QCD corrections to the hadronic production of a heavy quark pair and a W-boson including decay correlations

We perform an analytic calculation of the one-loop amplitude for the W-boson mediated process 0 \to d u-bar Q Q-bar l-bar l, retaining the mass for the quark Q. The momentum of each of the massive quarks is expressed as the sum of two massless momenta and the corresponding heavy quark spinor is expressed as a sum of two massless spinors. Using a special choice for the heavy quark spinors we obtain analytic expressions for the one-loop amplitudes which are amenable to fast numerical evaluation. The full next-to-leading order (NLO) calculation of hadron+hadron \to W(\to e nu) b b-bar with massive b-quarks is included in the program MCFM. A comparison is performed with previous published work.

FIG. 1: Example of a partonic process contributing to W bb production

I. INTRODUCTION
One of the most interesting channels currently under study at the Tevatron and the LHC is the final state containing a W -boson and jets, where some or all of the produced jets are tagged as containing a bottom quark. Several interesting partonic processes contribute to this final state, for example, A background to these processes involving heavy bosons and fermions, is the QCD and electroweak process occurring in the collision of hadrons H 1 and H 2 , An example of a partonic subprocess contributing to this process is shown in Fig. 1. Next-toleading corrections to this process were first considered in [1], working in the approximation in which the b-quark is considered massless. Since in many analyses the b-quark is required to have a minimum p T , typically 15 GeV or more, in order to be efficiently tagged, the neglect of the mass of the bottom quark is expected to be a good approximation.
After these initial studies, this same process was considered in refs. [2][3][4][5], without making the approximation m b = 0. These studies, performed without the inclusion of decay products of the W boson effectively confirmed that the m b = 0 approximation is good to a few percent, with the difference parametrically suppressed as m 2 b /p 2 T . Thus retaining the mass of the bquark extends the prediction to lower values of p T . In addition, it allows us treat the case where the two b-quarks end up in the same jet and the case when one b-quark is either too soft or too forward to be tagged. These kinematic configurations can be important for the Higgs search at the Tevatron and the LHC [6,7].
The purpose of this paper is to repeat the calculation of ref.
[2], i.e. including the effects of a finite b-quark mass, but also including the spin correlations present in the W -boson decay. The calculation is performed using the spinor helicity formalism, with the calculated amplitudes represented as analytic formulae. Although the formulae are not compact enough to be presented in their entirety here, they do lead to an efficient and numerically stable code.
Following a four-dimensional unitarity based approach [8,9], we will construct the logarithmic parts of the virtual amplitude using multiple cuts. Developments to this technique utilizing complex momenta [10][11][12][13] allow us to compute analytic expressions for the coefficients of the known scalar integrals.
The bulk of our paper is dedicated to a description of the calculation of the one-loop corrections to the process 0 → dūQQlℓ retaining the mass for the quark Q. The detailed plan of this paper is as follows. In section II we present our method for dealing with massive spinors using spinor helicity techniques. Section III gives a precise definition of the amplitude that we wish to calculate, including the decomposition into colour stripped amplitudes and the further decomposition into one-loop primitive amplitudes. Sections IV, V and VI illustrate our calculation of the leading colour primitive amplitude, A lc 6 , of the sub-leading colour primitive, A sl 6 , and of the primitive amplitudes containing a closed loop of fermions, A lf 6 and A hf 6 . Section VII presents the renormalization counterterms. After describing the implementation of the calculation into MCFM in section VIII and comparing with earlier work, we draw some conclusions in section IX.

