Non-cancellation of electroweak logarithms in high-energy scattering

We study electroweak Sudakov corrections in high energy scattering, and the cancellation between real and virtual Sudakov corrections. Numerical results are given for the case of heavy quark production by gluon collisions involving the rates $gg \to t \bar t, b \bar b, t \bar b W, t \bar t Z, b \bar b Z, t \bar t H, b \bar b H$. Gauge boson virtual corrections are related to real transverse gauge boson emission, and Higgs virtual corrections to Higgs and longitudinal gauge boson emission. At the LHC, electroweak corrections become important in the TeV regime. At the proposed 100 TeV collider, electroweak interactions enter a new regime, where the corrections are very large and need to be resummed.


I. INTRODUCTION
Electroweak corrections grow with energy due to the presence of Sudakov double logarithms α W ln 2 s/M 2 W , and are already relevant for LHC analyses with invariant masses in the TeV region. The corrections arise because of soft and collinear infrared divergences from the emission of electroweak bosons. The infrared singularities are cutoff by the gauge boson mass, and lead to finite α W ln 2 s/M 2 W corrections. Unlike in QCD, the electroweak logarithms do not cancel even for totally inclusive processes, because the initial states are not electroweak singlets [1][2][3].
In this paper, we discuss the cancellation (or lack thereof) between real and virtual corrections. We will use gg → tt, bb as an explicit numerical example. In this process, the initial state is an electroweak singlet, so the total cross section does not contain α W ln 2 s/M 2 W corrections. This allows us to compare the electroweak corrections in this process to the more familiar case of α s corrections to the R ratio for e + e − → hadrons. Even though electroweak Sudakov corrections cancel for the total cross section, they do not cancel for interesting experimentally measured rates, and are around 10% for invariant masses of ∼ 2 TeV. Electroweak corrections to processes involving electroweak-charged initial states, such as Drell-Yan production, qq → W W , or qq → tt, are larger than for gg → tt.
At present, omitted electroweak corrections are the largest error in many LHC cross section calculations, and are more important than higher order QCD corrections. Furthermore, the resummed electroweak corrections to all hard scattering processes at NLL order are known explicitly [4][5][6], and have a very simple form, so they can be incorporated into LHC cross section calculations.
Recently, there has been interest in building a hadron collider with an energy of around 100 TeV. For such a machine, electroweak corrections are no longer small, and resummed corrections must be included to get reliable cross sections. The numerical plots in this paper go out toŝ = 30 TeV to emphasize the importance of electroweak corrections at future machines.
We will make one simplification in this paper, by computing electroweak corrections in a pure SU (2) W gauge theory, neglecting the U (1) part. The reason is that in the Standard Model (SM), after spontaneous symmetry breaking, there is a massless photon. Electromagnetic corrections produce infrared divergences which are not regularized by a gauge boson mass. Instead they have to be treated by defining infrared safe observables, as done for QCD. Initial state infrared corrections can be absorbed into the parton distribution functions (PDFs). To implement this consistently requires electromagnetic corrections to be included in the parton evolution equations. These additional complications are separate from the main point of the paper, and can be avoided by using the SU (2) W theory.
The numerical results will be given for an SU (2) W gauge theory with α W equal to the Standard Model value α/ sin 2 θ W . We will treat W 1,2 as the SM W bosons, and W 3 as the SM Z 0 , and use the notation L ≡ ln s/M 2 W . The structure of electroweak corrections is discussed in Sec. II, and a summary of the SCET EW results for computing these is given in Appendix A. The cancellation of real and virtual electroweak corrections is discussed in Sec. III for an example where one can do the full computation analytically, and Sec. IV discusses the cancellation for heavy quark production, where the rates have to be computed numerically. Some subtleties for an unstable t-quark are discussed in Sec. IV C. The implications of electroweak corrections for experimental measurements is given in Sec. V.
