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Next-to-next-to-leading order calculations for heavy-to-light decays

We present technical aspects of next-to-next-to-leading order calculations for heavy-to-light decays such as top quark decay, semileptonic b quark decay into a u quark, muon decay, and radiative decays like b ->s gamma. Algebraic reduction of integrals to a set of master integrals is described, methods of determining the master integrals are presented, and a complete list of master integrals is given. As a sample application, the top quark decay width is calculated to O(alpha_s^2) accuracy.

. Analytical results are now known along the whole zero-recoil line [5,6] and at the lower left vertex of the triangle (this work). The dot at the left of the triangle indicates the kinematic configuration corresponding to the decay t → bW .

A. Optical theorem
At O α 2 s , there are three classes of radiative corrections which contribute to the decay t → bW : diagrams with virtual gluon loops, diagrams with real radiation, and diagrams with both a virtual gluon loop and a real gluon emission. The phase space calculations that would be required to calculate the decay rate in this approach are difficult even in the case that both masses m b and m W are neglected, and as we mentioned in the introduction, finding the functional dependence of the decay rate on m W is one of our objectives. Our task becomes manageable by using the optical theorem to relate the decay width to the imaginary part of top quark self-energy diagrams: The self-energy diagrams, Σ, that we must consider for the O α 2 s decay rate are three-loop diagrams (two gluon loops and a t → bW → t loop). The imaginary parts of these diagrams arise from the various cuts that can be made through the b, W , and g lines so that the two sides of the cut correspond to particular decay amplitudes. Despite the introduction of an additional loop, the calculation is feasible as a result of an assortment of techniques which have been developed to solve multiloop integrals analytically with the help of symbolic computation software.

B. Asymptotic expansions
With m b = 0, there are two scales in the problem: m t and m W . We define an expansion parameter ω = m 2 W /m 2 t so that the two scales can be expressed as hard and soft (O (1) and O ( √ ω), respectively) using m t as the unit of energy.
The asymptotic expansion in powers and logarithms of ω of the contributing integrals is performed using the method of asymptotic operation [7], and contributions associated with the two scales are identified using the following heuristic mnemonic. The loop momenta flowing through the gluon lines must be hard or else scaleless integrals (which vanish in dimensional regularization) will arise. This leaves us with two regions to consider, depending on the scale of the loop momentum flowing through the W line. In the first region, all the loop momenta are hard and the Euclidean-space W propagator can be expanded as a series of massless propagators: After the integration, powers of m 2 W /k 2 will become powers of ω. In the second region, the gluon momenta are hard but the loop momentum flowing through the W is soft. As a result, this loop momentum (k) decouples from the other momenta (ℓ) flowing through the b quark lines via the expansion Consequently, the diagrams in this region factor into a product of a two-loop self-energy-type integral and a one-loop vacuum bubble integral with a scale of m W . The leading contribution from this second region is O ω 2 ; the interplay between the two regions gives rise to terms with a formally large logarithm ln ω.

C. Single-scale integrals
With the asymptotic expansions in place to extract the functional dependence of the top decay width on m W , we have reduced the problem to that of solving single-scale self-energy loop integrals. The indispensable tools for doing this are integration-by-parts identities, used to reduce loop integrals with arbitrary powers of denominator factors to a small set of master integrals [8]. There are two approaches with which these identities can be applied to a collection of loop integrals. In the traditional method, one inspects the structure of the identities and rearranges them manually into the form of recurrence relations for a complete solution of the system. This method has proven to be very successful in numerous applications (e.g., [4,9,10,11,12,13]) but it requires much human work to implement.
Recently, Laporta [14] succeeded in developing a targeted solution of the linear system (for only those integrals that are needed in a given problem) [15]. This was not previously practical as it is much more expensive computationally. In our calculation we used the traditional complete approach (programmed in FORM [16]) as well as a modified version of the new targeted algorithm for which we implemented a dedicated computer algebra program. In both cases we independently obtained identical results. This serves as a check of correctness and also enables us to compare the two methods [17].

