Effective-Field-Theory Approach to
Top-Quark Production and Decay

The standard model (SM) of the strong and electroweak interactions has been very successful phenomenologically. However, there are reasons to believe that the SM is not a complete description of nature. It contains many arbitrary parameters with no connections, and provides no explanation for the symmetry-breaking mechanism that gives rise to the masses of gauge bosons and fermions. In addition, the existence of dark matter and dark energy are not accommodated by the SM. It also provides no explanation of the strong CP problem. Consequently, there could be further particles and interactions as one probes higher energy scales. These energy scales could be the Planck scale or an intermediate scale $\Lambda$.

A complete description of any new physics beyond the SM requires a fundamental theory. There are many models proposed for new physics. For example, the new particles could be those of supersymmetric theories or fermions that transform under different representations of the SM gauge group. The new interactions may also originate from quark substructure or preon exchange. Because of the diversity of these models, it is useful to introduce a model-independent approach.

If the physics beyond the SM lies at an energy scale $\Lambda$ less than 1 TeV, then we should be able to observe it directly at high-energy colliders. If it lies at a scale much greater than 1 TeV, then we can parameterize its effects via higher-dimension operators, suppressed by inverse powers of the scale $\Lambda$. If the new physics is too heavy to appear directly in low energy processes, then we can integrate it out from the Lagrangian. In this way, the effective Lagrangian is:

$$L_{\mathit{eff}}=L_{0}+\frac{1}{\Lambda}L_{1}+\frac{1}{\Lambda^{2}}L_{2}+\cdots$$

(1)

where $L_{0}$ is the SM Lagrangian of dimension four, $L_{1}$ is the new interaction of dimension five, $L_{2}$ is of dimension six, etc. The procedure is quite general and independent of the new interactions at scale $\Lambda$. The only constraint is that all $L_{i}$ are ${\rm SU(3)}_{C}\times{\rm SU(2)}_{L}\times{\rm U(1)}_{Y}$ invariant.

At dimension five, the only operator allowed by gauge invariance is [1]

$$L_{\mathit{eff}}=\frac{c^{ij}}{\Lambda}(L^{iT}\epsilon\phi)C(\phi^{T}\epsilon L^{j})+h.c.$$

(2)

where $L^{i}$ is the lepton doublet field of the $i^{th}$ generation and $\phi$ is the Higgs doublet field. When the Higgs doublet acquires a vacuum-expectation value, this term gives rise to a Majorana mass for neutrinos. Due to the tiny neutrino masses, the scale $\Lambda$ is probably around $10^{15}$ GeV. In contrast to this unique dimension-five operator, there are many independent dimension-six operators [2, 3].

Because the top quark is heavy relative to all the other observed SM fermions, we expect that the new physics at higher energy scales may reveal itself at lower energies through the effective interactions of the top quark, and deviations with respect to the SM predictions might be detectable. In this paper we will study the effect of these dimension-six operators on top quark interactions at hadron colliders. We focus on three different processes: top quark decay, single top production, and top pair production. If no deviation is observed experimentally, then one can place bounds on the coefficients of the dimension-six operators. The effects of non-standard interactions on top-quark physics at linear colliders and photon colliders can be found in Refs. [4, 5, 6, 7].

We choose the effective Lagrangian to realize the weak symmetry linearly, as the precision electroweak data favors a light Higgs boson. The situation where the weak symmetry is realized nonlinearly is studied in Ref. [8, 9, 10, 11, 12]. We will use the operator set introduced by Buchmuller and Wyler [3]. In their paper they categorize all possible gauge-invariant dimension-six operators, and use the equations of motion (EOM) to simplify them into 80 independent operators (for one generation). Subsequently it was found that several of these operators are actually not independent [13, 14, 15]. We focus on the operators that have an influence on the top quark.

We expect the leading modification to SM processes at order $\frac{1}{\Lambda^{2}}$. In this paper we don’t consider higher order contributions. We expect the scale $\Lambda$ to be large (at least larger than the scale we can probe directly) so $\frac{1}{\Lambda^{4}}$ contributions would be small compared to the uncertainty on top quark measurements. Thus we ignore all dimension-eight (and higher) operators, as well as effects involving two dimension-six operators.

For any physical observable, the $\frac{1}{\Lambda^{2}}$ contribution comes from the interference between dimension-six operators and the SM Lagrangian. This contribution might be suppressed for a variety reasons. For example, since all quark and lepton masses are negligible compared to the top quark mass, a new interaction that involves a right-handed quark or lepton (except for the top quark) has a very small interference with the SM charged-current weak interactions, which only involve left-handed fermions. It turns out that although there are a large number of dimension-six operators, only a few of them have significant effects at order $\frac{1}{\Lambda^{2}}$. We list these operators in Tables 1 and 2.