It is instructive to compare the SCET result with the vastly more difficult conventional fixed order approach to computing electroweak corrections. At fixed order one gets an expansion n,r c n,r α n W L r with r ≤ 2n, which breaks down at high energies. Furthermore, one has to do a very difficult multi-scale computation (with scales s, M Z , M H , m t ) for each new process being considered. The fixed order results are available only for a few cases, and often with the approximation that M W = M Z = M H . In contrast, the SCET result, Eq. (A2) has a simple form where all the pieces are known, so each new process can be computed by multiplying the appropriate factors, which are all known in closed form. The reason the fixed order calculation is much harder, of course, is that it includes the M 2 Z /s power corrections. However, these are negligible in the region where the electroweak corrections are large and experimentally important. We summarize the SCET EW results in Appendix A. More details can be found in Refs. [4][5][6][11][12][13][14][15].
An explicit numerical analysis comparing fixed order and SCET EW results is given in Sec. III.

III. CANCELLATION OF REAL AND VIRTUAL CORRECTIONS
Recall the familiar example of the total cross section for e + e − → hadrons, which has an expansion in α s (Q 2 ), with no large logarithms. At one-loop, the virtual correction to e + e − → qq are infrared divergent, as is the e + e − → qqg real radiation rate, but the sum of the two is infrared finite, and gives the correction to the ratio of the e + e − total cross section to its tree-level value, R = 1 + α s /π.
The electroweak corrections to gg → qq have a similar cancellation. Rather than study this process, we first start with the simpler case of J → qq, where J µ = qγ µ P L q is an external gauge invariant current that produces the doublet q L = (t, b) L , where we treat t and b as massless quarks. The main reason for doing this is to avoid complicated phase space integrals for real radiation, and fermion mass effects, and because it is closely related to the familiar QCD case of R. The gg → qq case with q L = (t, b) L will be studied numerically in Sec. IV.
The total cross section for J → qq can be written as the imaginary part of the vacuum bubbles Π(Q 2 ) in Fig. 1. Π(Q 2 ) at Euclidean Q 2 is infrared finite. Thus the analytic continuation to Minkowski space is also infrared finite, and the sum of the real and virtual rates, which is equal to the imaginary part of Π(Q 2 ), is infrared finite.
The virtual correction to J → qq is given by the graph in Fig. 2 and wave-function graphs, which gives the vertex form-factor at Euclidean momentum transfer q 2 = −Q 2 < 0, with where M W is the gauge boson mass. 1 Analytically continuing to time-like q 2 = s > 0, gives − 2r − (3 + 2r) ln r + (1 + r) 2 2Li 2 (−r) − ln 2 r + 2 ln r ln(1 + r) + π 2 3 + iπ (3 + 2r) + 2 (1 + r) 2 (ln r − ln(1 + r)) which for s → ∞ is The SCET EW computation gives radiative corrections to the Jqq operator neglecting M 2 /s power corrections, and gives precisely Eq. (5), when expanded out to order α W [11]. The one-loop virtual correction to the Jqq cross section is (neglecting power corrections) where σ 0 is the tree-level cross section. The − ln 2 r and −3 ln r terms lead to large corrections at high energy. The real radiation J → qqW arises from the graphs in Fig. 3, and is Expanding in r gives The total radiative correction is Plotted are the exact virtual correction (solid blue), the virtual corrections using SCETEW (dashed blue), real radiation (red), exact total rate (black) and the total rate using the SCETEW virtual correction (dashed black). and as r → 0 gives The ln 2 r and ln r terms cancel between σ R,V . The correction to R in QCD is given by Eq. (10) with the replacement α W → α s and C F → 4/3. The real and virtual corrections are shown in Fig. 4. Also shown is the virtual correction computed using the SCET EW result of Eq. (A2). The SCET EW and exact calculations for the virtual correction have only very small differences, which are below 1% for E > 2M W ∼ 160 GeV, and < 0.5% by 400 GeV, whereas the real and virtual corrections each exceed 5% by the time E > 15M W ∼ 1.2 TeV. This shows that in the regime where the electroweak corrections are relevant at the LHC, the SCET EW computation is sufficiently accurate. The figure also shows that the large real and virtual electroweak corrections cancel in the total cross section.
