Section 4.2.The system of differential equations is decoupled and can be solved as described in Sec-tion 4.2. To solve the differential equations we require the following boundary conditions,– 25 –hich can be obtained from ref. [7], Y S = (4 π ) ǫ − s − ǫ Γ(1 − ǫ )Γ(1 − ǫ ) Γ( ǫ + 1) ǫ Γ(2 − ǫ )Γ(2 − ǫ ) , (B.11) Y S = (4 π ) ǫ − ( − s ) − ǫ s − ǫ Γ(1 − ǫ ) Γ( ǫ + 1)2 ǫ Γ(2 − ǫ ) , (B.12) Y S = − (4 π ) ǫ − ( − s ) − ǫ s − − ǫ Γ(1 − ǫ )Γ(1 − ǫ ) Γ( ǫ + 1) ǫ Γ(1 − ǫ ) , (B.13)and all other cases vanish. These boundary conditions are given by the leading (soft) termof three master integrals in the limit of δ →
0. They are obtained using the methodsdescribed in Section 4.1. For convenience we will from now on set s = 1.The first two differential equations are homogeneous and can be easily solved to give Y ( δ ) = δ − ǫ Y S . (B.14) Y ( δ ) = δ − ǫ Y S . (B.15)We find that the homogeneous solution to the differential equation of master Y is vanishingand the inhomogeneous solution is according to eq. (4.46) given by Y ( δ ) == (1 − δ ) − ǫ (1 − ǫ ) ǫ δ Z dδ ′ (cid:16) (1 − ǫ )(1 − δ ′ ) ǫ − δ ′− ǫ Y S − (1 − ǫ )(1 − δ ′ ) ǫ − δ ′− ǫ Y S (cid:17) = (2 ǫ − (1 − δ ) − ǫ ǫ F (cid:16) − ǫ − ǫ (cid:17) (1; δ ) Y S − (2 ǫ − ǫ − − δ ) − ǫ ǫ F (cid:16) − ǫ − ǫ (cid:17) (1; δ ) Y S , (B.16)with F (cid:16) ab (cid:17) (1; δ ) = δ a a F ( a, b ; a + 1; δ ) . (B.17)To obtain this result we made use of eq. (A.1). As we proceed to solve the remaining twomaster integrals we find that the inhomogeneous solution to their differential equation isin turn dependent on Y . We are able to find solutions to the inhomogeneous equationmaking use of the definition of our iterated integrals in eq. (A.2).– 26 – ( δ ) = δ − − ǫ δ Z dδ ′ − ǫ )(1 − ǫ ) δ ′ ǫ (1 − δ ′ ) − ǫ − F (cid:16) − ǫ − ǫ (cid:17) (1; δ ′ ) Y S − δ − − ǫ δ Z dδ ′ − ǫ ) δ ′ ǫ (1 − δ ′ ) − ǫ − F (cid:16) − ǫ − ǫ (cid:17) (1; δ ′ ) Y S − δ − − ǫ δ Z dδ ′ (2 δ ′ − − ǫ )(1 − ǫ ) δ ′− ǫ − (1 − δ ′ ) ǫ Y S + δ − − ǫ δ Z dδ ′ − ǫ ) (1 − δ ′ ) − ǫ − ǫ Y S = 2 (cid:0) ǫ − ǫ + 1 (cid:1) δ − ǫ − F (cid:16) ǫ − ǫ ǫ − ǫ (cid:17) (1 , δ ) Y S − − ǫ ) δ − ǫ − F (cid:16) ǫ − ǫ ǫ − ǫ (cid:17) (1 , δ ) Y S + (cid:0) ǫ − ǫ + 1 (cid:1) δ − ǫ − ǫ F (cid:16) − ǫ (cid:17) (1; δ ) Y S − (cid:0) ǫ − ǫ + 1 (cid:1) δ − ǫ − ǫ Y S + 2(1 − ǫ ) ((1 − δ ) − ǫ − δ − ǫ − ǫ Y S (B.18) Y ( δ ) = δ − − ǫ − ǫ )(1 − ǫ ) δ Z dδ ′ δ ′ ǫ (1 − δ ′ ) − ǫ − F (cid:16) − ǫ − ǫ (cid:17) (1; δ ′ ) Y S − δ − − ǫ − ǫ ) δ Z dδ ′ δ ′ ǫ (1 − δ ′ ) − ǫ − F (cid:16) − ǫ − ǫ (cid:17) (1; δ ′ ) Y S + δ − − ǫ (cid:0) ǫ − ǫ + 1 (cid:1) ǫ δ Z dδ ′ δ ′ ǫ (1 − δ ′ ) Y S + δ − − ǫ Y S = 4 (cid:0) ǫ − ǫ + 1 (cid:1) δ − ǫ − F (cid:16) ǫ − ǫ ǫ − ǫ (cid:17) (1 , δ ) Y S − − ǫ ) δ − ǫ − F (cid:16) ǫ − ǫ ǫ − ǫ (cid:17) (1 , δ ) Y S + 2 (cid:0) ǫ − ǫ + 1 (cid:1) δ − ǫ − ǫ F (cid:16) ǫ (cid:17) (1; δ ) Y S + δ − − ǫ Y S (B.19)– 27 –he iterated integrals with two indices contributing to Y and Y