In Table 1, only one of the four-quark operators, $O_{qq}^{(1,3)}=(\bar{q}^{i}\gamma_{\mu}\tau^{I}q^{j})(\bar{q}\gamma^{\mu}\tau^{I}q)$, is listed explicitly. Here the superscripts $i,j$ denote the first two quark generations, while $q$ without superscript denotes the third generation. In single top production, this is the only (independent) four-quark operator that contributes. However, there are many other four-quark operators with different isospin and color structures [2, 3]. In the top pair production process $q\bar{q}\rightarrow t\bar{t}$, seven such operators contribute. The details are discussed in Section 4.

In Table 2, the CP-odd operators are listed. These interactions interfere with the SM only if the spin of the top quark is taken into account. The reason is that the SM conserves CP to a good approximation (the only CP violation is in the CKM matrix), and the inteference between a CP-odd operator and a CP-even operator is a CP violation effect. It was shown in Ref. [16] that, in the absence of final-state interactions, any CP violation observable can assume non-zero value only if it is $T_{N}$-odd, where $T_{N}$ is the “naive” time reversal, which means to apply time reversal without interchanging the initial and final states. Thus an observable is $T_{N}$-odd if it is proportional to a term of the form $\epsilon_{\mu\nu\rho\sigma}v^{\mu}v^{\nu}v^{\rho}v^{\sigma}$. If we don’t consider the top quark spin, $v$ must be the momentum of the particles, and such a term will not be present because the reactions we consider here involve at most three independent momenta. Therefore top polarimetry is essential for the study of CP violation. Since the top quark rapidly undergoes two-body weak decay $t\rightarrow Wb$ with a time much shorter than the time scale necessary to depolarize the spin, information on the top spin can be obtained from its decay products. CP violation will be discussed in Section 5.

There is an argument that can be used to neglect some of the new operators [17]. Some new operators can be generated at tree level from an underlying gauge theory, while others must be generated at loop order. In general the loop generated operators are suppressed by a factor of $1/16\pi^{2}$. However, the underlying theory may not be a weakly coupled gauge theory, or the loop diagrams could be enhanced due to the index of a fermion in a large representation. Furthermore, the underlying theory may not be a gauge theory at all. Fortunately, the effective field theory approach does not depend on the underlying theory. We will consider all dimension-six operators, without making any assumptions about the nature of the underlying theory.

We do not make any assumptions about the flavor structure of the dimension-six operators, although we don’t consider any flavor-changing neutral currents in this paper. The charged-current weak interaction of the top quark is proportional to $V_{tb}$, so the SM rate for top decay and single top production is proportional to $V_{tb}^{2}$. We write all dimension-six operators in terms of mass-eigenstate fields, so no diagonalization of the new interactions is necessary. Hence, in charged-current weak interactions, the interference between the SM amplitude and the new interaction is proportional to $V_{tb}C_{i}$, where $C_{i}$ is the (real) coefficient of the dimension-six Hermitian operator $O_{i}$ (also recall that $V_{tb}$ itself is purely real in the standard parameterization [18]). If the operator is not Hermitian, the coefficient $C_{i}$ is complex; CP-conserving processes are proportional to $V_{tb}{\rm Re}C_{i}$, while CP-violating processes are instead proportional to $V_{tb}{\rm Im}C_{i}$.

Deviations of top-quark processes from SM predictions have often been discussed using a vertex-function approach, where the $Wtb$ vertex is parameterized in terms of four unknown form factors [19]. Given our precision knowledge of the electroweak interaction, this approach is too crude. The effective field theory approach is well motivated; it takes into consideration the unbroken ${\rm SU(3)}_{C}\times{\rm SU(2)}_{L}\times{\rm U(1)}_{Y}$ gauge symmetry; it includes contact interactions as well as vertex corrections; it is valid for both on-shell and off-shell quarks; and it can be used for loop processes [20]. None of these virtues are shared by the vertex function approach [21].

When the fermion masses (except for the top quark) are ignored, there are only two independent dimension-six operators in [3] that contribute to top-quark decay at leading order:¹¹1The operator $O_{Dt}=(\bar{q}D_{\mu}t)D^{\mu}\tilde{\phi}$ listed in Ref. [3] can be removed using the EOM [14].