The above calculation demonstrates the usual cancellation of the L 2 and L terms between real and virtual graphs, if one sums over the entire SU (2) multiplet. This means that the cancellation holds provided one sums over the processes J → tt, J → bb, ttZ, bbZ, tbW − , and btW + . Most of the time, however, one is not interested in such a total cross section. This is because the particles of a given SU (2) multiplet can have very different experimental signatures. For example, the experimentally measured bb production rate only includes J → bb, and none of the other rates. The difference from QCD is that color charge is not an experimental observable, but electroweak charge is. Thus one can experimentally select on final states on the basis of electroweak charge.
The real and virtual cross sections are modified if one does not sum over all final states. In the simple example we are considering with degenerate fermions and bosons, the only change is that Eqs. (6,8) are modified by the replacement of the group theory factor N C F (N = 2) by G V and G R , which need not be equal, so that the total cross section can have large corrections at high energy. The dependence of the cross section on ln 2 r + 3 ln r is characteristic of the IR structure of a vector current [35].
To study this non-cancellation, we tabulate the group theory factors G V,R in Table I for some possible choices of final state, for an SU (N ) gauge theory. In Eq. (11), σ 0 = N σ 0 is the total tree-level rate, so thatσ 0 is Nindependent. The different cases are: 1. Any fermion with or without any gauge bosons, i.e. the full inclusive rate.
3. Specify one fermion with or without any gauge bosons, e.g. t + X, with X =t,tZ,bW − .
4. Specify one fermion and no gauge bosons, e.g. t+X, with X =t.
6. Specify both fermions and require no gauge bosons. Same as the previous case but X cannot contain gauge bosons.
One can see that for cases 1 and 3, the logarithmic terms are absent, while for all other cases, the logarithms survive and give rise to large corrections at high energies.

IV. HEAVY QUARK PRODUCTION
In this section, we study the real and virtual corrections to heavy quark production via gluon fusion, gg → qq. The tree-level graphs are given in Fig. 5. The real radiation is computed by numerical integration using MadGraph5 aMC@NLO [36]. The virtual corrections use the SCET results of Ref. [5]. Since the real emission rate is a fixed order result, the virtual correction is expanded out to order α W to study the real-virtual cancellation.
The gg → qq total cross-section has a t-channel singularity for forward scattering, and a u-channel singularity for backward scattering, from the graphs in Fig. 5. To avoid these singularities, we impose rapidity cuts. We require the particle with highest transverse momentum to have η > 1 or η > 3. We will refer to these as η = 1, 3 cuts, respectively. We also require that the particle with second highest p T satisfy η > 5. These cuts allow for collinear and soft W emission from energetic quarks, but avoid the forward and backward singularities. They are applied to both the gg → qq and gg → qqW rates.
The scattering cross section can depend on the collision energy s = E 2 CM , the rapidity cut η, and the particle masses {M }. If the cross section is infrared finite as {M } → 0, then it cannot contain ln s/M 2 terms. The Sudakov logarithms are a sign that the cross section is divergent in the massless limit. In the gg → qq case, the real and virtual corrections have Sudakov logarithms which cancel in the total rate.
We study the gg → qq, qqW rates for three cases: GeV and m t = 173 GeV.
Case (1) allows us to explain the structure of the gauge corrections without worrying about mass effects and Higgs corrections. Case (2) also involves Higgs radiative corrections, but has a stable t quark since m t < m b +m W . Finally case (3) is the physical case with an unstable t, which can decay via t → bW decay. The virtual corrections can be computed from the results in Refs. [5] (including also the y b terms), and are obtained by averaging the electroweak corrections for leftand right-handed quarks. The virtual corrections to the cross sections are where σ 0,t = σ(gg → tt), and σ 0,b = σ(gg → bb) are the corresponding tree-level rates, C F = 3/4 for SU (2), and y t,b are the quark Yukawa couplings. The corrections for u, d quarks are given by y t,b → 0. The tree-level cross section σ 0 depends on the η cut. The virtual rates depend on the η cut in the same way as the tree-level rates. The reason is that the virtual electroweak corrections for gg → qq do not depend on the kinematic variables (such as the scattering angle) in this case, so the radiative correction is an overall multiplicative factor. In other cases, such as qq → qq, the virtual electroweak corrections depend on kinematic variables, and have to be integrated over phase space. The gauge radiative corrections have both L 2 and L terms, whereas the Higgs radiative corrections are linear in L.