can be written as F (cid:16) a a b b (cid:17) (1 , , δ ) = δ a + a ( a + a ) a F , , a + a a + a + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a b b a + 1 1 − − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ, δ ! (B.20)= δ a + a ( a + a ) a ∞ X n,m =0 ( a + a ) n + m ( a + a + 1) n + m ( a ) n ( b ) n ( a + 1) n ( b ) m δ n n ! δ m m ! , where we introduced the Kamp´e de F´eriet function, F p,qp ′ ,q ′ α i α ′ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β i γ i β ′ i γ ′ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, y ! = ∞ X n,m =0 Q pi =1 ( α i ) n + m Q qi =1 ( β i ) n ( γ i ) m Q p ′ i =1 ( α ′ i ) n + m Q q ′ i =1 ( β ′ i ) n ( γ ′ i ) m x n n ! y m m ! . (B.21)Note that the generalized Kamp´e de F´eriet function S of Section 4.1 is a special case ofeq. (B.21), S ( a ; a , b ; a , b ; x , x ) = F , , a a b a − b x, y ! . (B.22) C. Matrix-elements A ggg ( s , s , s ) =Box( s , s , s ) N s s s (cid:16) − s s ( s + s + s ) ǫ + s (cid:0) s + ( s + s ) s + s + s + s s (cid:1) (cid:17) + Box( s , s , s ) N s s s (cid:16) − s s ( s + s + s ) ǫ + s (cid:0) s + ( s + s ) s + s + s + s s (cid:1) (cid:17) + Box( s , s , s ) N s s s (cid:16) − s s ( s + s + s ) ǫ + s (cid:0) s + ( s + s ) s + s + s + s s (cid:1) (cid:17) + A Bub ggg ( s , s , s ) + A Bub ggg ( s , s , s ) + A Bub ggg ( s , s , s ) (C.1)– 28 – Bub ggg ( s , s , s ) =Bub( s ) n s s ǫ ( ǫ − ǫ − N f + Ns s ( s + s ) ( ǫ − ǫ (2 ǫ − × h − s s ( s + s ) (cid:0) s + ( s + s ) s + s + s + s s (cid:1) + ( s + s ) (cid:0)(cid:0) s + 11 s s + 3 s (cid:1) s + 11 s s (cid:0) s + s s + s (cid:1)(cid:1) ǫ + s ( s + s ) (cid:0) s + 11 s s + 3 s (cid:1) ǫ − s s ( s + s ) (cid:0) s + 13 s s + 12 s (cid:1) ǫ − s (cid:0) s + 31 s s + 34 s s + 31 s s + 11 s (cid:1) ǫ − s ( s + s ) (cid:0) s + 31 s s + 34 s s + 31 s s + 11 s (cid:1) ǫ + s s ( s + s ) (cid:0) s + 5 s s + 4 s (cid:1) ǫ + 4 s (cid:0) s + 5 s s + 3 s s + 5 s s + 3 s (cid:1) ǫ + 4 s ( s + s ) (cid:0) s + 5 s s + 3 s s + 5 s s + 3 s (cid:1) ǫ − s ( s + s ) ( s + s + s ) (cid:0) s − s s + s (cid:1) ǫ io (C.2) B ggg ( s , s , s ) =Box( s , s , s ) s s N s (2 ǫ − (cid:16) − s ( s + s ) ( s + s )+ (cid:0) s + s ) s + s (2 s + s ) s − s s ( s + s ) (cid:1) ǫ (cid:17) + Box( s , s , s ) s s N s (2 ǫ − (cid:16) − s ( s + s ) ( s + s )+ (cid:0) − s s − ( s − s ) ( s + s ) s + 2 s s ( s + s ) (cid:1) ǫ (cid:17) + Box( s , s , s ) s s N s (2 ǫ − (cid:16) − s ( s + s ) ( s + s )+ (cid:0) s + s ) s + s s + ( s + s ) (2 s + s ) s (cid:1) ǫ (cid:17) + B Bub , ggg ( s , s , s ) + B Bub , ggg ( s , s , s ) + B Bub , ggg ( s , s , s ) (C.3) B Bub , ggg ( s , s , s ) =Bub( s ) n s s ǫ ( ǫ − ǫ − N f + Ns s ( s + s ) ( ǫ − ǫ (2 ǫ − × h − s ( s + s ) s ( s + s ) ( s + s ) − s s ( s + s ) ( s + s ) ǫ − ( s + s ) (cid:0) s s + (3 s − s ) ( s + s ) s − s s (11 s + 8 s ) s (cid:1) ǫ + s (cid:0) s − s s − s s − s s + 5 s (cid:1) ǫ − s s ( s + s ) ǫ + s (cid:0) s − s s − s s + 5 s (cid:1) ǫ + 5 s s ( s + s ) ǫ − s (cid:0) s − s s − s s − s s + s (cid:1) ǫ − s (cid:0) s − s s − s s + s (cid:1) ǫ io (C.4)– 29 – Bub , ggg ( s , s , s ) =Bub( s ) Ns s ( s + s ) ( ǫ − ǫ h s ( s + s ) s ( s + s ) − ( s + s ) (cid:0) s s + (cid:0) s + 3 s s + s (cid:1) s − s s ( s + s ) (cid:1) ǫ + s (2 s + s ) s ǫ − s s ( s + s ) ǫ + s s ǫ + ( s + s ) (2 s + s ) s ǫ + 4 s ( s + s ) s ǫ i (C.5) A q ¯ qg ( s , s , s ) = − Box( s , s , s ) s N s (2 ǫ − (cid:16) s ( s + s ) ( ǫ − ǫ + s (2 ǫ −
1) + s s ( ǫ − ǫ (cid:17) + Box( s , s , s ) s N s (2 ǫ − (cid:16) s ( s + s + s ) ǫ − s s ǫ + s s (cid:17) + Box( s , s , s ) s s N s (2 ǫ − (cid:16) ( s + s + s ) ǫ + 2 s ǫ − s (cid:17) + Bub( s ) n − s ( ǫ − s (2 ǫ − N f + N s s s ( s + s ) ( ǫ − ǫ (2 ǫ − h + 12 s s ( s + s ) + s ( s + s ) (cid:0) − s + 6 s s + 6 s ( s + s ) (cid:1) ǫ + 27 s s ( s + s ) ǫ − s ( s + s ) (cid:0) s + 13 s s − s s − s (cid:1) ǫ + s (cid:0) − s − s s + 8 s s + 6 s (cid:1) ǫ − s s ( s + s ) ǫ + 2 s (cid:0) s + 9 s s − s s − s (cid:1) ǫ + 2 s ( s + s ) (cid:0) s + 9 s s − s s − s (cid:1) ǫ − s ( s − s ) ( s + s ) ( s + s + s ) i + s (cid:0) ǫ − ǫ + 2 (cid:1) N s ǫ o + Bub( s ) n s N s ( s + s ) ǫ (cid:16) s ǫ + s (cid:0) ǫ + 2 ǫ − (cid:1) + s (cid:0) ǫ + 2 ǫ − (cid:1) (cid:17) + ( s + s + s ) ( s + s − s ǫ ) s ( s + s ) N o + Bub( s ) n ( s + 2 s ) ǫ + 2 s (cid:0) ǫ + 2 ǫ − (cid:1) N s ǫ + (2 s + s + 2 s ) ǫ s N o (C.6)– 30 – . Results: Threshold Expansion In this section, we present the results of our calculation for the functions η a ; jX of Section 2. D.1 gg initial state ˆ η (2;1) gg ( δ ; ǫ ) = N " (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) δ (1 − ǫ ) ( ǫ − ǫ + 16 (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) (1 − ǫ ) ǫ + 4 δ (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) ( ǫ − ǫ ( ǫ + 1)(2 ǫ − ǫ − × (cid:0) ǫ − ǫ − ǫ + 327 ǫ − ǫ + 338 ǫ − ǫ + 120 ǫ − (cid:1) + 4 δ (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) ǫ − ǫ ( ǫ + 1)(2 ǫ − ǫ − × (cid:0) ǫ − ǫ − ǫ + 639 ǫ − ǫ + 857 ǫ +646 ǫ − ǫ + 929 ǫ − ǫ + 36 (cid:1) + δ ǫ ( ǫ + 2) ( ǫ − (4 ǫ − ǫ + 3) × (cid:0) ǫ + 240 ǫ − ǫ + 8504 ǫ + 8669 ǫ − ǫ + 207893 ǫ +118784 ǫ − ǫ + 751815 ǫ + 694791 ǫ − ǫ + 614080 ǫ +312651 ǫ − ǫ + 301338 ǫ − ǫ + 24408 ǫ − (cid:1) + N N f " − δ (cid:0) ǫ − (cid:1) (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) ( ǫ − ǫ ( ǫ + 1)(2 ǫ − ǫ − − δ (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) ǫ (2 ǫ − ǫ − − δ ǫ ( ǫ + 2) ( ǫ − (4 ǫ − ǫ + 3) × (cid:0) ǫ + 6 ǫ − ǫ + ǫ + 310 ǫ − ǫ − ǫ +287 ǫ − ǫ − ǫ + 132 ǫ − (cid:1) + N f δ ǫ − ǫ + 3) + O ( δ ) (D.1)– 31 – η (3;1) gg ( δ ; ǫ ) = N " (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) (cid:0) ǫ − ǫ + 15 ǫ − (cid:1) (1 − ǫ ) ( ǫ − ǫ ( ǫ + 1)(2 ǫ − (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) δ (1 − ǫ ) ( ǫ − ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − (cid:0) ǫ − ǫ + 211 ǫ + 651 