T-odd transverse momentum dependent gluon fragmentation functions in a spectator model

In Quantum Chromodynamics (QCD), the nucleon emerges as a strongly interacting, relativistic bound state of quarks and gluon (collectively called partons). This concept of parton dynamics within the framework of QCD replies on two foundamental principles: Color confinement [1, 2] and asymptotic freedom [3, 4, 5, 6]. The former leads to the fact that quarks and gluon are tightly bound within hadrons, rendering the dynamics nonperturbative on the hadronic scale. Consequently, the exploration of the partonic structure of hadrons and the mechanisms underlying parton hadronization poses substantial challenges. Fortunately, the phenomenon of asymptotic freedom, enables the application of QCD factorization to analyze a variety of high-energy scattering processes. Within this framework, a collection of well-defined and fundamental functions provide valuable insights into the internal structure of the nucleon and the parton hadronization. These functions are known as the parton distribution functions (PDFs) and the parton fragmentation functions (FFs) respectively.

In general, leading-twist collinear PDFs/FFs are universal between different processes. These functions contain the information about 1-dimensional momentum structure. A much more comprehensive picture about the partonic structure of hadrons can be achieved by extending the PDFs to the objects in the 3-dimensional momentum space, namely, the transverse momentum dependent (TMD) PDFs and FFs. They explicitly depend on the parton transverse momenta and probe the 3-dimensional structure of hadrons [7, 8]. Furthermore, TMD PDFs and FFs play important roles in the high energy processes involving two hadrons, such as the semi-inclusive DIS, Drell-Yan process and hadron pair production in $e^{+}e^{-}$ annihilation. Different from the collinear PDFs/FFs, a proper treatment of the process dependent incoming or outgoing directions for the Wilson line [9, 7] of the TMD PDFs or FFs is necessary.

In this paper, we study the T-odd gluon TMD FFs in a spectator model. This model is based on the assumption that a time-link off-shell parton can fragment into a hadron and a single real spectator particle. This model was originally used to calculate the quark TMD PDFs of the nucleon [10, 11, 12, 13, 14, 15] and has been widely applied in different forms to calculate the quark TMD PDFs of the pion [16, 17, 18], the TMD FFs of the pion and kaon [19], and the gluon TMD PDFs [20, 21, 22, 23, 24]. The spectator model has also been extended [25] to calculate the T-even gluon TMD FFs $D_{1}^{h/g}(z,\bm{k}_{T}^{2})$, $G_{1L}^{h/g}(z,\bm{k}_{T}^{2})$, $G_{1T}^{h/g}(z,\bm{k}_{T}^{2})$, and $H_{1}^{\perp,h/g}(z,\bm{k}_{T}^{2})$ at the leading-twist level without considering the effects of the gauge-link. In this work, in order to obtain the necessary imaginary part required for T-odd gluon TMD FFs $D_{1T}^{\perp,h/g}(z,\bm{k}_{T}^{2})$, $H_{1T}^{h/g}(z,\bm{k}_{T}^{2})$, $H_{1L}^{\perp,h/g}(z,\bm{k}_{T}^{2})$ and $H_{1T}^{\perp,h/g}(z,\bm{k}_{T}^{2})$, we consider the effect of the gluon rescattering at one loop level. The spectator model has also been applied to calculate the Collins function for pions [26, 27, 28, 29, 30, 19] and kaons [19] by considering the pion loop or the gluon loop. The twist-3 fragmentation functions $\tilde{H}(z,\bm{k}_{T}^{2})$, $H(z,\bm{k}_{T}^{2})$, $\tilde{G}^{\perp}(z,\bm{k}_{T}^{2})$ and $G^{\perp}(z,\bm{k}_{T}^{2})$ [31, 32] have also been calculated within the spectator model.

The rest of the paper is organized as follows. In Section. II, we provide the formalism of the spectator model with the incorporation the gluon rescattering effect. We obtain the model result of the one loop order gluon-gluon correlator and the leading-twist T-odd gluon TMD FFs using proper projecting operators. In Section. III, we present the numerical results of the FFs $D_{1T}^{\perp,h/g}(z,\bm{k}_{T}^{2})$, $H_{1T}^{h/g}(z,\bm{k}_{T}^{2})$, $H_{1L}^{\perp,h/g}(z,\bm{k}_{T}^{2})$ and $H_{1T}^{\perp,h/g}(z,\bm{k}_{T}^{2})$ in the case the hadron is a proton. We summarize the paper in Section. IV.

We follow the conventions in Refs .[33, 25] and specify the kinematics of the final-state hadron and the fragmentation parton. In a reference frame in which the hadron has no transverse momentum, one can write

	$\displaystyle P_{h}$	$\displaystyle=P_{h}^{-}n_{-}+\frac{M_{h}^{2}}{2P_{h}^{-}}n_{+}\,,$		(1)
	$\displaystyle k$	$\displaystyle=\frac{P_{h}^{-}}{z}n_{-}+\frac{z\left(k^{2}+\bm{k}_{T}^{2}\right)}{2P_{h}^{-}}n_{+}+\bm{k}_{T}\,,$		(2)
	$\displaystyle S_{h}$	$\displaystyle=S_{hL}\frac{P_{h}^{-}}{M_{h}}n_{-}-S_{hL}\frac{M_{h}}{2P_{h}^{-}}n_{+}+\bm{S}_{hT}\,,$		(3)

where the light-cone vectors $n_{\pm}^{2}=0$ and $n_{+}\cdot n_{-}=1$ have been used, $M_{h}$ is the mass of the final-state hadron, $z=P_{h}^{-}/k^{-}$ is the momentum fraction carried by the hadron, $\bm{k}_{T}$ denotes the momentum component of the fragmentation parton, $S_{hL}$ and $S_{hT}$ describes longitudinal and transverse polarization of the hadron, respectively. The hadron is characterized by its 4-momentum $P_{h}$ and the covariant spin vector $S_{h}$.

The appropriate gauge-invariant gluon-gluon correlator for fragmentation is defined as [33, 34]:

$\displaystyle\Delta^{\mu\nu;\rho\sigma}\left(k;P_{h},S_{h}\right)$

$\displaystyle=\sum_{X}\int\frac{d^{4}\xi}{(2\pi)^{4}}e^{ik\cdot\xi}\left\langle 0\left|F^{\rho\sigma}(\xi)\right|P_{h},S_{h};X\right\rangle\left\langle P_{h},S_{h};X\left|\mathcal{U}(\xi,0)F^{\mu\nu}(0)\right|0\right\rangle\,,$

(4)

where $\mathcal{U}(\xi,0)$ is the gauge-link operator connecting space-times $\xi$ and 0 to ensure the gauge-invariance of the operator definition. In general, we introduce the correlation function integrated over $k^{+}$:

$\displaystyle\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};S_{h}\right)=\int dk^{+}\Delta^{-j;-i}\left(k;P_{h},S_{h}\right)\,,$

(5)

where $i$ and $j$ are transverse spatial indices.

We can decompose the gluon fragmentation correlator and define eight leading-twist TMD FFs of the gluon through the following projection [35, 7, 34]

	$\displaystyle\delta_{T}^{ij}\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};S_{h}\right)=$	$\displaystyle 2P_{h}^{-}\left[D_{1}^{h/g}\left(z,\bm{k}_{T}^{2}\right)+\frac{\varepsilon_{T}^{ij}k_{T}^{i}S_{hT}^{j}}{M_{h}}D_{1T}^{\perp h/g}\left(z,\bm{k}_{T}^{2}\right)\right]\,,$		(6)
	$\displaystyle i\varepsilon_{T}^{ij}\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};S_{h}\right)=$	$\displaystyle 2P_{h}^{-}\left[\Lambda_{h}G_{1L}^{h/g}\left(z,\bm{k}_{T}^{2}\right)+\frac{\vec{k}_{T}\cdot\vec{S}_{hT}}{M_{h}}G_{1T}^{h/77g}\left(z,\bm{k}_{T}^{2}\right)\right]\,,$		(7)
	$\displaystyle\hat{S}\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};S_{h}\right)=$	$\displaystyle 2P_{h}^{-}\hat{S}\left[\frac{k_{T}^{i}\varepsilon_{T}^{jk}S_{hT}^{k}}{2M_{h}}H_{1T}^{h/g}\left(z,\bm{k}_{T}^{2}\right)+\frac{k_{T}^{i}k_{T}^{j}}{2M_{h}^{2}}H_{1}^{\perp h/g}\left(z,\bm{k}_{T}^{2}\right)\right.$
		$\displaystyle\left.+\frac{k_{T}^{i}\varepsilon_{T}^{jk}k_{T}^{k}}{2M_{h}^{2}}\left(\Lambda_{h}H_{1L}^{\perp h/g}\left(z,\bm{k}_{T}^{2}\right)+\frac{\vec{k}_{T}\cdot\vec{S}_{hT}}{M_{h}}H_{1T}^{\perp h/g}\left(z,\bm{k}_{T}^{2}\right)\right)\right]\,,$		(8)