A. u, d Quark Production
The tree-level processes are gg → uū and gg → dd, and the real radiation processes are gg → uūZ, gg → ddZ, gg → udW − and gg → dūW + . Since we are working in an SU (2) W theory (with Z = W 3 ), custodial SU (2) implies that the σ(uū) = σ(dd), and σ(udW − ) = σ(dūW + ) = 2σ(uūZ) = 2σ(ddZ). Figure 6 shows the real and virtual corrections to the uū, dd production rate, as a function of E CM , for η = 1, 3 cuts. All rates have been normalized by dividing by the tree-level gg → uū rate for the corresponding η cut. This removes the overall 1/s dependence of the cross sections. The graph clearly shows that the virtual and real cross sections become large at high energy, and the L 2 dependence is reflected in the quadratic shape of the curves. The virtual correction is independent of the η cut, and as is typical of Sudakov effects, is negative. The real correction depends on the η cut. The L 2 , L corrections arise from soft and collinear radiation; the real radiation kinematics for the final state quarks in gg → qqW is similar to that for the tree-level gg → qq process. As a result, the L 2 , L terms do not depend on the η cut, and only the constant L 0 term does. This is reflected in the figure by the fact that the difference in cross sections between the two values of the η cut remains constant as E CM is changed. The L 2 , L terms cancel in the total cross section, as is evident by the curves for the total rate becoming horizontal for large energy, and only the constant terms survive. The electroweak corrections to the total cross section are at the 10% level. At partonic center-of-mass energies of about one TeV, the individual corrections from the real and virtual corrections are also at the 10% level, but they rise quickly as E CM is increased.
For a 100 TeV machine, partonic center-of-mass energies can exceed 10 TeV, and the corrections become large (factors of 2). For most experimentally relevant processes there is never a complete cancellation of the logarithms (since one is typically not measuring a totally inclusive rate, and furthermore the initial state is not an SU (2) singlet), the resummed expressions are needed.
The cancellation between real and virtual corrections is using the isospin relations mentioned earlier and Eqs. (12,13), where → 0 means that the L 2 , L dependence cancels, but there can be constant terms left over.
It is important to note that for initial states that are not electroweak singlets, such as for qq → qq, the real and virtual corrections have different L 2 , L dependence, We now consider the case of gg → tt, bb for m t = 173 GeV and m b = 100 GeV. An unphysical b mass has been chosen, so that the t → bW decay is forbidden. The case of unstable top is discussed in Sec. IV C. The virtual corrections for tt and bb production are given in Eq. (12). The real rates are computed using Mad-Graph5 aMC@NLO. All rates are divided by the corresponding gg → uū rate to remove an overall 1/s normalization factor. The tree-level rates gg → tt and gg → bb are essentially equal to gg → uū except very close to tt threshold, so each of these tree-level rates are 1 in the normalization of the plot, and have not been shown.
The real and virtual corrections are shown in Fig. 7 for the η = 1 cut. The η = 3 plots are very similar, with a small offset from the η = 1 curves, as for the u, d case in Fig. 6. The tbW − emission rate is the sum of the rates for transversely and longitudinally polarized gauge bosons. The rate for transversely polarized gauge bosons at high energies is the same as that for udW − production, since fermion mass effects are power suppressed. The rate for longitudinally polarized gauge bosons is the same as for emission of the unphysical scalar (by the equivalence theorem), and is related to the Higgs emission rate. The real and virtual rates can be written in terms of the udW − rate and the rate σ S to emit a scalar with unit Yukawa coupling, The σ(udW − ) terms in σ(tbW − ), etc. are for transverse W and Z emission and the σ S terms are for longitudinal W and Z emission. 2 One can verify that the real emission curves in Fig. 7 satisfy Eq. (15), so that five curves are given in terms of two quantities, σ(udW − ) determined already in Sec. IV A, and σ S . The Higgs emission curves σ(ttH), σ(bbH) are linear, which means they contain L terms but no L 2 terms. The sum of all the real radiation rates, as well as the total cross section, are shown in Fig. 8 section levels out at high energy (we have verified this by continuing the plot to even higher center of mass energies), which shows numerically that the L 2 and L terms cancel between the real and virtual corrections. The total real emission rate is and the total virtual rate is The cancellation σ R + σ V → 0 implies that The gauge and Higgs parts cancel separately. The gauge part cancels using Eq. (14), and From Eq. (13), we see that v t,b are linear in L, which explains the linearity of the Higgs emission cross section σ S .