ǫ − ǫ + 1644 ǫ − ǫ + 72 (cid:1) + δ (1 − ǫ ) (3 − ǫ ) ( ǫ − ǫ − ǫ ( ǫ + 1) ( ǫ + 2)( ǫ + 3)(3 ǫ − ǫ − × (cid:0) ǫ − ǫ − ǫ + 13014 ǫ − ǫ − ǫ +284685 ǫ + 5833 ǫ − ǫ + 1149456 ǫ − ǫ − ǫ +776466 ǫ − ǫ + 70488 ǫ − (cid:1) + δ − ǫ ) ( ǫ − ǫ − ǫ − ǫ − ǫ ( ǫ + 1) ( ǫ + 2) ǫ + 3)( ǫ + 4) × ǫ − ǫ − ǫ − (cid:0) ǫ − ǫ − ǫ + 1307 ǫ − ǫ + 139225 ǫ + 2358695 ǫ − ǫ − ǫ +32597314 ǫ + 23389539 ǫ − ǫ + 79086832 ǫ + 29647704 ǫ − ǫ + 54419616 ǫ − ǫ + 3272832 ǫ − (cid:1) + N f N " − (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) (1 − ǫ ) ( ǫ − ǫ (2 ǫ − (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) (cid:0) − ǫ + 43 ǫ − ǫ − ǫ + 8 (cid:1) δ (1 − ǫ ) ( ǫ − ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − δ (1 − ǫ ) (3 − ǫ ) ( ǫ − ǫ − ǫ ( ǫ + 1) ( ǫ + 2)( ǫ + 3)(3 ǫ − ǫ − × (cid:0) − ǫ − ǫ + 1488 ǫ + 2484 ǫ − ǫ − ǫ + 20203 ǫ +1086 ǫ − ǫ + 8697 ǫ + 3593 ǫ − ǫ + 1764 ǫ − (cid:1) − δ − ǫ ) ( ǫ − ǫ − ǫ − ǫ − ǫ ( ǫ + 2)( ǫ + 3)( ǫ + 4)(2 ǫ − × ǫ − ǫ − (cid:0) ǫ − ǫ − ǫ + 2723 ǫ + 77751 ǫ − ǫ − ǫ + 470019 ǫ + 375376 ǫ − ǫ + 721968 ǫ − ǫ + 48384 ǫ − (cid:1) + 2 N f δ (cid:0) ǫ + ǫ − ǫ − ǫ + 18 (cid:1) (1 − ǫ ) (3 − ǫ ) ( ǫ − ǫ − ( ǫ + 2)( ǫ + 3)(3 ǫ − ǫ − O ( δ ) (D.2)– 32 – η (4;1) gg ( δ ; ǫ ) = N " δ (cid:0) − ǫ + 111 ǫ − ǫ + 725 ǫ − ǫ + 72 (cid:1) (1 − ǫ ) (3 − ǫ ) ( ǫ − ǫ ( ǫ + 1) (4 ǫ − δ (cid:0) − ǫ + 26 ǫ + 78 ǫ + 286 ǫ − ǫ + 780 ǫ − (cid:1) (1 − ǫ ) ( ǫ − ǫ − ǫ ( ǫ + 1) ( ǫ + 2)(2 ǫ − ǫ − δ (1 − ǫ ) ( ǫ − ǫ − ( ǫ − ǫ ( ǫ + 1) ( ǫ + 2) ( ǫ + 3)(2 ǫ − × ǫ − ǫ − (cid:0) ǫ + 181 ǫ − ǫ − ǫ + 20965 ǫ − ǫ − ǫ + 118041 ǫ + 110903 ǫ − ǫ + 258264 ǫ − ǫ + 13896 (cid:1) + N N f " − δ (cid:0) ǫ + 8 ǫ − ǫ + 6 (cid:1) (1 − ǫ ) (3 − ǫ ) ( ǫ − ǫ ( ǫ + 1)(4 ǫ − δ (cid:0) − ǫ + 24 ǫ + 70 ǫ − ǫ + 28 (cid:1) (1 − ǫ ) ( ǫ − ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − − δ (cid:0) ǫ − ǫ − ǫ + 104 ǫ − ǫ + 897 ǫ − ǫ + 204 (cid:1) (1 − ǫ ) ( ǫ − ǫ − ǫ − ǫ ( ǫ + 1)( ǫ + 2)( ǫ + 3)(2 ǫ − ǫ − ǫ − − N f δ (1 − ǫ ) (3 − ǫ ) ( ǫ − (4 ǫ − O ( δ ) (D.3)ˆ η (4;2) gg ( δ ; ǫ ) = − N " − (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) δǫ (2 ǫ − − (cid:0) ǫ − ǫ + 7 ǫ − (cid:1) ǫ (2 ǫ − − δ (cid:0) ǫ − ǫ − ǫ + 266 ǫ − ǫ + 157 ǫ + 406 ǫ − ǫ + 147 ǫ − (cid:1) ǫ (16 ǫ − ǫ + 4 ǫ + 33 ǫ − ǫ + 3) − δ ǫ (16 ǫ − ǫ + 4 ǫ + 33 ǫ − ǫ + 3) (cid:0) ǫ − ǫ + 79 ǫ + 185 ǫ − ǫ + 1426 ǫ + 220 ǫ − ǫ + 1048 ǫ − ǫ + 36 (cid:1) − δ ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − ǫ − ǫ − × (cid:0) ǫ − ǫ + 580 ǫ + 1074 ǫ − ǫ + 2443 ǫ + 25643 ǫ − ǫ − ǫ + 23686 ǫ − ǫ + 11196 ǫ − ǫ + 432 (cid:1) – 33 – N N f " δǫ (16 ǫ − ǫ + 20 ǫ − − δ ǫ (16 ǫ − ǫ + 20 ǫ − δ (cid:0) ǫ − ǫ + 3 (cid:1) ǫ (64 ǫ − ǫ + 380 ǫ − ǫ + 81 ǫ − + O ( δ ) (D.4)ˆ η (4;3) gg ( δ ; ǫ ) = N " δ (cid:0) ǫ + 54 ǫ − ǫ + 363 ǫ − ǫ + 36 (cid:1) − ǫ ) (3 − ǫ ) ǫ ( ǫ + 1) (4 ǫ − − δ (cid:0) ǫ + 20 ǫ + 45 ǫ − ǫ − ǫ + 270 ǫ − ǫ + 12 (cid:1) ǫ ( ǫ + 1) ( ǫ + 2)(2 ǫ − ǫ − (4 ǫ − δ − ǫ ) ( ǫ − ǫ − ǫ ( ǫ + 1) ( ǫ + 2) ( ǫ + 3)(2 ǫ − ǫ − ǫ − × (cid:0) ǫ + 133 ǫ + 464 ǫ − ǫ − ǫ − ǫ + 11661 ǫ − ǫ − ǫ + 15852 ǫ − ǫ + 828 (cid:1) + N N f " δ (cid:0) ǫ − ǫ − ǫ + 9 ǫ − (cid:1) − ǫ ) (3 − ǫ ) ( ǫ − ǫ ( ǫ + 1)(4 ǫ − − δ (cid:0) ǫ + 17 ǫ + 8 ǫ − ǫ + 16 ǫ − (cid:1) ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − (4 ǫ − δ (cid:0) ǫ + 47 ǫ + 134 ǫ + 82 ǫ − ǫ + 111 ǫ − ǫ + 6 (cid:1) ǫ − ǫ ( ǫ + 1)( ǫ + 2)( ǫ + 3)(2 ǫ − ǫ − (4 ǫ − ǫ − + 2 N f δ ǫ (4 ǫ −
