Subtraction of the $t\bar{t}$ contribution in $tW\bar{b}$ production at the one-loop level

The leading order (LO) squared matrix element for the partonic process $A(p_{1})+B(p_{2})\to\bar{b}(p_{3})+W(p_{4})+t(p_{5})$ can be expressed as

where $\mathcal{M}^{(0)}_{1t}$ and $\mathcal{M}_{2t}^{(0)}$ denote the LO amplitudes for the single and double top resonant diagrams, respectively. Typical Feynman diagrams are shown in figure 3. The double resonant contribution should be considered as top quark pair production and decay, which has been calculated separately. Therefore, it needs to be subtracted in calculating the cross section of $tW$ production.

The most simple strategy is to remove the contribution from the diagrams of $\mathcal{M}^{(0)}_{2t}$ in eq. (1) Frixione:2008yi ,

A variant of this strategy has also been proposed Hollik:2012rc ,

Note that the above interference term may give a negative contribution. This kind of strategy, called diagram removal (DR), is easy to implement and thus has widely been used in experimental analyses. However, it breaks gauge invariance Frixione:2008yi .

One can also keep all the contributions in eq. (1) intact, but construct a subtraction term to cancel the double resonant contribution,

The subtraction term $\mathcal{R}_{\text{LO}}$ must have the same value as the double resonant diagrams near the on-shell region, which is defined by the limit $\Delta(\equiv s_{W\bar{b}}-m_{t}^{2}\equiv(p_{3}+p_{4})^{2}-m_{t}^{2})\to 0$, and decline fast enough in the off-shell regions. Schematically, we can write

where the subtraction kernel $R_{\text{LO}}$ has the same behavior as the amplitude squared of top quark pair production while the $S$ factor equals one in the on-shell region but suppresses the $R_{\text{LO}}$ contribution away from the on-shell region. The $S$ factor depends on the original momenta of the initial- and final-state particles, denoted by $\{p_{i}\}$, and the reshuffled ones, labeled by $\{\tilde{p}_{i}\}$. The reshuffled momenta are used in $R_{\text{LO}}$ so that the on-shell condition $(\tilde{p}_{3}+\tilde{p}_{4})^{2}=m_{t}^{2}$ and gauge invariance are preserved. We have performed the momentum reshuffling in the same way as in MC@NLO Frixione:2008yi and POWHEG-BOX Re:2010bp ; see appendix A for details.

A natural choice of the subtraction kernel is Frixione:2008yi

where the tilde means that the quantity is calculated with the reshuffled momenta. This subtraction scheme is called diagram subtraction (DS). In the limit of $\Delta\to 0$, the two terms in eq. (4) would cancel with each other. However, they are both divergent in this limit, leading to numerical instability. It is convenient to add an imaginary part in the denominator of the resonant propagator to regulate such divergences. The magnitude of this imaginary part is irrelevant as long as it is much smaller than $\Delta$ in the off-shell regions. We choose it to be $im_{t}\Gamma_{t}$ as indicated by the Breit-Wigner (BW) distribution Demartin:2016axk , where the top quark decay width $\Gamma_{t}$ has been calculated up to NNNLO in the SM Chen:2023dsi ; Chen:2023osm .

The suppression factor $S$ can be chosen as the ratio of the BW distributions before and after momentum reshuffling Frixione:2008yi ,

The above choice satisfies the requirement that $S_{1}\to 1$ as $\Delta\to 0$ and $S_{1}\to 0$ as $\Delta\gg m_{t}\,\Gamma_{t}$.

Therefore, we can express the LO squared amplitude with the subtraction term in the DS scheme as

where the subscript “Reg” denotes that we have added $im_{t}\Gamma_{t}$ in the anti-top quark propagator. In the second line of eq. (8), we have written explicitly the denominator of the resonant propagator with the coefficient $\widetilde{N}$. The third line is obtained by applying $\widetilde{\Delta}=0$.

The above methods can not be extended to higher-loop levels because one can not separate the double resonant diagrams ($\mathcal{M}_{2t}$) from the single resonant ones ($\mathcal{M}_{1t}$). It is possible that a single loop diagram contains contributions from both the double resonant and the single resonant channels. For example, the one-loop diagram in figure 4 can be considered as a virtual correction to the LO process shown in figure 3 (a) or figure 3 (b). As a result, it is not feasible to remove solely the double resonant diagrams, and the subtraction term defined in eq. (6) can not be well defined at loop levels.