Finally, we study the case of a physical b quark with m b = 4.7 GeV and an unstable t quark. The virtual corrections are still given by Eq. (12). There is, however, an important change in the tbW − decay rate because the process gg → tt followed byt →bW − contributes to this rate. The tbW − differential decay rate has a singularity when (pb + p W − ) 2 = m 2 t , and the cross section diverges when integrated over final state phase space. The standard way to resolve this singularity is to regulate it by the t-quark width using the replacement (the narrow width approximation, which is what is used in MadGraph5 aMC@NLO) for the t-quark propagator, where Γ t is the t-quark width. This is equivalent to summing a class of diagrams, the imaginary parts of W corrections to the t-quark propagator, shown in Fig. 11. This is not gauge invariant, and also formally mixes different orders in the α W expansion, since the t-quark width is O(α W m t ). The cut in the second graph of Fig. 1 is the same cut as occurs in summing the imaginary parts of Fig. 11, and the two cuts cannot be treated separately, as is done in the narrow width approximation.
If the t → bW − decay is kinematically forbidden, the tbW − real emission rate is order α W . When the decay is kinematically allowed, the tbW − rate becomes order 1. The reason is that in the resonance region, the rate is enhanced by a factor of 1/Γ t . The total tbW − rate includes what, in the kinematically forbidden case, is the O(1) tt rate. Once the tbW − decay is kinematically allowed, the approximation Eq. (20), while getting the correct O(1) rate, does not get the correct O(α W ) piece.
To understand how the infrared divergence cancellation occurs for an unstable t quark, consider the simpler case of tt production by a current J, as in Sec. III. The α W correction to the total rate can be computed from the imaginary part of the vacuum polarization graphs in Fig. 1. The vacuum polarization Π(q 2 ) has no singularities for Euclidean q 2 even if m t > m b + m W , so the analytic continuation to timelike q 2 does not either. The imaginary part for timelike q 2 is given by the real emis- sion and virtual correction cuts shown in Fig. 1, so the two contributions combined have no infrared divergence. The graphs in Fig. 1 are all order α W , and their total gives the O(α W ) correction to the total rate. The graphs are computed with the t-quark propagator on the l.h.s. of Eq. (20), rather than the narrow width approximation on the r.h.s. The real emission graph is singular because the t → bW − decay is kinematically allowed. A careful calculation shows that the virtual correction is also singular, and the sum is finite. The cancellation can be checked using the l.h.s. of Eq. (20) with the iǫ term acting as a regulator. The real and virtual graphs each have a piece proportional to 1/ǫ, which cancels in the sum.
The tbW − rate can be computed by adding the rates for two regions: A, which is a small region around where the t-quark is on-shell, and A ′ , which is the rest of phase space. In terms of the final state phase space variables m 2 bW = (p b + p W ) 2 , m 2 tW = (p t + p W ) 2 needed for threebody decay, A is the region m 2 t − ∆ ≤ m 2 bW < m 2 t + ∆, and A ′ is the remaining region. The phase space region is shown in Fig. 12, with A the region within the vertical band, and A ′ outside. For a stable t-quark, the vertical band moves outside the allowed phase space region, and there is no singularity in the phase space integral. For an unstable t-quark, the rate is non-singular in region A ′ , and can be computed by the propagator on the l.h.s. of Eq. (20). To correctly compute the O(α W ) terms, one must also use the propagator on the l.h.s. of Eq. (20), rather the the narrow width approximation on the r.h.s., for the integral over the singular region A.
The region A contribution has a singular 1/ǫ piece that must be subtracted, keeping only the finite O(α W ) part.
The 1/ǫ singular part of the rate becomes the O(1) contribution in the narrow width approximation, and the subleading O(α W ) is, unfortunately, not given correctly by the narrow width approximation.