1) (4 ǫ − ǫ + 3) + O ( δ ) (D.5)ˆ η (5;1) gg ( δ ; ǫ ) = N " (cid:0) ǫ − ǫ + 4 (cid:1) ǫ (4 ǫ − ǫ − ǫ + 3) − δ (cid:0) ǫ + 41 ǫ − ǫ − ǫ + 632 ǫ − ǫ + 36 (cid:1) ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − ǫ − δ ǫ − ǫ − ǫ − ǫ ( ǫ + 1)( ǫ + 2)( ǫ + 3)(2 ǫ − ǫ − ǫ − ǫ − × (cid:0) ǫ + 437 ǫ − ǫ − ǫ + 25611 ǫ − ǫ − ǫ +99655 ǫ − ǫ + 12588 ǫ − (cid:1) − δ ǫ − ǫ − ǫ − ǫ − ǫ ( ǫ + 1)( ǫ + 2)( ǫ + 3)( ǫ + 4)(2 ǫ − ǫ − × ǫ − ǫ − (cid:0) ǫ + 127 ǫ − ǫ − ǫ + 22950 ǫ + 58677 ǫ − ǫ − ǫ + 748092 ǫ − ǫ + 259068 ǫ − ǫ + 3168 (cid:1) – 34 – N N f " − ǫ (4 ǫ − ǫ + 11 ǫ − − δ ǫ (4 ǫ − ǫ + 3)+ 4 δ (3 ǫ − ǫ (20 ǫ − ǫ + 31 ǫ − + O ( δ ) (D.6)ˆ η (6;1) gg ( δ ; ǫ ) = N " + 2( ǫ − δǫ − ǫ − ǫ + 2 δ (cid:0) ǫ − ǫ + 1 (cid:1) ǫ (6 ǫ − − δ (cid:0) ǫ − ǫ + 1 (cid:1) ǫ (6 ǫ − − δ (3 ǫ − (cid:0) ǫ − ǫ + 35 ǫ − ǫ + 4 (cid:1) − ǫ ) ǫ (6 ǫ − δ ( ǫ − ǫ − − ǫ ) ǫ (6 ǫ − − δ ( ǫ − ǫ − − ǫ ) ǫ (6 ǫ − (D.7) D.2 gq initial state ˆ η (2;1) qg ( δ ; ǫ ) = N " (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) (1 − ǫ ) ( ǫ − ǫ + δ (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) ǫ − ǫ (2 ǫ − × (cid:0) ǫ + 3 ǫ − ǫ + 28 ǫ − (cid:1) + δ ǫ − ǫ ( ǫ + 1)(2 ǫ − (cid:0) ǫ + 196 ǫ − ǫ − ǫ +3420 ǫ − ǫ − ǫ + 1712 ǫ − ǫ + 424 ǫ − ǫ + 32 (cid:1) + δ ǫ − ǫ ( ǫ + 1)(2 ǫ − ǫ − × (cid:0) ǫ + 184 ǫ − ǫ − ǫ + 10512 ǫ − ǫ − ǫ +7782 ǫ + 2440 ǫ + 408 ǫ − ǫ + 2112 ǫ − (cid:1) − δ (cid:0) ǫ + 5 ǫ − (cid:1) (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) ǫ (2 ǫ − − δ (cid:0) ǫ + 56 ǫ − ǫ − ǫ + 417 ǫ − ǫ + 16 ǫ + 4 (cid:1) ǫ − ǫ (2 ǫ − − δ ǫ − ǫ ( ǫ + 1)(2 ǫ − ǫ − (cid:0) ǫ + 106 ǫ − ǫ − ǫ +1734 ǫ + 347 ǫ − ǫ + 348 ǫ + 5 ǫ + 86 ǫ − (cid:1) + 1 N " δ ( ǫ − (cid:0) ǫ + 5 ǫ − (cid:1) ǫ (2 ǫ − + δ ( ǫ − (cid:0) ǫ + 12 ǫ − ǫ − ǫ + 42 ǫ + 6 ǫ − (cid:1) ǫ (2 ǫ − ǫ − + O ( δ ) (D.8)– 35 – η (3;1) qg ( δ ; ǫ ) = N " (cid:0) ǫ − ǫ + 7 ǫ − (cid:1) − ǫ ) ǫ (2 ǫ −
3) + δ − ǫ ) ( ǫ − ǫ ( ǫ + 1)(2 ǫ − ǫ − × (cid:0) − ǫ − ǫ + 95 ǫ + 209 ǫ − ǫ + 761 ǫ − ǫ + 274 ǫ − (cid:1) + δ − ǫ ) ( ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − ǫ − × (cid:0) − ǫ − ǫ + 390 ǫ + 1195 ǫ − ǫ − ǫ + 8329 ǫ − ǫ + 1824 ǫ + 312 ǫ − ǫ − ǫ + 48 (cid:1) + δ − ǫ ) ( ǫ − ǫ ( ǫ + 1) ( ǫ + 2)( ǫ + 3)(2 ǫ − ǫ − ǫ − × (cid:0) − ǫ − ǫ + 218 ǫ + 4307 ǫ + 1359 ǫ − ǫ + 162 ǫ +66820 ǫ − ǫ − ǫ + 53626 ǫ + 16711 ǫ − ǫ − ǫ + 6744 ǫ − (cid:1) + N N f " − ( ǫ − (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) (1 − ǫ ) ǫ (2 ǫ − − ( ǫ − (cid:0) ǫ + ǫ − ǫ + 20 ǫ − (cid:1) δ − ǫ ) ǫ (6 ǫ − ǫ + 3) − ( ǫ − (cid:0) ǫ + 7 ǫ − ǫ + 5 ǫ + 45 ǫ + 8 ǫ − ǫ + 4 (cid:1) δ − ǫ ) ǫ ( ǫ + 1)(2 ǫ − ǫ − ǫ − δ − ǫ ) ǫ ( ǫ + 1)(2 ǫ − ǫ − ǫ − (cid:0) − ǫ − ǫ + 147 ǫ +49 ǫ − ǫ + 10 ǫ + 442 ǫ + 43 ǫ − ǫ + 48 (cid:1) + (cid:0) ǫ − ǫ + 2 (cid:1) (cid:0) ǫ + 2 ǫ − ǫ + 1 (cid:1) − ǫ ) ǫ + (cid:0) ǫ − ǫ − ǫ + 263 ǫ − ǫ + 288 ǫ − ǫ + 227 ǫ − ǫ + 18 (cid:1) δ − ǫ ) ( ǫ − ǫ (6 ǫ − ǫ + 3)+ δ − ǫ ) ( ǫ − ǫ − ǫ ( ǫ + 1)(2 ǫ − ǫ − ǫ − × (cid:0) ǫ − ǫ − ǫ + 1306 ǫ − ǫ − ǫ + 2752 ǫ + 2134 ǫ − ǫ + 3030 ǫ + 599 ǫ − ǫ + 960 ǫ − ǫ + 48 (cid:1) + δ − ǫ ) ( ǫ − ǫ − ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − ǫ − × (cid:0) ǫ − ǫ − ǫ + 7064 ǫ − ǫ − ǫ + 26560 ǫ +120391 ǫ − ǫ − ǫ + 291319 ǫ − ǫ + 2036 ǫ +14096 ǫ − ǫ + 15696 ǫ − (cid:1) – 36 – N f N " + ( ǫ − (cid:0) ǫ + 5 ǫ − (cid:1) δ − ǫ ) ǫ (6 ǫ − ǫ + 3) + ( ǫ − (cid:0) ǫ + 3 ǫ − ǫ + 4 ǫ + 1 (cid:1) δ − ǫ ) ǫ (2 ǫ − ǫ − ǫ − (cid:0) ǫ − ǫ + 53 ǫ − ǫ + 48 ǫ − ǫ + 16 (cid:1) δ − ǫ ) ǫ (2 ǫ − ǫ − ǫ − + 1 N " − ( ǫ − (cid:0) ǫ + 8 ǫ − ǫ + 12 ǫ − (cid:1) δ − ǫ ) ǫ (3 ǫ − (cid:0) − ǫ + 5 ǫ + 49 ǫ − ǫ + 11 ǫ − ǫ + ǫ + 2 (cid:1) δ − ǫ ) ǫ (9 ǫ − ǫ + 2)+ (cid:0) − ǫ + 18 ǫ + 145 ǫ − ǫ − ǫ + 394 ǫ + 74 ǫ − ǫ + 148 ǫ − (cid:1) δ − ǫ ) ( ǫ − ǫ (3 ǫ − ǫ − + O ( δ ) (D.9)ˆ η (4;1) qg ( δ ; ǫ ) = N " − ǫ − (cid:0) ǫ + 7 ǫ − (cid:1) δ − ǫ ) ǫ ( ǫ + 1)(2 ǫ − ǫ − (cid:0) − ǫ − ǫ + 128 ǫ + 248 ǫ − ǫ + 312 ǫ + 42 ǫ − (cid:1) δ ǫ − ǫ ( ǫ + 2)(2 ǫ − ǫ −
1) (2 ǫ + ǫ − + δ ǫ − ǫ − ǫ ( ǫ + 2)( ǫ + 3)(2 ǫ − ǫ − ǫ −
1) (2 ǫ + ǫ − × (cid:0) ǫ + 23 ǫ + 303 ǫ − ǫ − ǫ + 7977 ǫ +183 ǫ − ǫ + 2616 ǫ + 264 ǫ − (cid:1) + N N f " (cid:0) ǫ + 6 ǫ − ǫ + 6 (cid:1) δ − ǫ ) ǫ ( ǫ + 1)(2 ǫ − ǫ − − ( ǫ − (cid:0) ǫ + 16 ǫ + 31 ǫ + 2 ǫ − (cid:1) δ − ǫ ) ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − − ( ǫ − (cid:0) ǫ + 27 ǫ + 32 ǫ − ǫ − ǫ + 16 (cid:1) δ − ǫ ) ǫ ( ǫ + 1)( ǫ + 2)( ǫ + 3)(2 ǫ − ǫ − ǫ − − (cid:0) ǫ + 8 ǫ − ǫ + 91 ǫ − ǫ + 54 ǫ + 3 ǫ − (cid:1) δ − ǫ ) ( ǫ − ǫ ( ǫ + 1)(2 ǫ − ǫ − (cid:0) ǫ + 8 ǫ − ǫ − ǫ + 238 ǫ − ǫ − ǫ + 230 ǫ − ǫ + 24 (cid:1) δ − ǫ ) ( ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − δ − ǫ ) ( ǫ − ǫ − ǫ − ǫ ( ǫ + 2)( ǫ + 3)(2 ǫ − ǫ − ǫ − × (cid:0) ǫ + 16 ǫ − ǫ + 2247 ǫ − ǫ − ǫ +18007 ǫ − ǫ + 31832 ǫ − ǫ + 1992 (cid:1) – 37 – N f N " (cid:0) ǫ + 7 ǫ − ǫ + 2 (cid:1) δ − ǫ ) ǫ (8 ǫ − ǫ + 3)+ (cid:0) − ǫ + 2 ǫ + 19 ǫ − ǫ − (cid:1) δ − ǫ ) ( ǫ − ǫ (2 ǫ − ǫ − (cid:0) − ǫ + 28 ǫ + 18 ǫ − ǫ + 68 ǫ − (cid:1) δ − ǫ ) ( ǫ − ǫ − ǫ (2 ǫ − ǫ − ǫ − + 1 N " − (cid:0) ǫ + 13 ǫ − ǫ + 26 ǫ − ǫ + 4 (cid:1) δ − ǫ ) ( ǫ − ǫ (4 ǫ − (cid:0) ǫ − ǫ − ǫ + 27 ǫ − ǫ + 10 ǫ + 4 (cid:1) δ − ǫ ) ( ǫ − ǫ − ǫ (4 ǫ − (cid:0) ǫ − ǫ + 6 ǫ + 140 ǫ − ǫ + 278 ǫ − ǫ + 32 (cid:1) δ − ǫ ) ( ǫ − ǫ − ǫ − ǫ (4 ǫ − ǫ − + O ( δ ) (D.10)ˆ η (4;2) qg ( δ ; ǫ ) = − N " ǫ + 2 ǫ − ǫ + 12 ǫ − ǫ + δ (cid:0) − ǫ − ǫ + 33 ǫ − ǫ + 6 (cid:1) ǫ (8 ǫ − ǫ + 1)+ δ (cid:0) − ǫ − ǫ + 28 ǫ + 47 ǫ + 8 ǫ − ǫ − ǫ + 4 (cid:1) ǫ (8 ǫ + 2 ǫ − ǫ + 1)+ δ (cid:0) − ǫ − ǫ + 206 ǫ + 287 ǫ − ǫ − ǫ + 542 ǫ + 144 ǫ − ǫ + 48 (cid:1) ǫ − ǫ ( ǫ + 1)(2 ǫ − ǫ − ǫ − − δ (cid:0) ǫ + 3 ǫ − ǫ + 2 (cid:1) ǫ (8 ǫ − ǫ + 1) − δ (cid:0) ǫ + 2 ǫ − ǫ + 3 ǫ + 1 (cid:1) ǫ (8 ǫ − ǫ + 1) − δ (cid:0) ǫ + 10 ǫ − ǫ − ǫ + 45 ǫ + 28 ǫ − (cid:1) ǫ (32 ǫ − ǫ + 22 ǫ − O ( δ ) (D.11)– 38 – η (4;3) qg ( δ ; ǫ ) = N " ǫ − ( ǫ − ǫ (4 ǫ − ǫ + 3) + (cid:0) ǫ − ǫ + 125 ǫ − ǫ + 36 (cid:1) δ ǫ ( ǫ + 1)(4 ǫ −