and the four T-odd gluon TMD FFs can be projected into

	$\displaystyle\frac{\varepsilon^{ij}_{T}k^{i}_{T}S^{j}_{hT}}{M_{h}}D_{1T}^{\perp,h/g}\left(z,\bm{k}_{T}^{2}\right)=$	$\displaystyle\frac{1}{2P_{h}^{-}}\frac{\delta_{T}^{ij}}{2}\left[\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};S_{h}\right)-\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};-S_{h}\right)\right]\,,$		(9)
	$\displaystyle k_{T}^{i}k_{T}^{j}\hat{S}\frac{k_{T}^{i}\varepsilon_{T}^{jk}S_{hT}^{k}}{2M_{h}}H_{1T}^{h/g}\left(z,\bm{k}_{T}^{2}\right)=$	$\displaystyle k_{T}^{i}k_{T}^{j}\frac{1}{2P_{h}^{-}}\frac{\hat{S}}{2}\left[\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};S_{h}\right)-\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};-S_{h}\right)\right]\,,$		(10)
	$\displaystyle S_{T}^{i}S_{T}^{j}\hat{S}\frac{k_{T}^{i}\varepsilon_{T}^{jk}k_{T}^{k}}{2M_{h}^{2}}\left(\Lambda_{h}H_{1L}^{\perp h/g}\left(z,\bm{k}_{T}^{2}\right)+\right.$	$\displaystyle\left.\frac{\vec{k}_{T}\cdot\vec{S}_{hT}}{M_{h}}H_{1T}^{\perp h/g}\left(z,\bm{k}_{T}^{2}\right)\right)$
	$\displaystyle=$	$\displaystyle S_{T}^{i}S_{T}^{j}\frac{1}{2P_{h}^{-}}\frac{\hat{S}}{2}\left[\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};S_{h}\right)-\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};-S_{h}\right)\right]$
	$\displaystyle-$	$\displaystyle S_{T}^{i}S_{T}^{j}\hat{S}\frac{k_{T}^{i}\varepsilon_{T}^{jk}S_{hT}^{k}}{2M_{h}}H_{1T}^{h/g}\left(z,\bm{k}_{T}^{2}\right)\,.$		(11)

Here, we have used the symmetric transverse tensor $\delta_{T}^{ij}=-g_{T}^{ij}$ with $g_{T}^{ij}=g^{ij}-n_{+}^{i}n_{-}^{j}-n_{+}^{j}n_{-}^{i}$, the anti-symmetric transverse tensor $\epsilon_{T}^{ij}$ with $\epsilon_{T}^{12}=1$, and a symmetrization operator $\hat{S}$ for a generic tensor $O^{ij}$ which is defined as:

$\displaystyle\hat{S}O^{ij}\equiv\frac{1}{2}\left(O^{ij}+O^{ji}-\delta_{T}^{ij}O^{kk}\right)\,.$

(12)