III. LIST OF TOPOLOGIES AND MASTER INTEGRALS
In the "soft" loop-momentum region, the diagrams factor into a product of a two-loop self-energy-type integrals and a one-loop vacuum bubble integral with a scale of m W . The bubble integral is purely real and can be solved easily. The entire set of two-loop self-energy integrals was first solved in [18]; the contributions from this region are therefore quite straightforward.
In the "hard" loop-momentum region, we need to extract the imaginary parts of genuine three-loop integrals. We identify 9 master topologies in terms of which all 36 O α 2 s diagrams can be expressed. The diagrams which give rise to these master topologies are shown in Fig. 2. They are labeled by the letters A through I in analogy with the notation of [19] for the O α 2 contributions to muon decay. We use dimensional regularization (D ≡ 4 − 2ǫ) and take the top quark mass as the unit of energy, m t ≡ 1. Using the notation [d D k] = d D k/(2π) D we define the following Euclidean-space loop integrals (p is the incoming momentum), A(a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , B(a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , C(a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , D(a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , E(a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , F (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , G(a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , H(a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , For the decay rate calculation, we need the imaginary parts of these integrals, obtained by setting For the exponent a 9 we only need non-positive values; it refers to a product of momenta that may remain in the numerator after canceling scalar products against denominator factors. In order to specify the boundary conditions of the recurrence relations, we need 24 master integrals. They vary greatly in their complexity, but for completeness, we shall list them all. Some of these integrals -particularly those with six or fewer lines -are used by more than one of the topologies and thus any descriptions we provide using the integral labels A through I are not necessarily unique. F denotes the loop factor Γ(1 + ǫ)/(4π) D/2 .