	$\displaystyle O_{\phi q}^{(3)}$	$\displaystyle=$	$\displaystyle i(\phi^{+}\tau^{I}D_{\mu}\phi)(\bar{q}\gamma^{\mu}\tau^{I}q)$		(3)
	$\displaystyle O_{tW}$	$\displaystyle=$	$\displaystyle(\bar{q}\sigma^{\mu\nu}\tau^{I}t)\tilde{\phi}W^{I}_{\mu\nu}$		(4)

The operators $O_{\phi q}^{(3)}$ and $O_{tW}$ modify the SM $Wtb$ interaction. Upon symmetry breaking, they generate the following terms in the Lagrangian:

	$\displaystyle L_{\mathit{eff}}=\frac{C_{\phi q}^{(3)}}{\Lambda^{2}}\frac{gv^{2}}{\sqrt{2}}\bar{b}\gamma^{\mu}P_{L}tW^{-}_{\mu}+h.c.$		(5)
	$\displaystyle L_{\mathit{eff}}=-2\frac{C_{tW}}{\Lambda^{2}}v\bar{b}\sigma^{\mu\nu}P_{R}t\partial_{\nu}W^{-}_{\mu}+h.c.$		(6)

where $v=246$ GeV is the vacuum expectation value (VEV) of $\phi$. The operator $O_{\phi q}^{(3)}$ simply leads to a rescaling of the SM $Wtb$ vertex by a factor of $(1+\frac{C_{\phi q}^{(3)}v^{2}}{\Lambda^{2}V_{tb}})$, so it does not affect any distributions, and is therefore impossible to detect in angular distributions of top-quark decays. The vertex-function approach to top-quark decay is pursued in Refs. [22, 23].

These operators interfere with the SM amplitude, as is shown in Figure 1. We can compute their correction to the SM amplitude. The $t\rightarrow be^{+}\nu$ squared amplitude is:

$$\frac{1}{2}\Sigma|M|^{2}=\frac{V_{tb}^{2}g^{4}u(m_{t}^{2}-u)}{2(s-m_{W}^{2})^{2}}+\frac{C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\frac{g^{4}u(m_{t}^{2}-u)}{(s-m_{W}^{2})^{2}}+\frac{4\sqrt{2}{\rm Re}C_{tW}V_{tb}m_{t}m_{W}}{\Lambda^{2}}\frac{g^{2}su}{(s-m_{W}^{2})^{2}}$$

(7)

where $C_{i}$ is the coefficient of operator $O_{i}$, and $s,t,u$ are generalizations of the usual Mandelstam variables ($s=(p_{t}-p_{b})^{2},\quad t=(p_{t}-p_{\nu})^{2},\quad u=(p_{t}-p_{e^{+}})^{2}$). $C_{\phi q}^{(3)}$ is real.

Using the narrow width approximation for the $W$ boson, the differential decay rate is

	$\displaystyle\frac{d\Gamma}{d\cos\theta}$	$\displaystyle=$	$\displaystyle\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{4}}{4096\pi^{2}m_{t}^{3}m_{W}\Gamma_{W}}(m_{t}^{2}-m_{W}^{2})^{2}[m_{t}^{2}+m_{W}^{2}+(m_{t}^{2}-m_{W}^{2})\cos\theta](1-\cos\theta)$		(8)
			$\displaystyle+\frac{{\rm Re}C_{tW}V_{tb}g^{2}}{128\sqrt{2}\pi^{2}\Lambda^{2}m_{t}^{2}\Gamma_{W}}m_{W}^{2}(m_{t}^{2}-m_{W}^{2})^{2}(1-\cos\theta)$

Here $\theta$ is the angle between the momenta of top quark and the neutrino in the $W$ rest frame, and $\Gamma_{W}$ is the width of the $W$ boson. In the SM, at tree level $\Gamma_{W}$ is given by:

$$\Gamma_{W}=\frac{3\alpha_{W}}{4}m_{W}\;.$$

(9)

The angular dependence is shown in Figure 2. The curves are normalized to have equal areas. The contribution from $O_{\phi q}^{(3)}$ is the same as the SM contribution, because $O_{\phi q}^{(3)}$ simply rescales the SM $Wtb$ vertex. It therefore does not affect angular distributions. The angular dependence of the contribution from $O_{tW}$ is not dramatically different from the SM.

The partial width is given by

$$\Gamma=\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{4}(m_{t}^{6}-3m_{W}^{4}m_{t}^{2}+2m_{W}^{6})}{3072\pi^{2}\Gamma_{W}m_{t}^{3}m_{W}}+{\rm Re}C_{tW}V_{tb}\frac{g^{2}m_{W}^{2}(m_{t}^{2}-m_{W}^{2})^{2}}{64\sqrt{2}\pi^{2}\Lambda^{2}\Gamma_{W}m_{t}^{2}}\;.$$

(10)

Both dimension-six operators affect the partial width. The total width is given by the above expression times a factor of nine. Unfortunately, it is not known how to measure the partial or total widths in a hadron collider environment.