The interpretation of the leading-twist TMD FFs in Eqs. (6)-(8) is summarized in Tab. 1. Here $D_{1}^{h/g}$ is the well-known unpolarized FF which describes the number density of unpolarized hadron in an unpolarized gluon. $G_{1L}^{h/g}$, $G_{1T}^{h/g}$, and $H_{1}^{h/g}$ denote the longitudinally polarized, longi-transversely polarized, and linearly polarized FFs, respectively. These functions are T-even, while $D_{1T}^{h/g}$, $H_{1T}^{h/g}$, $H_{1L}^{\perp,h/g}$, $H_{1T}^{\perp,h/g}$ are naive T-odd functions. In the spectator model, the tree level gluon-gluon correlator is modeled as:

$\displaystyle\Delta^{ij}\left(z,\bm{k}_{T}^{2};S_{h}\right)\sim$

$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{2(1-z)k^{-}}\left[\overline{\mathcal{M}}_{0}^{j}(z,\bm{k}_{T}^{2};S_{h})\mathcal{M}_{0}^{i}(z,\bm{k}_{T}^{2};S_{h})\right]\,.$

(13)

The tree level diagrams lead to vanishing result for T-odd FFs because of lack of the imaginary phases [36, 30]. T-odd functions typically require the interference between two amplitudes with different imaginary parts to exist. In order to obtain the necessary imaginary part in the scattering amplitude, one has to consider the diagrams at loop levels. In this paper, we will take into account the contribution of gluon rescattering at the one loop order. There are four different diagrams (and their hermitian conjugates) that may contribute to the correlator $\Delta^{h/g,ij}\left(z,\bm{k}_{T}^{2};S_{h}\right)$, as shown in Fig. 1, including the self-energy diagram (Fig. 1a), the vertex diagram (Fig. 1b), the hard vertex diagram (Fig. 1c), and the box diagram (Fig. 1d).

We can write the expressions of the correlator:

	$\displaystyle\Delta^{ij}_{(a)}=$	$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{2(1-z)k^{-}}\int\frac{d^{4}l}{(2\pi)^{4}}\operatorname{Tr}\left[\left(\not{P_{h}}+M_{h}\right)\frac{1+\gamma^{5}\not{S}}{2}\mathcal{Y}_{v,b^{\prime}c^{\prime}}^{*}\frac{G^{j\nu\ast}_{ab^{\prime}}(k,k)}{k^{2}}\left(\not{k}-\not{P_{h}}+M_{X}\right)_{cc^{\prime}}\mathcal{Y}_{\mu,bc}\frac{-ig^{\mu\rho}\delta_{bd}}{k^{2}+i\varepsilon}\right.$
		$\displaystyle\left.(-gf_{d^{\prime}e^{\prime}f^{\prime}}V^{\rho^{\prime}\alpha^{\prime}\beta^{\prime}}(k,-l,l-k))(-gf_{dfe}V^{\rho\beta\alpha}(-k,k-l,l))\frac{-ig^{\alpha\alpha^{\prime}}\delta_{ee^{\prime}}}{l^{2}+i\varepsilon}\cdot\frac{-ig^{\beta\beta^{\prime}}\delta_{ff^{\prime}}}{(k-l)^{2}+i\varepsilon}\cdot\frac{G^{i\rho^{\prime}}_{ad^{\prime}}(k,k)}{k^{2}}\right]+H.c.\,,$		(14)
	$\displaystyle\Delta^{ij}_{(b)}=$	$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{2(1-z)k^{-}}\int\frac{d^{4}l}{(2\pi)^{4}}\operatorname{Tr}\left[\left(\not{P_{h}}+M_{h}\right)\frac{1+\gamma^{5}\not{S}}{2}\mathcal{Y}_{v,b^{\prime}c^{\prime}}^{*}\frac{G^{j\nu\ast}_{ab^{\prime}}(k,k)}{k^{2}}\left(\not{k}-\not{P_{h}}+M_{X}\right)_{c^{\prime}f^{\prime}}(g\gamma^{\alpha}f_{fe^{\prime}f^{\prime}})\right.$
		$\displaystyle\left.\frac{i\left(\not{k}-\not{P_{h}}-\not{l}+M_{X}\right)_{cf}}{(k-P_{h}-l)^{2}-M_{X}^{2}+i\varepsilon}\mathcal{Y}_{\mu,bc}\frac{-ig^{\rho\rho^{\prime}}\delta_{bd}}{(k-l)^{2}+i\varepsilon}(-gf_{d^{\prime}ed}V^{\rho^{\prime}\alpha^{\prime}\rho}(k,-l,l-k))\frac{-ig^{\alpha\alpha^{\prime}}\delta_{ee^{\prime}}}{l^{2}+i\varepsilon}\cdot\frac{G^{i\rho^{\prime}}_{ad^{\prime}}(k,k)}{k^{2}}\right]+H.c.\,,$		(15)
	$\displaystyle\Delta^{ij}_{(c)}=$	$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{2(1-z)k^{-}}\int\frac{d^{4}l}{(2\pi)^{4}}\operatorname{Tr}\left[\left(\not{P_{h}}+M_{h}\right)\frac{1+\gamma^{5}\not{S}}{2}\mathcal{Y}_{v,b^{\prime}c^{\prime}}^{*}\frac{G^{j\nu\ast}_{ab^{\prime}}(k,k)}{k^{2}}\left(\not{k}-\not{P_{h}}+M_{X}\right)_{cc^{\prime}}\mathcal{Y}_{\mu,bc}\frac{-ig^{\mu\rho}\delta_{bd}}{k^{2}+i\varepsilon}\right.$
		$\displaystyle\left.(-gf_{dd^{\prime}f}V^{\rho\rho^{\prime}\alpha}(-k,k-l,l))\frac{-ig^{\alpha\rho}\delta_{ff^{\prime}}}{l^{2}+i\varepsilon}\cdot\frac{ign_{+}^{\rho}f_{eaf^{\prime}}}{-l\cdot n_{+}\pm i\varepsilon}\cdot\frac{G^{i\rho^{\prime}}_{ad^{\prime}}(k-l,k-l)}{(k-l)^{2}+i\varepsilon}\right]+H.c.\,,$		(16)
	$\displaystyle\Delta^{ij}_{(d)}=$	$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{2(1-z)k^{-}}\int\frac{d^{4}l}{(2\pi)^{4}}\operatorname{Tr}\left[\left(\not{P_{h}}+M_{h}\right)\frac{1+\gamma^{5}\not{S}}{2}\mathcal{Y}_{v,b^{\prime}c^{\prime}}^{*}\frac{G^{j\nu\ast}_{ab^{\prime}}(k,k)}{k^{2}}\left(\not{k}-\not{P_{h}}+M_{X}\right)_{c^{\prime}f^{\prime}}\frac{ign_{+}^{\rho}f_{fd^{\prime}f^{\prime}}}{-l\cdot n_{+}\pm i\varepsilon}\right.$
		$\displaystyle\left.\frac{-ig^{\alpha\rho}\delta_{ad^{\prime}}}{l^{2}+i\varepsilon}(g\gamma^{\alpha}f_{fd^{\prime}f^{\prime}})\frac{i\left(\not{k}-\not{P_{h}}-\not{l}+M_{X}\right)_{cf}}{(k-P_{h}-l)^{2}-M_{X}^{2}+i\varepsilon}\mathcal{Y}_{\mu,bc}\frac{G^{i\mu}_{ab}(k-l,k-l)}{(k-l)^{2}+i\varepsilon}\right]+H.c.\,,$		(17)

where $a,b,c,d,e,f$ are the color indices. Details of the Feynman rules for the eikonal propagator and the eikonal vertex can be found in Ref. [37]. Here, $g$ is the coupling of the three-gluon vertex and the spectator-gluon-spectator vertex, and $M_{X}$ the mass of the spectator. We use the notation

$\displaystyle V^{\mu\nu\rho}(p,q,r)=(p-q)^{\rho}g^{\mu\nu}+(q-r)^{\mu}g^{\nu\rho}+(r-p)^{\nu}g^{\rho\mu}\,.$

(18)

The same Feynman rules are used here as in Ref. [25]. The term

$\displaystyle G_{ab}^{i\mu}(k,k)$

$\displaystyle=-i\delta_{ab}k^{-}\left(g^{i\mu}-\frac{k^{i}n_{+}^{\mu}}{k^{-}}\right)$