1) (4 ǫ − ǫ + 3) + δ − ǫ ) ( ǫ − ǫ ( ǫ + 1) ( ǫ + 2)(2 ǫ − (4 ǫ − × (cid:0) ǫ + 120 ǫ + 87 ǫ − ǫ + 2977 ǫ + 2461 ǫ − ǫ + 11725 ǫ − ǫ + 9895 ǫ − ǫ − ǫ + 144 (cid:1) − δ ǫ − ǫ ( ǫ + 1) ( ǫ + 2)( ǫ + 3)(2 ǫ − ǫ − (4 ǫ − ǫ − × (cid:0) ǫ + 189 ǫ + 438 ǫ − ǫ − ǫ + 2448 ǫ + 9364 ǫ − ǫ +3369 ǫ + 8725 ǫ − ǫ + 1668 ǫ + 1152 ǫ − (cid:1) + N N f " − ( ǫ − ǫ − ǫ (4 ǫ − ǫ + 3) + (cid:0) − ǫ + 32 ǫ − ǫ + 28 ǫ − (cid:1) δ ǫ ( ǫ + 1)(4 ǫ −
1) (4 ǫ − ǫ + 3) + ( ǫ − (cid:0) ǫ + 161 ǫ − ǫ − ǫ + 82 ǫ + 12 ǫ − (cid:1) δ − ǫ ) ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − (4 ǫ − − ( ǫ − (cid:0) ǫ + 163 ǫ + 287 ǫ − ǫ − ǫ + 48 (cid:1) δ ǫ ( ǫ + 1)( ǫ + 2)( ǫ + 3)(2 ǫ − ǫ − (4 ǫ − ǫ − + N f " ( ǫ − ǫ (4 ǫ − ǫ + 3) − δ ( ǫ − − ǫ ) (3 − ǫ ) ǫ (4 ǫ − δ ( ǫ − ǫ − − ǫ ) ǫ (2 ǫ − (4 ǫ − + ( ǫ − (cid:0) ǫ − ǫ + 4 ǫ − (cid:1) − ǫ ) ǫ (2 ǫ −
3) + (cid:0) ǫ − ǫ + 75 ǫ − ǫ + 44 ǫ − (cid:1) δ − ǫ ) ǫ ( ǫ + 1)(2 ǫ − ǫ − δ ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − (4 ǫ − (cid:0) ǫ + 81 ǫ + 201 ǫ − ǫ − ǫ + 683 ǫ + ǫ + 380 ǫ − ǫ − ǫ + 48 (cid:1) − δ ǫ ( ǫ + 2)( ǫ + 3)(2 ǫ − (4 ǫ − ǫ − (cid:0) ǫ + 219 ǫ + 825 ǫ +950 ǫ − ǫ − ǫ + 1276 ǫ − ǫ + 720 ǫ − (cid:1) + N f N " − (cid:0) ǫ − ǫ + 2 (cid:1) ( ǫ − − ǫ ) ǫ (2 ǫ −
3) + δ (cid:0) ǫ − ǫ + 2 (cid:1) ( ǫ − − ǫ ) ǫ (8 ǫ − ǫ + 3) − δ (cid:0) ǫ − ǫ + 4 ǫ − (cid:1) ( ǫ − ǫ (2 ǫ − (8 ǫ − ǫ + 3) – 39 – 1 N " ( ǫ − (cid:0) − ǫ + ǫ − (cid:1) − ǫ ) ǫ − ( ǫ − (cid:0) − ǫ + ǫ − (cid:1) δ − ǫ ) ǫ (4 ǫ − (cid:0) ǫ + 15 ǫ + 194 ǫ − ǫ + 225 ǫ − ǫ + 82 ǫ − ǫ + 8 (cid:1) δ ǫ − ǫ (2 ǫ − (4 ǫ − (cid:0) − ǫ − ǫ − ǫ + 28 ǫ + 2 ǫ − ǫ + 2 (cid:1) δ ǫ − ǫ (2 ǫ − (4 ǫ − ǫ − + O ( δ ) (D.12)ˆ η (5;1) qg ( δ ; ǫ ) = N " ( ǫ − ǫ − ǫ (4 ǫ − ǫ + 3) − δ (cid:0) ǫ + 33 ǫ − ǫ + 131 ǫ − ǫ + 24 (cid:1) ǫ (20 ǫ − ǫ − ǫ + 20 ǫ − δ (cid:0) ǫ + 33 ǫ − ǫ − ǫ + 2135 ǫ − ǫ − ǫ + 440 ǫ + 120 ǫ − (cid:1) ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − ǫ − ǫ − − δ ǫ − ǫ − ǫ − ǫ ( ǫ + 2)( ǫ + 3)(2 ǫ − ǫ − ǫ − ǫ − × (cid:0) ǫ − ǫ + 78 ǫ + 1886 ǫ − ǫ + 929 ǫ + 14430 ǫ − ǫ + 13788 ǫ − ǫ + 576 (cid:1) + N N f " − ( ǫ − ǫ (4 ǫ − ǫ + 3) + δ (cid:0) ǫ − ǫ + 2 (cid:1) ǫ (20 ǫ − ǫ + 23 ǫ − δ ( ǫ − (cid:0) ǫ + 4 ǫ − ǫ + 2 (cid:1) ǫ (2 ǫ − ǫ − ǫ − ǫ − + ( ǫ − (cid:0) ǫ − ǫ + 2 (cid:1) ǫ (2 ǫ − δ (cid:0) − ǫ − ǫ + 15 ǫ + 2 ǫ − (cid:1) ǫ (10 ǫ − ǫ + 8 ǫ − δ ( ǫ + 1) (cid:0) ǫ + 25 ǫ − ǫ + 240 ǫ − ǫ + 56 ǫ − (cid:1) ǫ − ǫ − ǫ (2 ǫ − ǫ − ǫ − δ (cid:0) − ǫ + 63 ǫ − ǫ + 296 ǫ + 34 ǫ − ǫ + 106 ǫ − (cid:1) ǫ − ǫ − ǫ − ǫ (2 ǫ − ǫ − ǫ − ǫ − O ( δ ) (D.13)ˆ η (6;1) qg ( δ ; ǫ ) = N " ǫ − ǫ + δ ǫ (6 ǫ − − δ ( ǫ + 1) (cid:0) ǫ + 12 ǫ − ǫ + 2 (cid:1) ǫ (12 ǫ − ǫ + 1)+ δ ( ǫ − ǫ (12 ǫ − ǫ + 