IV. EVALUATION OF MASTER INTEGRALS
The eleven master integrals in Eqs. (26) through (36) were evaluated on a case-by-case basis by integrating the phase space of massless propagators severed by cuts. In this section, we will illustrate the procedure with a few examples. We will compute master integrals having two-, three-, and four-particle cuts, respectively.
The general procedure is as follows. For any particular integral, we can use the optical theorem to obtain its imaginary part via where M 1 and M 2 are the scalar amplitudes arising from the uncut propagators on the two sides of the cut, the cut propagators are assumed to be on mass shell, and dΠ cut is the integral over the phase space of the cut propagators. Working in dimensional regularization, the phase space integral involves a factor of for every particle involved in a given cut, along with a D-dimensional δ-function to conserve overall four-momentum. Much of the phase space integration can be done for a general case without assuming anything about the amplitudes M 1 and M 2 , so that the phase space integrals relevant to an n-particle cut can be reduced to simpler integrals involving a small number of scalar parameters. We shall see this explicitly in the following three examples, each with a different number of cut particle lines.
A. Example 1: two-particle cut Our first example will be the integral of Eq. (30). As can be seen in the sketch of Fig. 3, there is a single cut (through the two rightmost propagators) that can be made through massless lines in the diagram. The two-body phase space integral for massless particles is especially simple, since the two particles must emerge back-to-back while partitioning the total energy equally. Defining the general phase-space factor by  1, 1, 1, 1, 2, 0, 0, 0). The imaginary part of this integral can be obtained from a two-particle cut through the massless lines on the right side of the diagram. Solid and dashed lines depict massive and massless propagators, respectively.
an explicit calculation leads to the two-body phase space factor where, as before, F denotes the loop factor Γ(1 + ǫ)/(4π) D/2 . There are no remaining integrals in this expression, and so we have where L 1 is the factor arising from the two-loop integral for the three-point function to the left of the cut in Fig. 3. Explicitly, where k and l are the loop momenta, p is the incoming momentum from the left (with the onshell condition p 2 = −1), q emerges onto the massless line on the upper-right (q 2 = 0), and (p − q) emerges onto the massless line on the lower-right (such that (p − q) 2 = 0). Applying the Feynman parameters x, y, and z in three consecutive stages, we can evaluate the l-and k-integrals so that we are left with the parametric integral The y-integral can be solved explicitly, but the remaining x-and z-integrals are more difficult. The integral is finite (as a result of our use of a second factor of one of the massive lines) and thus we can expand the integrand in ǫ using expressions like and then evaluate the integrals order by order in ǫ. The leading (ǫ 0 ) term is relatively simple, but the presence of products of logarithms in the subsequent terms quickly escalates the level of difficulty. The FORM packages Summer [21] and Harmpol [22] have proven to be very useful for evaluating some of these integrals; some of the related formulas are conveniently tabulated in [23]. For the first two terms of L 1 , we obtain and, in conjunction with Eqs. (40) and (41), thereby establish the result in Eq. (30).
B. Example 2: three-particle cut An example of a master integral requiring a three-particle cut is Eq. (35). This integral is sketched in Fig. 4, where it is seen that a diagonal cut through the massless lines leaves a one-loop amplitude to the left of the cut and a  F (1, 1, 0, 1, 0, 1, 1, 1, 0). The imaginary part of this integral can be obtained from a three-particle cut through the massless lines from the upper left side of the diagram towards the lower right.
single massive propagator to the right. The three massless particles that we cut through can be put on shell with a range of relative kinematic configurations, meaning that the three-body phase space integral can only be partially solved in advance. The remainder of the integral will be over two kinematic parameters (such as the energies of two of the particles) upon which the amplitudes M 1 and M 2 might also depend. A very convenient formulation of the three-body phase space involves the use of the relativistic invariants as the integration variables. This leads to the expression where B 11 = B(1−ǫ, 1−ǫ). Returning to the master integral in Fig. 4, we observe that the massive propagator (m = 1) in the upper right part of the diagram splits directly into two onshell massless particles, and thus the momentum q flowing through this propagator is related to one of our invariants: In addition, we must include a factor in the phase space integral resulting from the one-loop integral in the lower left portion of Fig. 4. The momentum l flowing through this diagram is related to the other invariant via l 2 = −u, and the loop integral can quickly be evaluated with a Feynman parameter: Substituting this expression into Eq. (47), we have Im F (1, 1, 0, 1, 0, 1, 1, where J denotes the 3-parameter integral: As in the previous example, this integral is finite, and hence we can safely expand the integrand in ǫ and evaluate it term by term. The result, once folded in with the other factors in Eq. (50), leads to the expression of Eq. (35).

C. Example 3: four-particle cut
Our final example illustrates how a four-particle cut can be used to obtain Eq. (34). This integral is sketched in Fig. 5. On each side of the cut we have a massive propagator. As might be expected when cutting through four lines  F (1, 1, 0, 0, 1, 1, 1, 1, 0). The imaginary part of this integral can be obtained from a four-particle cut through the massless lines.
in a planar three-loop diagram, there are no loop integrals remaining. The most general expression for a four-body phase space integral is extremely complicated, but it happens that we can obtain a simpler expression suitable for our purposes by modifying the three-body result. Specifically, the generalization of the three-body phase space integral in Eq. (47) from the case of three massless particles to the case of two massless particles and a particle of squared-mass y = q 2 2 (such that 0 ≤ y ≤ 1) leads to When y = 0, this reduces to Eq. (47), but if we interpret y as the relativistic invariant of a pair of massless particles in a four-body phase space, we obtain a four-body phase space integral that is applicable to certain integrals where two of the massless particles form a subloop. In order to evaluate the integral in Fig. 5, we will need to replace the invariant u = (q 1 + q 3 ) 2 with v = (q 1 + q 2 ) 2 since the massless loop corresponds to a massive q 2 and thus the massive propagators beside the cut carry momenta (q 1 + q 2 ) and (q 2 + q 3 ). With Eq. (55) can be rewritten as The integral we need to evaluate has the additional propagator factors (1 − v) −1 and (1 − z) −1 , and as before, can be expanded in ǫ and evaluated term by term, thereby leading to the result of Eq. (34).