We also consider the energy dependence of the leptons in the top quark rest frame. The SM computation can be found in [24, 25]. The correction from dimension-six operators at leading order is:

	$\displaystyle\frac{d\Gamma}{dE_{e^{+}}}$	$\displaystyle=$	$\displaystyle\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{4}E_{e^{+}}(m_{t}-2E_{e^{+}})}{128\pi^{2}m_{W}\Gamma_{W}}+\frac{{\rm Re}C_{tW}V_{tb}g^{2}m_{W}^{2}(m_{t}-2E_{e^{+}})}{16\sqrt{2}\pi^{2}\Lambda^{2}\Gamma_{W}}$
	$\displaystyle\frac{d\Gamma}{dE_{\nu}}$	$\displaystyle=$	$\displaystyle\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{4}(-4E_{\nu}^{2}m_{t}^{2}+2E_{\nu}(m_{t}^{3}+2m_{W}^{2}m_{t})-m_{W}^{2}(m_{t}^{2}+m_{W}^{2}))}{256\pi^{2}m_{t}^{2}m_{W}\Gamma_{W}}$		(11)
			$\displaystyle+\frac{{\rm Re}C_{tW}V_{tb}g^{2}m_{W}^{2}(2E_{\nu}m_{t}-m_{W}^{2})}{16\sqrt{2}\pi^{2}\Lambda^{2}m_{t}\Gamma_{W}}$

where $m_{W}^{2}/2m_{t}<E_{e^{+}},E_{\nu}<m_{t}/2$ are the energies of the electron and neutrino, respectively. We don’t list the energy dependence of the bottom quark, because the narrow width approximation for the $W$ boson is used and the energy of the bottom quark is given by $E_{b}=(m_{t}^{2}-m_{W}^{2})/2m_{t}$. These results are shown in Figure 4 and 4. Again the curves are normalized so that the areas are the same. Compared to Figure 2, the two curves are more distinct, which implies the effect of $O_{tW}$ would be more apparent in the energy distribution of the leptons.

The angular distribution and the energy distribution are not independent. The energy of the leptons are fixed in the $W$ rest frame. Therefore their energy in the top quark rest frame is given by a boost, which only depends on the angle $\theta$:

	$\displaystyle E_{\nu}=\frac{1}{2}(E+|q|\cos\theta)$		(12)
	$\displaystyle E_{e^{+}}=\frac{1}{2}(E-|q|\cos\theta)$		(13)

where $E=(m_{t}^{2}+m_{W}^{2})/2m_{t}$ and $|q|=(m_{t}^{2}-m_{W}^{2})/2m_{t}$ are the energy and momentum of the $W$ boson in the top quark rest frame. Furthermore, both the angular distribution and energy distribution can be expressed using the $W$ helicity fractions [22]:

$$\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta}=\frac{3}{8}(1+\cos\theta)^{2}F_{R}+\frac{3}{8}(1-\cos\theta)^{2}F_{L}+\frac{3}{4}\sin^{2}\theta F_{0}$$

(14)

and

$$\frac{1}{\Gamma}\frac{d\Gamma}{dE_{e^{+}}}=\frac{1}{(E_{max}-E_{min})^{3}}\left(3(E_{e^{+}}-E_{min})^{2}F_{R}+3(E_{max}-E_{e^{+}})^{2}F_{L}+6(E_{max}-E_{e^{+}})(E_{e^{+}}-E_{min})F_{0}\right)$$

(15)

where $E_{max}=m_{t}/2$ and $E_{min}=m_{W}^{2}/2m_{t}$, $F_{i}=\Gamma_{i}/\Gamma$ are the $W$ boson helicity fractions, corresponding to positive (R), negative (L), or zero (0) helicity. The helicity fraction is affected by the operator $O_{tW}$:

	$\displaystyle F_{0}$	$\displaystyle=$	$\displaystyle\frac{m_{t}^{2}}{m_{t}^{2}+2m_{W}^{2}}-\frac{4\sqrt{2}{\rm Re}C_{tW}v^{2}}{\Lambda^{2}V_{tb}}\frac{m_{t}m_{W}(m_{t}^{2}-m_{W}^{2})}{(m_{t}^{2}+2m_{W}^{2})^{2}}$
	$\displaystyle F_{L}$	$\displaystyle=$	$\displaystyle\frac{2m_{W}^{2}}{m_{t}^{2}+2m_{W}^{2}}+\frac{4\sqrt{2}{\rm Re}C_{tW}v^{2}}{\Lambda^{2}V_{tb}}\frac{m_{t}m_{W}(m_{t}^{2}-m_{W}^{2})}{(m_{t}^{2}+2m_{W}^{2})^{2}}$
	$\displaystyle F_{R}$	$\displaystyle=$	$\displaystyle 0$		(16)

These equations make manifest the earlier observation that the operator $O_{\phi q}^{(3)}$, which simply rescales the SM vertex, cannot affect any distributions. Thus top-quark decay is sensitive only to the operator $O_{tW}$, and can be used to measure (or bound) its coefficient.