In this work, we propose a new subtraction scheme, dubbed power subtraction (PS), which can be generalized to loop diagrams. We note that the squared amplitude can be power expanded around $\Delta=0$ as

In all the coefficients $B^{(i)}$, the relation $s_{W\bar{b}}=m_{t}^{2}+\Delta$ has been used. The first term $\frac{B^{(2)}}{\Delta^{2}}$ is not equal to $|\mathcal{M}^{(0)}_{2t}|^{2}$ because the latter contains also subleading $\Delta$ terms. We introduce the following subtraction kernel

where $\widetilde{B^{(2)}}$ is obtained from $B^{(2)}$ after momentum reshuffling. The denominator $\widetilde{\Delta}$ is vanishing in principle. However, we use it here as a regulator, which will cancel after combination with the suppression factor. We do not add an imaginary part in the propagator to avoid numerical instability as above because this would change the infra-red divergence structure of the loop diagrams for on-shell top quark production. Instead, we define a new suppression factor as the ratio of the denominators of the resonant propagator before and after momentum reshuffling,

The numerator of $S_{2}$ cancels the pole of $\tilde{\Delta}$ in $(R_{\text{LO}})_{\text{2}}$. Then the subtracted LO squared amplitude is

The two terms will cancel with each other in the limit $\Delta\to 0$. To ensure numerical stability, we set a cut $|\sqrt{s_{W\bar{b}}}-m_{t}|/m_{t}>\delta$ in phase space integration. The final cross section should not depend on this cut parameter $\delta$ if it is chosen small enough. Note that $\widetilde{B^{(2)}}$ coincides with the coefficient $\tilde{N}$ in eq. (8). Because the power expansion can be performed for all the results of loop diagrams, this subtraction scheme can be applied beyond the tree level. The gauge invariance is clearly preserved in this scheme.

When extending the previous scheme to one-loop level, we first expand the squared amplitude around $\Delta=0$,

where we have considered the interference between the one-loop and the tree diagrams. The first term is what we want to subtract. We have used the renormalized one-loop amplitude $\mathcal{M}^{(1)}_{\text{Ren}}$, and thus all the coefficients $C^{(i)}$ contain only infra-red divergences.

In a standard NLO calculation, the infra-red divergences of one-loop virtual corrections are canceled with a proper counter-term, such as that in the dipole subtraction formalism Catani:1996vz ; Catani:2002hc , i.e., the sum

is finite. Here the counter-term can be written as

where the singular factor $\mathbf{I}$ is factorized from the LO squared amplitude and depends only on the kinematics and color structure of the external particles. Expanding this counter-term gives

where $I^{(i)}$ are the expansion coefficients. It is easy to find that each sum of $C^{(i)}+I^{(i)},i=2,1,0$ is finite, i.e., free of $1/\epsilon$ poles. Now our task is to construct a resonance subtraction term for the combination in eq. (14). To ensure gauge invariance, we need to perform momentum reshuffling, and thus derive the resonance subtraction kernel

with

The coefficient $C^{(2)}$ receives contributions from the loop diagrams with an anti-top quark propagator; see figure 8 for sample diagrams. In figure 8(a), the anti-top quark is not involved in the loop integral and therefore the result can be Taylor expanded around $\Delta=0$, as at the tree level. However, the anti-top quark propagator exists inside a loop integral in figure 8(b) and figure 8(c). As a consequence, the result of the squared amplitude contains $\log\Delta$ terms¹¹1The double $\log^{2}\Delta$ terms may appear in a single diagram but cancel in the sum of all one-loop diagrams., and we can express

Some of the logarithmic terms serve as a regulator for the soft divergence in the diagrams with an on-shell anti-top quark as an external leg, such as the diagram shown in figure 8(b). The correspondence between these logarithms and the dimensional divergences is explained in detail in appendix B. Then the $\widetilde{C^{(2)}}$ term in eq. (17) is defined as