II. TREATMENT OF MASSIVE SPINORS
A method for dealing with a massive particle in the context of the spinor helicity method has been given by Kleiss and Stirling [14]. A massive momentum can always be represented as a sum of two massless momenta. Thus the spinor solution for a massive particle can be expressed in terms of massless spinors by decomposing the physical momentum in terms of the two massless momenta. If we are ultimately going to sum over the spin degrees of freedom, the only constraint that the massive spinors must satisfy is that they should give the standard result for the spin sum after averaging over polarizations, namely, s=± u s (p, m)ū s (p, m) = p + m , s=± v s (p, m)v s (p, m) = p − m . (2) We can decompose a massive vector into two massless vectors by introducing an arbitrary massless reference vector, η, In this equation p ♭ is a massless vector. The details of our spinor product notation are given in Appendix A. The definitions of massive external spinor wave functions are, where the subscripts ± label the spin degrees of freedom. In the massless limit these labels correspond to the helicity quantum numbers, but in the massive case they have no such interpretation. Treating the spinors as independent functions of p ♭ and η i.e. ignoring the constraint in Eq.
(3) we can show using simple manipulations that, This has the attractive feature that amplitudes with different spin labels can be obtained from one another by exchanging p ♭ and η. This method has been used in the calculation of one-loop corrections to top production [15,16].

A. Special choice for massive spinors
In this paper, however, we adopt a different approach. By making a specific choice for the vector η in terms of other vectors in the problem we can simplify the calculation of individual amplitudes. In addition, we shall find that for our particular choice of the vector η, one-loop results for the colour suppressed primitive amplitude can be obtained directly from the corresponding massless amplitude.
The implementation of the Kleiss-Stirling scheme appropriate for the case where we have pairs of massive particles is due to Rodrigo [17]. The two massive momenta p 2 and p 3 with equal masses, corresponding to the momenta of the antiquark and quark respectively, are written in terms of lightlike momenta k 2 and k 3 , where, and s 23 = (p 2 + p 3 ) 2 ≡ 2k 2 · k 3 . The decomposition of Eq. (6) has the advantage that momentum conservation is preserved, p 2 + p 3 = k 2 + k 3 . The inverse transformation is given by, In the rest of this paper we shall denote massive vectors by p i , (p 2 i = 0) and massless vectors by k i , (k 2 i = 0). For the massless vectors k i we shall further define massless spinors, In terms of the two massless momenta in Eq. (8) the explicit results for the massive solutions of the Dirac equation are, This corresponds to choosing, in Eqs. (3,4).
With this choice, the results for the massive quark current with spin labels {+, −} and {−, +} have the same form as they would have in the massless limit (i.e. in the limit in The results for spin labels {−, −} and {+, +} on the heavy quark current are, where the overall normalization is given by, Contracting these equations with a Dirac matrix we obtain, As an example of the use of these spinors, in Appendix B 1 we outline the calculation of the tree-level amplitudes for gQQg scattering, which will appear later as an ingredient in the calculation of our one-loop amplitudes.

III. SETUP
We shall consider the process, both at tree level and including the one-loop QCD corrections. The process is mediated by the exchange of a W -boson which decays into an antilepton-lepton pair with momenta labelled by k 5 and k 6 , as shown in Fig. 2. The V-A structure of the charged weak interaction ensures that the lepton and massless-quark lines will have fixed helicity. Thus the outgoing lepton (6) will be always left-handed and the outgoing antilepton (5) always right-handed.
With this understanding we can often drop the lepton labels from the specification of the amplitude.

A. Colour decomposition
As noted above we shall suppress the labels for the outgoing leptons 5 and 6 which play no role in the colour decomposition. The colour decomposition for the tree graphs is, In Eq. (20) we have introduced the strong coupling constant, g, and the weak coupling, g W , defined through, The Breit-Wigner factor P W (s) is given by, The indices j 1 , j 3 , (j 2 ,j 4 ) denote the colour labels of the corresponding quark (antiquark) lines.
At one loop the colour decomposition is given by, where the overall factor c Γ is, The interference of the one-loop amplitude with the lowest order, summed over initial and final colours, is given by, Therefore A 6;2 plays no role in the calculation of NLO corrections.