Finally, we investigate the polarized differential decay rate. In the rest frame of the top quark, the angular distribution of any top quark decay product is given by [24, 25]

$$\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta_{i}}=\frac{1+\alpha_{i}\cos\theta_{i}}{2}$$

(17)

where $\theta_{i}=\theta_{b},\theta_{v},\theta_{e^{+}}$ is the angle between the spin axis of the top quark and the momentum of the bottom quark, neutrino or positron. The “analyzing power” $\alpha_{i}$ measures the degree to which the direction of the decay product $i$ is correlated with the top spin. If dimension-six operators are added, the relation still holds, but the coefficient $\alpha_{i}$ will be affected by the new operators. Since $O_{\phi q}^{(3)}$ is just a rescaling of the SM interaction, the only correction is from $O_{tW}$. This could be an independent way to determine the coefficient ${\rm Re}C_{tW}$. At leading order, the correction is given by:

	$\displaystyle\alpha_{b}$	$\displaystyle=$	$\displaystyle-\frac{m_{t}^{2}-2m_{W}^{2}}{m_{t}^{2}+2m_{W}^{2}}+\frac{{\rm Re}C_{tW}v^{2}}{\Lambda^{2}V_{tb}}\frac{8\sqrt{2}m_{t}m_{W}(m_{t}^{2}-m_{W}^{2})}{(m_{t}^{2}+2m_{W}^{2})^{2}}$
	$\displaystyle\alpha_{v}$	$\displaystyle=$	$\displaystyle\frac{m_{t}^{6}-12m_{t}^{4}m_{W}^{2}+3m_{t}^{2}m_{W}^{4}(3+8\ln(m_{t}/m_{W}))+2m_{W}^{6}}{m_{t}^{6}-3m_{t}^{2}m_{W}^{4}+2m_{W}^{6}}$
			$\displaystyle-\frac{{\rm Re}C_{tW}v^{2}}{\Lambda^{2}V_{tb}}\frac{12\sqrt{2}m_{t}m_{W}(m_{t}^{6}-6m_{t}^{4}m_{W}^{2}+3m_{t}^{2}m_{W}^{4}(1+4\ln(m_{t}/m_{W}))+2m_{W}^{6})}{(m_{t}^{2}+2m_{W}^{2})^{2}(m_{t}^{2}-m_{W}^{2})^{2}}$
	$\displaystyle\alpha_{e^{+}}$	$\displaystyle=$	$\displaystyle 1$		(18)

The same equations hold for hadronic top decay, with $\alpha_{u}=\alpha_{\nu}$, $\alpha_{\bar{d}}=\alpha_{e^{+}}$. The coefficient $\alpha_{e^{+}}$ is not affected by dimension-six operators. This is consistent with the results in Ref. [26].

The measurement of these coefficients requires a source of polarized top quarks. This is addressed in the next section.

Single top quarks are produced through the electroweak interaction. There are three separate processes: $s$-channel [27], $t$-channel [28, 29, 30], and $Wt$ production [31]. An effective field theory approach to the $s$- and $t$-channel processes was advocated in Ref. [32]. We update that analysis by including an additional operator, which was neglected in that study because it is loop-suppressed if the underlying theory is a gauge theory. We also perform an effective field theory analysis of the $Wt$ process. The vertex-function approach to single-top production is pursued in Refs. [33, 34, 35].

Single top production contains four distinct channels: the $s$-channel process $u\bar{d}\rightarrow t\bar{b}$, the $t$-channel processes $ub\rightarrow dt$ and $\bar{d}b\rightarrow\bar{u}t$, and the $Wt$ associated production channel $gb\rightarrow Wt$. We first consider the $s$ and $t$ channels. The following operators contribute [32]:

	$\displaystyle O_{\phi q}^{(3)}$	$\displaystyle=$	$\displaystyle i(\phi^{+}\tau^{I}D_{\mu}\phi)(\bar{q}\gamma^{\mu}\tau^{I}q)$		(19)
	$\displaystyle O_{tW}$	$\displaystyle=$	$\displaystyle(\bar{q}\sigma^{\mu\nu}\tau^{I}t)\tilde{\phi}W^{I}_{\mu\nu}$		(20)
	$\displaystyle O_{qq}^{(1,3)}$	$\displaystyle=$	$\displaystyle(\bar{q}^{i}\gamma_{\mu}\tau^{I}q^{j})(\bar{q}\gamma^{\mu}\tau^{I}q)$		(21)