Adopting the suppression factor in eq. (11), we obtain the squared amplitude with both the infra-red and resonant divergences subtracted,

The LO diagrams of the $d\bar{d}\to\bar{b}Wt$ process are shown in figure 3. The left diagram in figure 3 contains a resonant anti-top quark which decays into $\bar{b}W$. Its contribution has to be removed in the DR scheme or is used to construct a subtraction term in the DS scheme. In the PS scheme, the subtraction term is given by

where $C_{A}=3$ and $C_{F}=4/3$ in QCD. The Mandelstam variables are defined by

The sign $\sigma_{ij}=1$ if the $i$- and $j$-th particles are both initial or both final states; and $\sigma_{ij}=-1$ in the other cases. After phase space integration and convolution with the PDFs, the results of the cross section in different schemes are presented in table 1. Note that, in the PS scheme, we have imposed a cut, $|\sqrt{s_{W\bar{b}}}-m_{t}|/m_{t}>\delta$, for numerical stability when implementing the Monte Carlo integration, since large number cancellation appears near the resonant region. The value of $\delta$ does not affect the numerical result significantly as long as $\delta\ll 1$. We have checked that in the interval for $\delta$ between $10^{-2}$ and $10^{-6}$ the integration result is stable. For the cross section in the PS scheme in table 1, we have used $\delta=0.006$.

It can be seen that the results in the DR1, DS, and PS schemes are similar in size. A negative result is obtained in the DR2 scheme because of the contribution from the interference term. This may be not a problem if one considers this process as a real correction to the $tW$ production, in which much larger positive corrections come from the virtual corrections and the real corrections without $\bar{b}Wt$ final states. However, given that the $d\bar{d}\to\bar{b}Wt$ process does not require subtraction of infra-red divergences, it is more reasonable to have a positive value for its contribution.

To see more clearly the subtraction pattern, we show the invariant mass distribution for the $W\bar{b}$ system in figure 9. In the bin including the resonance, a cancellation of five digits occurs, illustrating the validity of the subtraction term. The cross sections in the two adjoining bins drop by an order of magnitude and the cancellation between the cross section and the subtraction term is still obvious. The differential cross sections in the other bins are even smaller and the subtraction term plays a less significant role. From the lower panel of figure 9, we see that the subtracted cross section still contains a single pole of $\Delta$. The contributions on the two sides of the resonance peak are almost the same but with opposite signs. The eleven bins around the resonance peak, i.e., $117.5{\rm~{}GeV}<m_{W\bar{b}}<227.5$ GeV contribute less than 1% of total cross section. Therefore, these resonant contributions to the differential distributions other than $m_{W\bar{b}}$ would be very small.

The one-loop diagrams can be separated into three types as shown in figure 8. The amplitudes of the first two types of diagrams depend on four-point (or less point) scalar integrals, of which the analytic results can be found in the literature Ellis:2007qk ; Patel:2016fam . Therefore it is straightforward to expand the results in a series of $\Delta$. In order to obtain the analytic result of the pentagon integral, denoted by $I_{5}$, in the Feynman diagram of the third type (figure 8(c)), we need to firstly reduce it to a linear combination of five box integrals Bern:1993kr ,

where the coefficients $c_{i}$ depend on the kinematic variables and masses, and the box integrals $I_{4}^{(i)}$ can be obtained after removing one of the propagators in the pentagon integral. Then one can power expand these box integrals analytically. We find that only the following integral contains $\mathcal{O}(1/\Delta)$ contribution,

with $D=4-2\epsilon$.

Actually, the expansion of the amplitude in $\Delta$ can be described systematically by using the effective field theory of unstable particles Beneke:2003xh . The idea is that the anti-top quark in a loop diagram can only have soft momentum fluctuation near the resonant region. As a result, the momentum of the gluon connecting the two partons in the production and decay processes must be soft. We can extract the leading power contribution just by making use of the eikonal approximation of the gluon interaction. In other words, we consider such gluons arising from the soft Wilson lines associating with the corresponding partons. Such soft loop integrals are usually vanishing in QCD amplitudes because they are scaleless. But they are non-vanishing in our case because the anti-top quark propagator, which introduces a scale of the offshellness, should also be taken into account. As an example, when calculating the Feynman diagram in figure 8(c), we only need to evaluate the following soft loop integral

Note that the soft gluon momentum $l^{\mu}$ is of the order $\lambda^{2}\sim\Delta/m_{t}^{2}$. This kind of integrals is general for the soft gluon correction in Feynman diagrams with a resonant propagator. Its analytical result, along with other soft loop integrals, is shown in appendix C.