D. Structure of the calculation
In the presence of propagators with vanishing masses on internal lines the one-loop amplitude contains infrared and collinear divergences. In addition, the amplitude contains ultraviolet divergences. We regulate all of these divergences using dimensional regularization, continuing the loop integration to D = 4−2ǫ dimensions. The divergences then appear as poles in ǫ. For the primitive amplitudes the divergence structure is quite simple and we can separate the amplitude as follows, where V j contains all the divergent pieces. In this equation A j 6 is any of the primitive amplitudes and the index j runs over {lc, cb, sl, lf, hf}. As we shall see later the simplicity The calculation proceeds by noting that the primitive amplitudes can be expressed as a sum of scalar box, triangle, bubble and tadpole integrals and a rational part, where the label j runs over the values {lc, cb, sl, lf, hf}. In this equation, illustrated in using the QCDLoop program [19]. By using four-dimensional unitarity methods [10][11][12][13]20], we obtain the coefficients, a, b, c, d in the four-dimensional helicity (FDH) scheme [21]. The rational terms require information from beyond four dimensions.

A. Divergent parts
For the leading colour primitive amplitude the divergent term which enters in Eq. (34) is given by [22],

B. Calculation of the box coefficients
The five scalar box integrals that enter the leading colour amplitude are enumerated in Table I. Of these, we directly calculate the coefficients of boxes 1-3 and obtain the remaining two by symmetry: where h 3 and h 2 represent the spin labels of the quark and antiquark respectively and the operation flip is defined in Eq. (27). When using the Rodrigo choice to decompose p 2 and p 3 , box 3 only gives a non-zero contribution for the spin-labels {−, −} and {+, +}.
The coefficients are computed using quadruple cuts with complex momenta [10]. For boxes 1 and 2 the presence of a massive propagator leads to a more complicated parametrization of the loop momentum than in the purely massless case. The particular parameterizations that we have used are spelled out below.

Calculation of d 1|2|3
The cuts used to isolate this coefficient are depicted in Fig. 7, where we remind the reader that we work in the case that k 2 1 = k 2 4 = 0 and p 2 2 = p 3 3 = m 2 . The loop momentum is subject to the following constraints, Denoting l 1 = l − k 1 we have l 1 · k 1 = 0 so we must have either |l 1 ∼ |1 or |l 1 ] ∼ |1], depending on the helicity of the internal gluons, in such a way that the 3-point qqg vertex is non-vanishing.
In the case [l 1 | ∼ [1| we can write where the boldface momenta denote massive vectors. By inspection, it is clear that this parametrization of the loop momentum l µ satisfies the constraints in Eq. (38). Correspondingly in the case l 1 | ∼ 1| we can write menta, As an example, we will calculate the coefficient of this box integral for the spin label choice {−, +}. The box coefficient is given by the formula, with the loop momenta fixed according to the constraints above, i.e. l µ is given by either Eq. (39) or (40) above. In this formula we have for brevity suppressed the gluon particle labels. Note that we have not explicitly cut the heavy quark propagator but, equivalently, have simply written the full gQQg amplitude and multiplied by the cut propagator. Since the gQQg amplitudes with the same helicity gluons vanish (c.f. Eq. (B13)), the coefficient receives only two contributions.
Let us first inspect the assignment h a = −h b = +1. We note that it is simplest to manipulate the gQQg amplitude, (Eq. (B13) first line), using the Schouten identity, Eq. (A6), into an alternative form in order to take full advantage of the vanishing of the cut propagator, On shell, (l 1 − p 2 ) 2 − m 2 ≡ l 1 | 2|l 1 ] = 0 and we can simply discard the second term in this equation. Using results for the other amplitudes presented in Appendix B we find that, where we have simplified the gQQg amplitude further using the decomposition of p 2 in terms of k 2 and k 3 . Since the propagator l does not have a simple decomposition in terms of external momenta, it is simplest to eliminate it by multiplying this expression in numerator and denominator by a factor 4 l l 123 l . After some simplification this yields, This is now in a form where we can use the loop momenta definitions given in Eq. (40) with spinors identified by, for example, using the identity, l µ = 1 2 l|γ µ |l]. We can thus make the replacements: This results in the final expression, Adding in the contribution from the other helicity configuration (h a = −h b = −1), computed in a similar fashion, yields the full result for this coefficient, This is in agreement with the coefficient presented in Ref. [18] in the limit that the heavy quark mass is taken to zero.