1) − δ ( ǫ − ǫ (12 ǫ − ǫ + 1) + O ( δ ) (D.14)– 40 – .3 q ¯ q initial state ˆ η (2;1) q ¯ q ( δ ; ǫ ) = + N δ (cid:0) ǫ + 5 ǫ − (cid:1) ( ǫ − ǫ (4 ǫ − ǫ + 3) − N N f δ (cid:0) ǫ + 5 ǫ − (cid:1) ( ǫ − ǫ (4 ǫ − ǫ + 3) + 2 N f δ ( ǫ − ǫ (4 ǫ − ǫ + 3) − δ (cid:0) ǫ + 17 ǫ − ǫ + 52 (cid:1) ( ǫ − − ǫ ) ǫ (2 ǫ − + N f δ (5 ǫ − ǫ − N (3 − ǫ ) ǫ (2 ǫ − + δ (4 − ǫ ) ( ǫ − N ǫ (2 ǫ − ǫ − + O ( δ ) (D.15)ˆ η (3;1) q ¯ q ( δ ; ǫ ) = N " − δ ( ǫ + 1) (cid:0) ǫ + 5 ǫ − (cid:1) − ǫ ) ǫ (2 ǫ − ǫ − δ (cid:0) − ǫ + 2 ǫ + 837 ǫ − ǫ − ǫ + 5379 ǫ − ǫ + 1914 ǫ − (cid:1) − ǫ ) ( ǫ − ǫ (2 ǫ − ǫ − ǫ − + 1 N " δ (cid:0) ǫ − ǫ + 4 (cid:1) − ǫ ) ǫ (3 ǫ − δ (cid:0) ǫ + 15 ǫ − ǫ − ǫ + 241 ǫ − ǫ − ǫ + 80 ǫ − (cid:1) − ǫ ) ǫ (9 ǫ − ǫ + 2) + N N f " δ (cid:0) ǫ − (cid:1) (1 − ǫ ) ǫ (2 ǫ − ǫ − − δ (cid:0) ǫ + 7 ǫ − ǫ + 15 ǫ − (cid:1) ǫ (36 ǫ − ǫ + 107 ǫ − ǫ + 6) + N f N " δ ( ǫ − (1 − ǫ ) ǫ (2 ǫ − ǫ − − δ (cid:0) ǫ + 10 ǫ − ǫ + 11 ǫ − (cid:1) ǫ (36 ǫ − ǫ − ǫ + 64 ǫ − ǫ + 6) + δ (cid:0) − ǫ + 7 ǫ + 5 ǫ − ǫ + 12 (cid:1) − ǫ ) ǫ (6 ǫ − ǫ + 6)+ δ − ǫ ) ( ǫ − ǫ ( ǫ + 1)(2 ǫ − ǫ − ǫ − × (cid:0) ǫ + 200 ǫ + 31 ǫ − ǫ + 2162 ǫ − ǫ − ǫ + 2730 ǫ − ǫ + 1608 ǫ − (cid:1) + O ( δ ) (D.16)– 41 – η (4;1) q ¯ q ( δ ; ǫ ) = + N " δǫ ( ǫ + 1) − ǫ ) ( ǫ − (4 ǫ − − δ (cid:0) ǫ + 3 ǫ − ǫ + 4 ǫ + 11 ǫ − ǫ + 2 (cid:1) − ǫ ) ( ǫ − ǫ − ǫ (4 ǫ − δ (cid:0) ǫ − ǫ + 10 ǫ + 348 ǫ − ǫ + 1442 ǫ − ǫ + 328 ǫ − (cid:1) ǫ − ǫ − ( ǫ − ǫ (2 ǫ − ǫ − ǫ − + δǫ ( ǫ + 1)4(1 − ǫ ) ( ǫ − ǫ − δ (cid:0) − ǫ − ǫ + 33 ǫ − ǫ + 4 (cid:1) − ǫ ) ( ǫ − ǫ − ǫ (4 ǫ − δ (cid:0) ǫ + ǫ − ǫ + 395 ǫ + 799 ǫ − ǫ + 400 ǫ + 600 ǫ − ǫ + 96 (cid:1) ǫ − ǫ − ǫ − ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − ǫ − ǫ − N " δǫ − ǫ ) (4 ǫ − δ (cid:0) − ǫ − ǫ + ǫ + 3 ǫ − (cid:1) − ǫ ) ǫ ( ǫ + 1)(4 ǫ − δ (cid:0) ǫ + 32 ǫ + 39 ǫ − ǫ − ǫ + 19 ǫ + 8 ǫ − (cid:1) ǫ ( ǫ + 1) ( ǫ + 2)(2 ǫ − ǫ − ǫ − + O ( δ ) (D.17)ˆ η (4;2) q ¯ q ( δ ; ǫ ) = − N " δ (5 ǫ − ǫ (8 ǫ − ǫ + 1) − δ (cid:0) − ǫ − ǫ + 34 ǫ + 21 ǫ − ǫ + ǫ + 22 ǫ − (cid:1) ǫ ( ǫ + 1)(2 ǫ − ǫ − ǫ − − N f N " δ ( ǫ − ǫ (16 ǫ − ǫ + 20 ǫ −
3) + δ (cid:0) − ǫ + 3 ǫ + 6 ǫ − ǫ + 2 (cid:1) ǫ (64 ǫ − ǫ + 188 ǫ − ǫ + 9) − δ (cid:0) − ǫ − ǫ + 13 (cid:1) ǫ (16 ǫ − ǫ + 20 ǫ − − δ (cid:0) − ǫ + 8 ǫ + 280 ǫ − ǫ − ǫ + 225 ǫ − (cid:1) ǫ (64 ǫ − ǫ + 188 ǫ − ǫ + 9)+ O ( δ ) (D.18)– 42 – η (4;3) q ¯ q ( δ ; ǫ ) = N " δ ( ǫ + 1) − ǫ ) ( ǫ − ǫ − − δ ( ǫ + 1) (cid:0) ǫ + 7 ǫ − ǫ + 7 ǫ − (cid:1) ǫ − ǫ (2 ǫ − (4 ǫ − δ (cid:0) ǫ + 20 ǫ − ǫ − ǫ + 686 ǫ − ǫ + 816 ǫ − ǫ + 80 ǫ − (cid:1) ǫ − ǫ − ǫ (2 ǫ − (4 ǫ − ǫ − + δ ( ǫ + 1)4(1 − ǫ ) (4 ǫ − − δ (cid:0) ǫ + 19 ǫ − ǫ + 2 (cid:1) ǫ (2 ǫ − (4 ǫ − δ (cid:0) ǫ + 119 ǫ + 277 ǫ − ǫ − ǫ + 534 ǫ + 76 ǫ − ǫ + 112 ǫ − (cid:1) ǫ ( ǫ + 1)( ǫ + 2)(2 ǫ − (4 ǫ − ǫ − N " δ ( ǫ − − ǫ ) (4 ǫ − δ (cid:0) − ǫ − ǫ + 12 ǫ + 5 ǫ − ǫ + 1 (cid:1) ǫ ( ǫ + 1)(2 ǫ − (4 ǫ − δ ( ǫ − (cid:0) ǫ + 39 ǫ + 156 ǫ + 177 ǫ − ǫ − ǫ + 26 ǫ + 24 ǫ − (cid:1) ǫ ( ǫ + 2)(4 ǫ − ǫ −
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