Making use of this method, we compute the leading power expansion of the one-loop amplitude and obtain

where we have only written the divergent parts explicitly. We see that the divergent part is proportional to the LO analogue $B^{(2)}$ and does not contain any $\log\Delta$ terms²²2It is worth noting that there may appear the logarithmic term $\log\Delta$ in a single diagram.. This feature is expected because the sum $C^{(2)}+I^{(2)}$ should be finite and $I^{(2)}$ does not depend on $\Delta$. In real calculation, it results from color conservation and the structure of soft-collinear divergences. The relevant contribution can be written as

where ${\bf T}_{i}$’s are color operators Catani:1996vz and we have used color conservation ${\bf T}_{\bar{t},f}={\bf T}_{3}=-{\bf T}_{\bar{t},i}$ and ${\bf T}_{1}+{\bf T}_{2}+{\bf T}_{5}+{\bf T}_{\bar{t},f}=0$ in the second line. The coefficient of the color operators in each term is determined by the soft-collinear divergence associating with the corresponding external states; see appendix C for the results of soft loop integrals.

The ellipsis in eq. (3.2) denotes the finite piece. The complete form is too lengthy to be shown here. The $\log\Delta$ terms in the finite piece are also proportional to the LO $B^{(2)}$ and can be expressed by

where we have introduced $p_{\bar{t}}=p_{3}+p_{4}$ and $\beta_{t}=\sqrt{1-4m_{t}^{2}/s_{12}}$ is the velocity of the top or anti-top quark in the center-of-mass frame in the limit $\Delta\to 0$. The above expression in the bracket is equal to the difference between the infra-red divergences of the matrix elements for $d\bar{d}\to\bar{b}Wt$ and $d\bar{d}\to\bar{t}t$ with $\bar{t}\to\bar{b}W$ normalized by their respective Born results and $\alpha_{s}/2\pi\epsilon$. To be specific, the infra-red divergence for $d\bar{d}\to\bar{b}Wt$ with an off-shell anti-top propagator can be written schematically as

while the infra-red structure for $d\bar{d}\to\bar{t}t$ with on-shell anti-top decaying $\bar{t}\to\bar{b}W$ is

In the limit $\Delta\to 0$, the two results should coincide, which explains the same coefficients $d_{2}$ and $d_{1}$ in the above two equations. With small but non-vanishing $\Delta$, expansion in $\epsilon$ as shown in the second equation of (30) gives rise to the $\log(|\Delta|)$ terms in the finite piece in (3.2). Because of this relation, the gauge invariance of $\widetilde{C^{(2)}}$ is also obvious.

We have imposed a phase space cut $|\sqrt{s_{W\bar{b}}}-m_{t}|/m_{t}>\delta$ in the Monte Carlo integration. The one-loop corrections with resonance contribution subtracted for different values of $\delta$ are shown in figure 10. We find that the results become insensitive to $\delta$ when it becomes smaller than 0.1. It is ideal that one can take the limit of $\delta\to 0$. However, this is not possible in practice because the numerical uncertainties due to Monte Carlo integration grow with decreasing $\delta$. Given that the central values almost remain the same, it is reasonable to choose the result at $\delta=0.006$. The LO cross section and the one-loop correction are shown in table 2. We see that the virtual correction decreases the LO result by 41%.

The invariant mass distribution for the $W\bar{b}$ system is shown in figure 11. A large cancellation can be observed near the onshell region. The resonance subtraction term drops fast away from the onshell region, as in the LO. The two bins adjacent to the resonance peak provide opposite contributions, illustrating the remaining $1/\Delta$ structure in the squared amplitude. The contributions from the nine bins around the resonance peak amount to only 10% of the total cross section, indicating a good cancellation effect of the resonance contributions in the PS scheme at the one-loop level.