For the four-quark operator $O_{qq}^{(1,3)}$, the superscripts $i,j$ denote the first two quark generations. Another four-quark operator that could contribute is $(\bar{q}^{i}\gamma_{\mu}q)(\bar{q}\gamma^{\mu}q^{j})$. However, using the Fierz identity, this can be turned into a linear combination of $O_{qq}^{(1,3)}$ and some other four-quark operators with different isospin and color structures which do not contribute to this process. Four-quark operators are neglected in the vertex-function approach to the $Wtb$ vertex.

The Feynman diagrams are shown in Figure 5. Since the operator $O_{tW}$ will be measured (or bounded) from studies of top-quark decay, the $s$- and $t$-channel production of single top quarks can be used to measure (or bound) the operators $O_{\phi q}^{(3)}$ and $O_{qq}^{(1,3)}$.

Now we turn to consider the $gb\rightarrow Wt$ process. The contributing operators are

	$\displaystyle O_{\phi q}^{(3)}$	$\displaystyle=$	$\displaystyle i(\phi^{+}\tau^{I}D_{\mu}\phi)(\bar{q}\gamma^{\mu}\tau^{I}q)$		(22)
	$\displaystyle O_{tW}$	$\displaystyle=$	$\displaystyle(\bar{q}\sigma^{\mu\nu}\tau^{I}t)\tilde{\phi}W^{I}_{\mu\nu}$		(23)
	$\displaystyle O_{tG}$	$\displaystyle=$	$\displaystyle(\bar{q}\sigma^{\mu\nu}\lambda^{A}t)\tilde{\phi}G^{A}_{\mu\nu}$		(24)

Again, the first two operators $O_{\phi q}^{(3)}$ and $O_{tW}$ will affect the $Wtb$ coupling. The “chromomagnetic moment” operator $O_{tG}$ modifies the $gtt$ coupling:

$$L_{\mathit{eff}}=\frac{{\rm Re}C_{tG}}{{\sqrt{2}}\Lambda^{2}}v\left(\bar{t}\sigma^{\mu\nu}\lambda^{A}t\right)G_{\mu\nu}^{A}$$

(25)

This interaction is neglected in the vertex-function approach to the $Wtb$ vertex.

The Feynman diagrams are shown in Figure 6. Since the operators $O_{\phi q}^{(3)}$ and $O_{tW}$ will be measured (or bounded) from single-top production and top-quark decay, respectively, the $Wt$ associated production process can be used to measure (or bound) the operator $O_{tG}$, which is also present in $t\bar{t}$ production (see Section 4).

Here we list all the corrections to the SM amplitudes and cross sections. The squared amplitude of the three channels are:

	$s$-channel:
	$\displaystyle\frac{1}{4}\Sigma|M_{u\bar{d}\rightarrow t\bar{b}}|^{2}\!\!\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!\!\!\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{4}u(u-m_{t}^{2})}{4(s-m_{W}^{2})^{2}}-\frac{2\sqrt{2}{\rm Re}C_{tW}V_{tb}m_{t}m_{W}}{\Lambda^{2}}\frac{g^{2}su}{(s-m_{W}^{2})^{2}}+\frac{2C_{qq}^{(1,3)}V_{tb}}{\Lambda^{2}}\frac{g^{2}u(u-m_{t}^{2})}{s-m_{W}^{2}}$		(26)
	$t$-channel:
	$\displaystyle\frac{1}{4}\Sigma|M_{ub\rightarrow dt}|^{2}\!\!\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!\!\!\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{4}s(s-m_{t}^{2})}{4(t-m_{W}^{2})^{2}}-\frac{2\sqrt{2}{\rm Re}C_{tW}V_{tb}m_{t}m_{W}}{\Lambda^{2}}\frac{g^{2}st}{(t-m_{W}^{2})^{2}}+\frac{2C_{qq}^{(1,3)}V_{tb}}{\Lambda^{2}}\frac{g^{2}s(s-m_{t}^{2})}{t-m_{W}^{2}}$		(27)
	$\displaystyle\frac{1}{4}\Sigma|M_{\bar{d}b\rightarrow\bar{u}t}|^{2}\!\!\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!\!\!\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{4}u(u-m_{t}^{2})}{4(t-m_{W}^{2})^{2}}-\frac{2\sqrt{2}{\rm Re}C_{tW}V_{tb}m_{t}m_{W}}{\Lambda^{2}}\frac{g^{2}ut}{(t-m_{W}^{2})^{2}}+\frac{2C_{qq}^{(1,3)}V_{tb}}{\Lambda^{2}}\frac{g^{2}u(u-m_{t}^{2})}{t-m_{W}^{2}}$		(28)