Calculation of d 1|2|34
This coefficient is obtained from the cuts shown in Fig. 8, where the loop momentum is subject to the conditions, An explicit solution for l 1 = l − k 1 is given by, The other solution, corresponding to [l 1 | ∼ [1| can be obtained from Eq. (50) by complex conjugation. We note that both of these parameterizations can be obtained from Eqs. (39) and (40) by simply making the replacements, 3 → ( 3 + 4) and s 23 → s 234 . From these forms it is straightforward to compute the coefficients of this box.

Calculation of d 1|23|4
The cuts used to isolate this coefficient are depicted in Fig. 9. This box with massless internal lines and two opposite massive external lines is often referred to as the easy box [23].
The heavy quark does not participate in the loop, and the loop momentum is subject to the following constraints, The explicit solutions for l 1 = l − k 1 and l 123 = l − k 1 − p 2 − p 3 are given either by, or by, Both of these clearly satisfy the constraints in Eq. (52).
As an example, we will calculate the coefficient of this box integral for the spin label choice {−, +}. For this spin label choice it turns out that the box coefficient vanishes. The argument is quite general and may be useful in explaining the absence of the easy box in other contexts, e.g. ref. [24]. The result for the box coefficient is calculated from, with the loop momenta fixed according to the constraints above, i.e. l µ is given by either

C. Calculation of triangle coefficients
The leading colour amplitude receives contributions from the ten triangle scalar integrals listed in Table II. The coefficients of Triangles 1-4 are calculated directly as detailed in the subsection below. Triangle 5 can be simply obtained by symmetry, The coefficients of the five remaining triangles are then uniquely determined by the known divergence structure [22] of the amplitude, Eq. (36). The terms in Eq. (36) proportional to log(−2k 1 .p 2 )/ǫ and log(−2k 4 .p 3 )/ǫ fix two of the triangle coefficients, while the absence of single poles of the form log(s)/ǫ for s ∈ {s 23 , s 1234 , s 123 , s 234 } requires the following relations, .
These expressions are written in terms of already-calculated box and triangle coefficients, the leading order amplitude, and the other triangle coefficients discussed below.

Forde method for triangle coefficients
We will calculate the coefficients of the triangle integrals using the method of Forde [13].
Triangles 2, 3 and 4 will require a slight extension of the formalism to include one of the internal propagators with a mass. We first review the case of three massless internal momenta, shown in Fig. 10 in order to introduce our notation which differs from that of Forde.
Defining l 1 and l 2 as follows, the cut loop momenta (l 2 i = 0), i = 0, 1, 2 may be written in the following general form, All momenta can be expanded in terms of massless momenta, K ♭ 1 and K ♭ 2 , The inverse relations are, The dot product of K 1 and K 2 in Eq. (63) produces a quadratic equation for γ, the solutions of which express γ in terms of the external momenta, From Eq. (62) the massless vectors l i can be expressed as a linear combination of the spinor solutions for the vectors K ♭ 1 and K ♭ 2 , The on-shell conditions l 2 i = 0 for i = 0, 1, 2 allow us to derive the coefficients, x i and y i , The spinor products can be expressed as follows, Turning now to the case where the propagator with momentum l 2 has a mass, we have that l 2 0 = l 2 1 = 0 and l 2 2 = m 2 . Since l 2 is no longer massless it does not have an expansion of the form Eq. (66). The results for the coefficients in the expansion defined in Eq. (66) for l 0 and l 1 are, With these results in hand we can then compute the coefficients for triangles 2-4 with the loop momentum assignments as shown in Figure 11 by making the replacements: • Triangle 2: K 1 = −p 123 , K 2 = p 3 .
• Triangle 4: For the case of triangles 2 and 4, we see that the coefficients are particularly simple since We shall sketch the calculation of the coefficient of triangle 4 because it is the simplest.
We have S 1 = p 2 23 = s 23 and S 2 = p 2 3 = m 2 and hence see that,  (1, 2, 3, 4) and the notation used in the text to denote their coefficients.
The lightlike momenta, Eq. (64), for the two solutions for γ reduce to, Using these assignments we find that the formulae for the coefficients of this triangle are relatively compact. For example, for the {−, +} spin labels the result is,