The phase space integral of the decay distribution over A has the form where the denominator is from the absolute value squared of the propagator in Eq. (20), f contains all non-singular factors in the decay distribution, and ∆ is the width of the integration region. Expanding around m 2 t , gives The first term is the singular 1/ǫ piece that must be subtracted, and the remaining terms are the finite O(α W ) terms. Relative to the contribution from region A ′ , they are smaller by a factor ∆, i.e. the width of the vertical band relative to the width of the full phase space region.
Since we are only interested in the O(α W ) contribution to the rate, we can get a good estimate of this by simply using the contribution from region A ′ , and ignoring A.
The O(α W ) term from A is a small correction, since the size of A is much smaller than A ′ . A practical way to do this in MadGraph5 aMC@NLO is to use the $t tag, which excludes a region of width 15Γ t around the on-shell t-quark.
The results of this computation are shown in Figs. (9,10), and are very similar to those for m b = 100 GeV. The main difference is the Yukawa correction is smaller, since y b is now almost zero. The entire discussion of Sec. IV B holds, and will not be repeated again.

V. DISCUSSION AND CONCLUSIONS
We have presented the electroweak radiative corrections to gg → tt, gg → bb production in Sec. IV. The individual processes that contribute have large electroweak corrections that depend on L 2 and L, but these cancel in the total rate. The virtual corrections are around −10% for E CM ∼ 2 TeV, and grow with energy.
The electroweak corrections to the individual processes are relevant for measurements at the LHC. For example, suppose one is interested in measuring the tt production rate. The virtual corrections to tt contribute to this rate. If one has a perfect detector, then one can exclude the real emission final states tbW , ttZ, bbZ, ttH, bbH. In this case, the cross section is given by the blue dots in Fig. 9, and there are large electroweak radiative corrections. In a more realistic case, there will be some leakage from the real radiation processes into the tt channel. For example ttZ with Z → νν could be mistaken for tt, or ttZ with Z → qq, where the Z decay products cannot be separated from the t-quark decay jets. If some fraction of the real radiation is included, then there will be some cancellation with the electroweak corrections to the virtual rate, so that the overall electroweak correction is somewhat smaller. A realistic calculation of the measured rates is beyond the scope of this work. To do such a calculation requires taking the corrections discussed in this paper, integrating over the gluon PDFs, and then putting the parton processes through a showering algorithm and detector cuts. In addition, one should also include the quark production rates qq → tt, which were included in the analysis of Ref. [37]. As noted earlier, the electroweak corrections to qq → tt do not cancel even for the totally inclusive rate. It should be clear that even in a complete calculation, the electroweak corrections do not cancel, and a significant correction remains.
The electroweak radiative corrections start to become measurable at LHC energies, and their importance grows with energy. We have numerically studied the gg → tt process in this paper. Most processes have much larger electroweak corrections than this process, because they typically contain more particles with electroweak interactions. (The gluon does not have electroweak interactions at leading order.) The corrections for qq → tt are approximately twice as large, because the initial and final states both have electroweak interactions. Processes such as qq → W W which involve electroweak gauge bosons have even larger corrections, since the group theory factor C F = 3/4 is replaced by C A = 2 in the amplitude.
The effective theory method breaks the electroweak correction into the high-scale matching C, the running γ and the low-scale matching D. The L 2 term arise from γ, and the L terms from γ and D. All terms are known to NLL order, as are the most important terms at NNLL order (see appendix).
In addition to the electroweak corrections, there are of course, QCD corrections, which are much larger, and have been included in existing calculations and implemented in Monte Carlo code. The QCD and electroweak corrections factor in A and D L to two-loop order and in B, D 0 and C to one-loop order [4,13], so that the total radiative correction to NLL order can be written as the product R QCD R EW . R QCD has been included in existing calculations, so the electroweak corrections can be included to NLL order simply by reweighing the QCD results by R EW . This has to be done before integrating over the final state phase space, since R EW can depend on kinematic variables such as scattering angles. One complication is that R EW depends on the helicities of the partons, since the weak interactions are chiral.