(19)

is the specific Feynman rule for the field strength tensor of the form $-i\left(p^{\mu}g^{\nu\rho}-p^{\nu}g^{\mu\rho}\right)\delta_{ab}$ [38, 7], and the gluon-hadron-spectator vertex $\mathcal{Y}_{bc}^{\mu}$ is modeled as [21]:

$\displaystyle\mathcal{Y}_{bc}^{\mu}$

$\displaystyle=\delta_{bc}\left[g_{1}\left(k^{2}\right)\gamma^{\mu}+g_{2}\left(k^{2}\right)\frac{i}{2M_{h}}\sigma^{\mu\nu}k_{\nu}\right]\,,$

(20)

where $\sigma^{\mu\nu}=i\left[\gamma^{\mu},\gamma^{\nu}\right]/2$, $g_{1}(k^{2})$ and $g_{2}(k^{2})$ are the gluon-hadron-spectator couplings. Here, the vertex is modeled by two Dirac structures $\gamma^{\mu}$ and $\sigma^{\mu\nu}$ which mimics the conserved electromagnetic current of a free nucleon obtained by applying a standard Gordon decomposition. The kind of coupling $g_{1,2}(k^{2})$ has several different choices in the literature [14]. Following Refs. [21, 25], we apply the dipolar form factor

$\displaystyle g_{1,2}\left(k^{2}\right)$

$\displaystyle=\kappa_{1,2}\frac{k^{2}}{\left|k^{2}-\Lambda_{X}^{2}\right|^{2}}\,,$

(21)

where $\kappa_{1,2}$ are free parameters, and $\Lambda_{X}=k_{T}^{\textrm{max}}$ is cut-off parameter in order to regularize the divergence. The spectator-gluon-spectator vertex is also an important term which connects an anti-octet spectator to an octet gluon and an anti-octet spectator, while the gluon-hadron-spectator vertex connects a color-neutral hadron to an octet gluon and an anti-octet spectator. In this study, the spectator-gluon-spectator vertex is modeled as a Dirac structure $g\gamma^{\mu}$ with the color structure constant $f_{abc}$. The three-gluon vertex and the $f$-type eikonal vertex of Eqs. (16) and (17) will make sense in this case.

Since the spectator is on-shell $(k-P_{h})^{2}=M_{X}^{2}$, we can obtain the following expression for the virtuality $k^{2}$ of the fragmentation gluon:

$\displaystyle k^{2}=\frac{z}{1-z}\vec{k}_{T}^{2}+\frac{M_{X}^{2}}{1-z}+\frac{M_{h}^{2}}{z}\,.$

(22)

In Eqs. (16) and (17) we have applied the Feynman rule $i/(-l\cdot n_{+}\pm i\varepsilon)$ for the eikonal propagator. It should be noted that the sign of the factor $i\varepsilon$ in the eikonal propagator is different for SIDIS $(+)$ and $e^{+}e^{-}$ annihilation $(-)$. However, in the calculation of T-odd functions, we utilize the Cutkosky cut rule to put certain internal lines on the mass shell to obtain the necessary imaginary phase. For the result of Fig. 1, the only way of the cuts corresponds to the following replacements on the propagators by using the Dirac delta functions

$\displaystyle\frac{1}{l^{2}+i\varepsilon}\rightarrow-2\pi i\delta(l^{2})\,,\quad\frac{1}{(k-l)^{2}+i\varepsilon}\rightarrow-2\pi i\delta((k-l)^{2})\,.$

(23)

The other combinations (cutting through the eikonal line or the spectator line) do not contribute, as shown in Refs. [39, 31].