D. Calculation of bubble and tadpole coefficients
The bubble integrals present in the leading colour amplitude are shown in Table III. The coefficients of bubbles 1-4 are computed by using the method of spinor integration [11,12,20]. This is straightforward to apply for bubbles 1-3 but requires a small modification for bubble 4 due to the effect of the massive propagator. The modified method is described in the subsection below. Coefficients of the bubble integrals 5 and 6 are then obtained by symmetry, All occurrences of B 0 (p 3 ; 0, m) have been replaced by B 0 (p 2 ; 0, m), so we only need to determine the coefficient b 2 . A linear combination of the coefficient b 2 and a is determined from the known form of the single pole in ǫ, (i.e. the single pole with no associated logarithm in the expansion of Eq. (36)), Eq. (74) is sufficient to fix the poles and the logarithms, but since, it leaves the constant term undetermined. In order to separately fix the coefficients a and b 2 we perform a direct Feynman diagram computation of the tadpole coefficient a to find, These results for a are valid in any covariant gauge.

Bubble integral with one massive propagator
To set up the formalism we consider the scalar bubble integral with one massive propagator, as shown in Fig. 12 (bubbles 4 and 5 are of this type). The evaluation of the coefficient of this bubble integral starts from the identity, so that the formalism follows through as in the massless case, except that t is now frozen at the value, The discontinuity of the scalar integral is given by, Using the standard formula [11,20] we can express the integrand as a total derivative, and perform one of the spinor integrations, We obtain, where η is an arbitrary massless momentum, We perform the final integral over l dl by inspecting the residues of the pole in the integrand when l| = [η|P to find, So, in contrast to the massless case, when applying the spinor integration approach an additional rescaling factor of P 2 /(P 2 −m 2 ) must be applied in order to obtain the coefficient of the scalar bubble integral.

E. Calculation of the rational terms
The purely rational terms are calculated using a traditional Feynman diagram approach, with tensor integrals handled using Passarino-Veltman reduction [25]. This might seem like a retrograde step, ineluctably leading to the algebraic complexity that we have been trying to avoid in this amplitude calculation. However two details of our particular case make it quite simple.
First, in the calculation of the rational part we only need to consider diagrams that violate the cut-constructibility condition [9]. This states that if n-point integrals (n > 2) have at most n − 2 powers of the loop momentum in the numerator of the integrand, and the two-point integrals have at most one power of the loop momentum, they will be cutconstructible. For our particular calculation the pentagon diagrams are all cut-constructible, and, in leading colour, there is only one box diagram, shown in Fig. 13, which is not cutconstructible and hence gives a rational part. The calculation of the rational parts from lower point diagrams, n = 3, 2 does not lead to great algebraic complexity. Second, the rational terms are generated by terms in the Passarino-Veltman decomposition that involve the metric tensor, g µν . Therefore any terms that are not proportional to g µν may be discarded at an early stage of the computation.
The results for the rational terms turn out to be quite simple. For the {−, +} case we find, where the flip operation is defined in Eq. (27). The contribution for the opposite spin labels on the heavy quark line is simply related to this one, For the {−, −} spin labels the result is, where the flip symmetry applies to all terms inside the curly brackets {. . .} but not the prefactor N −− , defined in Eq. (14). The final combination of spin labels is obtained by symmetry, which we note is equivalent to replacing N −− by N ++ in Eq. (86).