The experimental energy reach at the LHC is high enough that electroweak corrections should be included in measurements that are approaching 10% accuracy. Recently, there have been studies of a possible 100 TeV hadron collider. At these high energies, the electroweak corrections are large, and must be resummed to have reliable cross sections. theory. A single gauge invariant operator breaks up into different components because the weak interaction symmetry is broken. For example, each of the operators O i in Eq. (A1) breaks up into a SU (3) invariant gg → tt and gg → bb operator. 4. The operators in the theory below µ l are then used to compute the scattering cross sections.
The final result is that the scattering amplitudes M can be written as Eq. (A2) gives the scattering amplitude in resummed form. Explicit formulae for all the pieces can be found in Ref. [5]. The high-scale matching C(µ h , L Q ) is a n dimensional column vector with a perturbative expansion in α i (µ h ), with i = 1, 2, 3 being the U (1), SU (2) and SU (3) couplings. It also depends on L Q = ln s/µ 2 h , which is not a large logarithm if one picks µ 2 h ∼ s. For Eq. (A1), n = 3 since there are 3 gauge invariant amplitudes.
The SCET anomalous dimension γ(µ) is an n × n anomalous dimension matrix which can be written as the sum of a collinear and soft part where the collinear part is diagonal and linear in logn r · p r = E r , the energy of the parton, to all orders in perturbation theory [4,35]. The sum on r is over all partons in the scattering process, and A r (µ) and B r (µ) have a perturbative expansion in α i (µ). γ S at one-loop order is where the sum is over all parton pairs rs , and n r = (1, n r ) is a null vector in the direction of parton r for each incoming parton, and n r = −(1, n r ) for each outgoing parton. T (i) r is the gauge generator for the ı th gauge group acting on parton r.
The low-scale matching has a collinear part D C and a soft part d S . The soft part d S is an m×n matrix, where m is the number of amplitudes produced after SU (2)×U (1) breaking. In gg → qq, if q is an electroweak doublet of left-handed quarks (t, b) L , then starting with the operators in Eq. (A1) gives m = 6 operators after SU (2)×U (1) breaking, where q 4 q 3 → t 4 t 3 , or q 4 q 3 → b 4 b 3 . If q in Eq. (A1) is an electroweak singlet, such as b R or t R , then m = 3. and J r and H r are functions of α S,W,EM (µ l ), and can depend on electroweak scale masses and µ l via dimensionless ratios such as M W /M Z and L M = ln M Z /µ l . The sum on r is over all particles in operator O i produced after electroweak symmetry breaking, and D C is linear in lnn · p to all orders in perturbation theory [4,35]. The exponent contains at most a double-log given by integrating the A i terms in the collinear anomalous dimension. The low-scale matching contains a single-log term. This a new feature of SCET EW first pointed out in Ref. [11]. One can show that the low-scale matching contains at most a single-log to all orders in perturbation theory [5,11]. As a consequence, resummed perturbation theory remains valid even at high energy, because α n ln s/M 2 W ≪ 1 for large enough n. A i , γ S , and J i are related to the cusp anomalous dimension.
The log term in the matching Eq. (A6) is needed for proper factorization of scales. A typical Sudakov doublelog term at one loop has the form (dropping the overall α) The first term is the high-scale matching C, the second term arises from integrating the ln Q 2 /µ 2 anomalous dimension from µ h to µ l , and the third term is the low-scale matching D. The existence of the log term in the matching also follows from the consistency condition that the theory is independent of µ l . Since changes in the running between µ h and µ l contain a single log from the anomalous dimension, there must be a single log in the matching. What is non-trivial is that Eq. (A2) only requires a single-log in the matching to all orders in perturbation theory [5,11]. The resummed electroweak corrections can be grouped as LL, NLL, etc. in the usual way, and the precise definition for SCET EW can be found in Ref. [4]. All terms needed for a NLL computation are known, so all processes can be computed to resummed NLL order. Refs. [5,6] computed the one-loop d S and C terms, for all 2 → 2 processes.
The three-loop cusp anomalous dimension A and twoloop non-cusp anomalous B are known, except for the scalar Higgs contributions, which are numerically small. The two-loop contribution to D C is not known. The