$Wt$ associated production:

	$\displaystyle\frac{1}{96}\Sigma|M_{gb\rightarrow Wt}|^{2}$	$\displaystyle=$	$\displaystyle\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{2}g_{s}^{2}}{24m_{W}^{2}s(t-m_{t}^{2})^{2}}\left(m_{t}^{8}-(2s+t)m_{t}^{6}+((s+t)^{2}-2tm_{W}^{2}-2m_{W}^{4})m_{t}^{4}\right.$		(29)
			$\displaystyle\left.-(t(s+t)^{2}-2(s^{2}-st+2t^{2})m_{W}^{2}+2tm_{W}^{4}-4m_{W}^{6})m_{t}^{2}\right.$
			$\displaystyle\left.-2tm_{W}^{2}(s^{2}+t^{2}-2(s+t)m_{W}^{2}+2m_{W}^{4})\right)$
			$\displaystyle+\frac{2{\rm Re}C_{tW}V_{tb}g_{s}^{2}m_{t}m_{W}}{3\sqrt{2}\Lambda^{2}s(t-m_{t}^{2})^{2}}\left(3m_{t}^{6}-(2s+3t+6m_{W}^{2})m_{t}^{4}\right.$
			$\displaystyle\left.-(s^{2}+2st-3t^{2}-6m_{W}^{4})m_{t}^{2}+t(s^{2}-2st-3t^{2}+6(s+t)m_{W}^{2}-6m_{W}^{4})\right)$
			$\displaystyle+\frac{{\rm Re}C_{tG}V_{tb}^{2}g^{2}g_{s}m_{t}v}{3\sqrt{2}\Lambda^{2}(m_{t}^{2}-t)}(m_{t}^{2}+2s-t)$

As before, $C_{i}$ is the coefficient of operator $O_{i}$ and $s,t,u$ are the usual Mandelstam variables. We have set $V_{ud}=1$ for simplicity. The differential cross sections are as follows:

	$\displaystyle\frac{d\sigma_{u\bar{d}\rightarrow t\bar{b}}}{d\cos\theta}$	$\displaystyle=$	$\displaystyle\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{4}(s-m_{t}^{2})^{2}}{512\pi s^{2}(s-m_{W}^{2})^{2}}(1+\cos\theta)\left(s+m_{t}^{2}+(s-m_{t}^{2})\cos\theta\right)$		(30)
			$\displaystyle+{\rm Re}C_{tW}V_{tb}\frac{g^{2}m_{t}m_{W}(s-m_{t}^{2})^{2}}{16\sqrt{2}\pi\Lambda^{2}s(s-m_{W}^{2})^{2}}(1+\cos\theta)$
			$\displaystyle+C_{qq}^{(1,3)}V_{tb}\frac{g^{2}(s-m_{t}^{2})^{2}}{64\pi\Lambda^{2}s^{2}(s-m_{W}^{2})}(1+\cos\theta)(s+m_{t}^{2}+(s-m_{t}^{2})\cos\theta)$

with $\theta$ the angle between up quark and top quark momenta in the center of mass frame;

	$\displaystyle\frac{d\sigma_{ub\rightarrow dt}}{d\cos\theta}\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{4}(s-m_{t}^{2})^{2}}{32\pi s(2m_{W}^{2}+(s-m_{t}^{2})(1-\cos\theta))^{2}}$		(31)
			$\displaystyle\!\!\!\!\!+{\rm Re}C_{tW}V_{tb}\frac{g^{2}m_{t}m_{W}(s-m_{t}^{2})^{2}(1-\cos\theta)}{4\sqrt{2}\pi\Lambda^{2}s(2m_{W}^{2}+(s-m_{t}^{2})(1-\cos\theta))^{2}}-C_{qq}^{(1,3)}V_{tb}\frac{g^{2}(s-m_{t}^{2})^{2}}{8\pi\Lambda^{2}s(2m_{W}^{2}+(s-m_{t}^{2})(1-\cos\theta))}$