V. THE RESULTS FOR PRIMITIVE AMPLITUDE
The subleading colour primitive amplitude is shown in Fig. 4. Fortunately with our choice of massive spinors, Eq. (10), the entire result for all spin labels of the massive quark line can be obtained from the one-loop results of Bern, Dixon and Kosower in Eqs. (12.10,12.11) of ref. [18], after some replacements and manipulation. The calculation reported in ref. [18] was for zero mass quarks and leptons with the following helicity assignments, According to Fig. 2 our standard labelling of the graphs (for one of the amplitudes whose spin labelling corresponds to a non-zero amplitude in the massless case) is, To establish the correspondence between our massive amplitudes and the massless ones of ref. [18], we first consider the tree graphs. In our notation the tree amplitudes are given by Eqs. (26,28). Note that we must perform the interchanges (1 ↔ 3), ↔ []), to compare with Eq. (12.9) of ref. [18]. As a consequence of our Eq. (12) the massive {−, +} treelevel amplitude, Eq. (26), calculated from diagrams of Fig. 2 and expressed in terms of the massless vectors, k i , is identical to the massless result presented in Eq. (12.9) of ref. [18] after performing the above interchange.
This same transformation, flipping the sign of all helicities and interchanging 1 and 3, can be used at one-loop level to obtain the bulk of the results for the massive theory. This is true for the upper two diagrams of Fig. 4, because the massive quark enters only in the form of the heavy quark current see, Eqs. (12,13). The primitive amplitude shown in Fig. 4 can be split into divergent (V sl ) and finite (F sl ) pieces as follows After performing the interchanges to reduce it to our notation, the singular part of the one-loop amplitude as reported by Bern, Dixon and Kosower in ref. [18], Eq. (12.10) is, where the two contributions correspond to the upper and lower row of Fig. 4 in the massless theory, where s ij = 2k i · k j . V vertex in Eq. (93) is the complete correction to external vertex for a massless line in the FDH scheme. For the massive case this must be replaced by the vertex correction for a massive line, i.e. the result for the lower two diagrams of Fig. 4. The result is, where β and β ± are given in Eq. (7), x = −β − /β + , and, The self energy corrections on the external massive lines will be accounted for separately in association with the wave function renormalization. This concludes our description of the divergent parts and the lower two graphs of Fig. 4.
We The only remaining issue is whether we can also obtain the massive {−, −} and {+, +} amplitudes from the massless results. Note that in ref. [18] the finite parts of the massless amplitudes are written in terms of certain symmetry operations in order to make the amplitudes more compact. As a first step we write out the amplitudes explicitly. With this result in hand we want to address the issue of whether the results obtained with an external fermionic current, Eq. (12) (i.e. the massless one-loop amplitude), can be used to obtain the results with the fermionic currents of Eq. (13). Thus expressed in our notation, the massless amplitude for helicity choice (2 + Q , 3 − Q ) must contain one 3| and one |2]. All other dependence on k 2 or k 3 can only enter in the combination k 2 + k 3 which can be eliminated by momentum conservation. After the amplitude has been recast in this form, we can obtain the required result by replacing the current of Eq. (12) with the current of Eq. (13).
An example may help to clarify the procedure. We shall consider a particular box which contributes to the one-loop amplitude, The result for this box coefficient in the massless helicity-conserving amplitude, (adapted from the first line of Eq. (12.11) of ref. [18]) is, The unrenormalized contribution of the fermion loop diagram, shown in Fig. 14, for a quark of mass m is, where (not including the minus sign for a fermion loop), and T R = 1 2 . For n lf quarks which can be considered massless this becomes, Thus in our notation, (c.f. Eq. (31)) the result for the unrenormalized fermion-loop primitive is, where A tree 6 are given in Eqs. (26,28).

VII. RENORMALIZATION
The amplitudes presented so far are bare amplitudes, which require ultraviolet renormalization. The renormalization scheme is slightly more complicated in the presence of massive particles [29] so we specify it in detail here. The requirements for our renormalization scheme are, • The decoupling of heavy quarks should be manifest.
• The evolution equations for the running coupling and for the parton distribution functions should be the same as the equations in the theory without the heavy quark.
Both the strong coupling and the parton distribution functions should run with the coefficients appropriate for the MS scheme in the absence of the massive particles.
• The mass parameter should correspond to a pole mass.
These three requirements completely specify the renormalization scheme. If the diagram in question contains no heavy internal loops of heavy particles we use the MS scheme. If on the other hand the diagram contains heavy loops we will perform subtraction at zero momentum, p = 0. The resultant renormalized Green's function will be a function of p 2 /m 2 and hence exhibit decoupling as the mass of the heavy quarks becomes large.
The full renormalized amplitude, A R 6;1 is obtained by adding an overall counterterm, We will now justify the contributions in the counterterm term by term. The first term in In the high mass limit Π R simplifies to, Thus after renormalization the contribution of the heavy quark is given by (c.f. Eq. (31)) We must also perform a finite renormalization of the gauge coupling [27] to translate from the FDH coupling to the normal MS coupling, This explains the third term in Eq. (105). The last term in Eq. (105) represents the wave function renormalization for the two external massive fermions, calculated in Appendix D.
In the FDH scheme we have from Eq. (D10), independent of the gauge-fixing parameter in any covariant gauge.
Since our calculation is performed in the four-dimensional helicity scheme there is a further finite renormalization [27] required to arrive at the 't Hooft-Veltman scheme. We shall work consistently in the FDH scheme, so this will not be required.