Now we can obtain the expressions of four the leading-twist T-odd gluon TMD FFs by projecting $\Delta^{ij}$ with $\delta_{T}^{ij}$, $\varepsilon_{T}^{ij}$ and $\hat{S}$ in Eqs. (6)-(8). However, using the correlator from Fig. 1a or Fig. 1c, we obtain the following result

$\displaystyle X^{h/g}\left(z,\bm{k}^{2}_{T}\right)\propto\int\frac{d^{4}l}{(2\pi)^{4}}\left(k^{\alpha}l^{-}-k^{-}l^{\alpha}\right)\delta(l^{2})\delta((k-l)^{2})\,,$

(24)

Using the decomposition of the integral

$\displaystyle\int d^{4}l~{}\delta(l^{2})\delta((k-l)^{2})l^{\mu}=\mathcal{F}k^{\mu}\,,$

(25)

one can conclude that Eqs. (14) and (16) should generate zero contribution to the T-odd TMD FFs. Then the result of Eq. (15) reads

	$\displaystyle D_{1T(b)}^{\perp,h/g}\left(z,\bm{k}_{T}^{2}\right)=$	$\displaystyle\frac{\left(-2C^{2}_{A}C_{F}\right)g^{2}\left(k^{2}\left(1-2z\right)+M_{h}^{2}-M_{X}^{2}\right)}{256\pi^{5}k^{4}\left(1-z\right)}D_{g}\left(z,\bm{k}_{T}^{2}\right)\,,$		(26)
	$\displaystyle H_{1T(b)}^{h/g}\left(z,\bm{k}_{T}^{2}\right)=$	$\displaystyle-\frac{\left(-2C^{2}_{A}C_{F}\right)g^{2}\left(k^{2}\left(1-2z\right)+M_{h}^{2}-M_{X}^{2}\right)}{128\pi^{5}k^{4}\left(1-z\right)}D_{g}\left(z,\bm{k}_{T}^{2}\right)\,,$		(27)
	$\displaystyle H_{1T(b)}^{\perp h/g}\left(z,\bm{k}_{T}^{2}\right)=$	$\displaystyle\frac{\left(-2C^{2}_{A}C_{F}\right)g^{2}M_{h}^{2}z}{64\pi^{5}k^{4}\left(1-z\right)}D_{g}\left(z,\bm{k}_{T}^{2}\right)\,,$		(28)
	$\displaystyle H_{1L(b)}^{\perp h/g}\left(z,\bm{k}_{T}^{2}\right)=$	$\displaystyle\frac{\left(-2C^{2}_{A}C_{F}\right)g^{2}M_{h}^{2}}{64\pi^{5}k^{4}\left(1-z\right)}D_{g}\left(z,\bm{k}_{T}^{2}\right)\,,$		(29)

	$\displaystyle D_{g}\left(z,\bm{k}_{T}^{2}\right)=$	$\displaystyle 16g_{1}^{2}M_{h}^{2}\left[\left(\mathcal{B}+\mathcal{Y}\right)+\frac{M_{h}+M_{X}}{2M_{h}}\left(\mathcal{A}+I_{2}+2\mathcal{Z}\right)\right]-2g_{1}g_{2}\left[\left(2\mathcal{A}+\mathcal{B}+4\mathcal{Y}+5\mathcal{Z}\right)k^{2}\right.$
		$\displaystyle+\left.\left(2\mathcal{A}+5\mathcal{B}+2I_{2}+4\mathcal{Y}+3\mathcal{Z}\right)\left(M_{h}^{2}-M_{X}^{2}\right)+\left(4\mathcal{B}-6\mathcal{Y}\right)M_{h}M_{X}\right]$
		$\displaystyle-\frac{g_{2}^{2}}{M_{h}}\left[\mathcal{B}\left(k^{2}\left(M_{h}-5M_{X}\right)-\left(M_{h}-M_{X}\right)\left(M_{h}+M_{X}\right)^{2}\right)-\mathcal{Y}\left(k^{2}\left(3M_{h}-4M_{X}\right)\right.\right.$
		$\displaystyle+\left.\left.\left(M_{h}^{2}-5M_{h}M_{X}+4M_{X}^{2}\right)\left(M_{h}+M_{X}\right)\right)-2\mathcal{Z}k^{2}\left(2M_{h}-3M_{X}\right)\right]\,,$		(30)