	$\displaystyle\frac{d\sigma_{\bar{d}b\rightarrow\bar{u}t}}{d\cos\theta}$	$\displaystyle=$	$\displaystyle\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{4}(s-m_{t}^{2})^{2}(1+\cos\theta)(s+m_{t}^{2}+(s-m_{t}^{2})\cos\theta)}{128\pi\Lambda^{2}s^{2}(2m_{W}^{2}+(s-m_{t}^{2})(1-\cos\theta))^{2}}$		(32)
			$\displaystyle-{\rm Re}C_{tW}V_{tb}\frac{g^{2}m_{t}m_{W}(s-m_{t}^{2})^{3}\sin^{2}\theta}{8\sqrt{2}\pi\Lambda^{2}s^{2}(2m_{W}^{2}+(s-m_{t}^{2})(1-\cos\theta))^{2}}$
			$\displaystyle-C_{qq}^{(1,3)}V_{tb}\frac{g^{2}(s-m_{t}^{2})^{2}(1+\cos\theta)(s+m_{t}^{2}+(s-m_{t}^{2})\cos\theta)}{32\pi\Lambda^{2}s^{2}(2m_{W}^{2}+(s-m_{t}^{2})(1-\cos\theta))}$

with $\theta$ the angle between bottom quark and top quark momenta in the center of mass frame;

	$\displaystyle\frac{d\sigma_{gb\rightarrow Wt}}{d\cos\theta}\!\!\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!\!\!\left(V_{tb}^{2}+\frac{2C_{\phi q}^{(3)}V_{tb}v^{2}}{\Lambda^{2}}\right)\frac{g^{2}g_{s}^{2}\lambda^{1/2}}{1536\pi s^{3}m_{W}^{2}(s+m_{t}^{2}-m_{W}^{2}-\lambda^{1/2}\cos\theta)^{2}}\left((m_{t}^{2}+10m_{W}^{2})s^{3}\right.$		(33)
			$\displaystyle\left.+(3m_{t}^{4}+19m_{t}^{2}m_{W}^{2}-22m_{W}^{4})s^{2}-(9m_{t}^{6}+8m_{t}^{4}m_{W}^{2}+5m_{t}^{2}m_{W}^{4}-22m_{W}^{6})s+5(m_{t}^{2}-m_{W}^{2})^{3}(m_{t}^{2}+2m_{W}^{2})\right.$
			$\displaystyle\left.-(m_{t}^{2}+2m_{W}^{2})\lambda^{3/2}\cos^{3}\theta+\left((6m_{W}^{2}-m_{t}^{2})s-m_{t}^{4}-m_{t}^{2}m_{W}^{2}+2m_{W}^{4}\right)\lambda\cos^{2}\theta\right.$
			$\displaystyle\left.-\left((14m_{W}^{2}-m_{t}^{2})s^{2}-2(m_{t}^{4}-7m_{t}^{2}m_{W}^{2}+6m_{W}^{4})s+3(m_{t}^{6}-3m_{t}^{2}m_{W}^{4}+2m_{W}^{6})\right)\lambda^{1/2}\cos\theta\right)$
			$\displaystyle-{\rm Re}C_{tW}V_{tb}\frac{g_{s}^{2}m_{t}m_{W}\lambda^{1/2}}{96\sqrt{2}\pi\Lambda^{2}s^{3}\left(s+m_{t}^{2}-m_{W}^{2}-\lambda^{1/2}\cos\theta\right)^{2}}\left(5s^{3}-9(m_{t}^{2}-m_{W}^{2})s^{2}\right.$
			$\displaystyle\left.+(19m_{t}^{4}+10m_{t}^{2}m_{W}^{2}-29m_{W}^{4})s-15(m_{t}^{2}-m_{W}^{2})^{3}+3\lambda^{3/2}\cos^{3}\theta-(5s-3m_{t}^{2}+3m_{W}^{2})\lambda\cos^{2}\theta\right.$
			$\displaystyle\left.-\left(3s^{2}-10(m_{t}^{2}-m_{W}^{2})s-9(m_{t}^{2}-m_{W}^{2})^{2}\right)\lambda^{1/2}\cos\theta\right)$
			$\displaystyle+{\rm Re}C_{tG}V_{tb}^{2}\frac{g^{2}g_{s}vm_{t}\lambda^{1/2}\left(m_{t}^{2}-m_{W}^{2}+5s-\lambda^{1/2}\cos\theta\right)}{96\sqrt{2}\pi\Lambda^{2}s^{2}\left(m_{t}^{2}-m_{W}^{2}+s-\lambda^{1/2}\cos\theta\right)}$

with $\theta$ the angle between gluon and top quark momenta in the center of mass frame, and
$\lambda=s^{2}+m_{t}^{4}+m_{W}^{4}-2sm_{t}^{2}-2sm_{W}^{2}-2m_{t}^{2}m_{W}^{2}$. The angular dependence at $\sqrt{s}=2m_{t}$ (recall that the kinematic threshold is $\sqrt{s}=m_{t}$) is shown in Figures 8-10 (areas are normalized).