V. RESULTS
The QCD corrections to the decay width for t → bW can be written as where is the tree-level decay width in the limit that m W = 0. The results for X 0 and X 1 are known analytically [24,25].
The two-loop contribution with which we are presently concerned, X 2 , has been estimated numerically [26,27]. Detailed comparisons of our analytical results with those studies is given in [1]. We divide X 2 into four gauge-invariant color structures, (60) Here we use SU (N ) factors (with N = 3) C F = (N 2 − 1)/(2N ), C A = N , and T R = 1/2. N L denotes the number of massless quark species and N H that of quarks with the mass equal to that of the decaying quark. Defining the parameter ω = m 2 W /m 2 t , our results for the four classes of contributions in Eq. (60) are: As discussed in [1], we obtain the two-loop QCD correction to the top quark decay width by substituting ω ≃ 0.213 into these expressions. There are already sufficiently many terms so as to render the theoretical uncertainty 20 times smaller than the experimental uncertainty induced by the mass measurements of the top quark. Furthermore, these expressions can be smoothly matched to the corresponding expressions [4] for the two-loop QCD corrections to the semileptonic decay b → uℓν in the zero recoil limit. The result of such a matching procedure is depicted in Fig. 6.

VI. CONCLUSIONS
We have presented an analytical approach to next-to-next-to-leading order studies of heavy-to-light decays. We have used top quark decay as an illustrative example here; in [1] we also studied the distribution of the lepton invariant mass in the semileptonic b → u decay and have reproduced the O α 2 corrections to the muon lifetime. A number of other phenomenologically interesting studies become possible with the results presented here. For example, corrections to radiative b → sγ decays and to B d,s meson mixing require matrix elements of very similar if not identical kinematic structure. (For the mixing, one can expand in the four-momentum carried by the light quark and obtain propagator-like diagrams studied in the present paper.) We employed two approaches to reducing Feynman integrals to a basis set of primitive ones. In both we use integration-by-parts recurrence relations that we either solve completely, or for a targeted set of needed integrals.
The most appealing feature of the targeted approach is that the core of the program is universal for all topologies. Unlike the complete solution method, where each new topology necessitates a by-hand construction of a new solution algorithm, the targeted approach needs only a minimal specification of the topology under consideration. As a result, once the program core is written and tested, very little programming is needed to incorporate new topologies, thereby saving time, effort, and minimizing the likelihood of programming bugs.
Another major difference between the two approaches is the use of subtopologies. In the traditional method, the removal of a line will often facilitate a mapping of a loop integral from a more complex topology onto a simpler topology, for which a new set of identities is used. The drawback to this intuitively reasonable process is a proliferation in the number of topologies to be programmed. For example, the 9 master topologies that we encounter in this work (discussed in section III) require approximately 40 additional subtopologies. The targeted approach, on the other hand, only needs to deal with the master topologies, as it is able to solve a given topology completely using only one set of identities. At the three-loop level, it would be very difficult for a human to program an algorithm in this way.
A key feature of the targeted approach is the generation of large tables containing all possible integrals in a topology with a predetermined range of powers. Once this table is generated, it is accessed by the programs responsible for assembling a particular physical calculation. In the complete method, conversely, the algorithms are typically applied in real-time, and the generation of dynamic tables will likely be used in the near future to improve the performance of this approach. Given the large range of powers which can arise when performing a calculation using asymptotic expansions, these tables and intermediate expressions can become quite large.
In summary, the targeted approach appears to be superior by virtue of saving human time. The burden is shifted to the computer and thus an efficient implementation of the algorithm is essential. We developed the project-specific software based on the BEAR package [28]. The targeted approach has also recently been implemented in MAPLE [29].
The availability of the targeted approach greatly simplifies multiloop calculations. The remaining bottleneck involves the evaluation of master integrals. Further progress in perturbative calculations may depend on a radically new approach addressing this issue.