VIII. IMPLEMENTATION INTO MCFM
The one-loop matrix elements, computed using the methods described above, have been included in a full next-to-leading order calculation of the W QQ process. The amplitude for the lowest order process is given in Eqs. (26,28). In order to complete the NLO calculation the Born level amplitude and the one-loop amplitude must be supplemented with results for the real radiation diagrams and a method for cancelling infrared singularities between the two contributions. The tree-level real radiation process has been computed using the diagrams shown in Fig. 15, adopting the same choice of massive spinors as used in the virtual contribution. Infrared singularities are handled using the subtraction method [32] implemented using the dipole formulation [33] and extended to the case of massive emitters and spectators [34]. The full calculation will be made available as part of the the MCFM code [30,31].
To  2. The cancellation of infrared singularities is performed using a slight extension of the original dipole formulation in which the extent of each subtraction region is controlled by an additional parameter [36,37]. This parameter also appears in the integrated form of the dipole counterterms in such a way that the sum of real and virtual radiation does not depend upon its value. We have checked that this independence is indeed manifest in our calculation.  For convenience, we choose the same set of input parameters here as reported in Ref.
[3], which are summarized in Table IV. The final state is defined by the following cuts on the b-jets, where the jets are identified using the k T clustering algorithm with pseudo-cone size R = 0.7.
The results presented here are inclusive of the additional jet that may be present at NLO.
As already discussed previously, although we treat the b-quark as a massive particle when it appears in the final state, we use n lf = 5 light flavours in the running of α s and the PDF evolution. This is primarily for comparison with previous work [3]. We note that because of the smallness of V cb and V ub the b-quark distributions in the initial state make a negligible contribution. However because s 23 > 4m 2 b it is more appropriate to have a strong coupling constant running with n lf = 5 active flavours.
We first present the cross sections for this process at the LHC, for center-of-mass energies √ s = 7, 8 and 14 TeV. The results at LO and NLO are shown in Table V In this equation the azimuthal angles and pseudorapidities (in the lab-frame) of the lepton and jets are denoted by φ and η respectively. We see that the effect of the NLO corrections on the shape of this distribution is relatively minor, visible only for R min jl > 1.5.

IX. CONCLUSIONS
We have presented the first computation of the W bb cross section with massive b-quarks, including the lepton correlations present in the decay of the W -boson. This calculation required knowledge of the one-loop virtual corrections to the qQQqℓℓ process retaining the mass for the heavy quarks Q. The one-loop amplitude was obtained using the spinor helicity formalism. This method has been extensively used for one-loop calculations with massless quarks, but rarely with massive quarks. The calculation required a number of modifications of standard techniques to cope with the presence of the mass.
Although our results are analytic we have not yet simplified them sufficiently to publish them in a journal article. Our results for the one-loop amplitudes will be included in the released version of MCFM, which is an appropriate method of publishing such results. Our analytic results did lead to a code which is fast and numerically stable. Using this code we intend to study the detailed phenomenology of this process in a future publication. For the case of a massless momentum k j we may write, but for the case of a massive momentum, (in our notation the momenta p 2 and p 3 ), this separation in no longer possible. As a compact notation we therefore write in this case i| j|k], denoting the massive momentum by a bold-face symbol.