and the result of Eq. (17) has the form

		$\displaystyle D_{1T(d)}^{\perp,h/g}\left(z,\bm{k}_{T}^{2}\right)$
	$\displaystyle=$	$\displaystyle\frac{\left(-2C^{2}_{A}C_{F}\right)g^{2}}{128\pi^{5}k^{2}\left(1-z\right)z}\left[8g_{1}^{2}M_{h}\left[z\left(\mathcal{B}M_{h}\left(z-1\right)-\mathcal{A}M_{X}\right)+\mathcal{C}k_{-}\left(z-1\right)\left(M_{h}\left(z-1\right)+zM_{X}\right)\right.\right.$
		$\displaystyle+\left.I_{2}\left(M_{h}\left(z-1\right)^{2}+z^{2}M_{X}\right)\right]+2g_{1}g_{2}\left[-z\left(\mathcal{A}k^{2}\left(2z-1\right)+\mathcal{B}\left(2k^{2}\left(z-1\right)+M_{h}^{2}\left(2z-1\right)\right.\right.\right.$
		$\displaystyle+\left.\left.\left.3M_{h}M_{X}-2M_{X}^{2}\left(z-1\right)\right)\right)+\left(\mathcal{C}k_{-}+I_{2}\right)\left(z-1\right)\left(k^{2}-M_{h}^{2}\left(2z-1\right)-4M_{h}M_{X}-M_{X}^{2}\left(z-1\right)\right)\right]$
		$\displaystyle+\frac{g_{2}^{2}}{M_{h}}\left[z\left(\mathcal{A}k^{2}\left(M_{h}+3M_{X}\right)+\mathcal{B}\left(2k^{2}M_{X}+M_{h}^{3}+2M_{h}^{2}M_{X}-M_{h}M_{X}^{2}-2M_{X}^{3}\right)\right)\right.$
		$\displaystyle-\left.\left.\left(\mathcal{C}k_{-}+I_{2}\right)\left(z-1\right)\left(M_{h}-M_{X}\right)\left(k^{2}-\left(M_{h}+M_{X}\right)^{2}\right)-2I_{2}M_{X}k^{2}z\right]\right]\,,$		(31)
		$\displaystyle H_{1T(d)}^{h/g}\left(z,\bm{k}_{T}^{2}\right)$
	$\displaystyle=$	$\displaystyle-\frac{\left(-2C^{2}_{A}C_{F}\right)g^{2}}{128\pi^{5}k^{2}\left(1-z\right)z}\left[16g_{1}^{2}M_{h}z\left[\mathcal{A}M_{h}+\mathcal{B}M_{h}z+\mathcal{C}k_{-}\left(M_{h}\left(z-1\right)+zM_{X}\right)+I_{2}\left(M_{h}\left(z-2\right)+zM_{X}\right)\right]\right.$
		$\displaystyle+4g_{1}g_{2}\left[-z\left(\mathcal{A}\left(k^{2}\left(2z-1\right)+2\left(M_{h}^{2}-M_{X}^{2}\right)\right)+\mathcal{B}\left(2z\left(k^{2}-M_{X}^{2}+M_{h}^{2}\right)+M_{h}^{2}-M_{h}M_{X}\right)\right)\right.$
		$\displaystyle+\mathcal{C}k_{-}\left(\left(1-z\right)\left(k^{2}+\left(2z+1\right)\left(M_{h}^{2}-M_{X}^{2}\right)\right)-4zM_{h}M_{X}\right)+I_{2}\left(\left(z+1\right)\left(k^{2}-\left(2z-1\right)\left(M_{h}^{2}-M_{X}^{2}\right)\right)\right.$
		$\displaystyle+\left.\left.4z\left(M_{h}^{2}-M_{X}^{2}-M_{h}M_{X}\right)\right)\right]+\frac{2g_{2}^{2}}{M_{h}}\left[z\left(\mathcal{A}k^{2}\left(3M_{h}+M_{X}\right)+\mathcal{B}\left(2k^{2}+M_{h}^{2}-M_{X}^{2}\right)\right)\right.$
		$\displaystyle+\left.\left.\left(\mathcal{C}k_{-}+I_{2}\right)\left(M_{h}-M_{X}\right)\left(k^{2}\left(z-1\right)+\left(M_{h}+M_{X}\right)\left(M_{h}\left(z-1\right)+M_{X}\left(z+1\right)\right)\right)-2I_{2}M_{h}k^{2}z\right]\right]\,,$		(32)
		$\displaystyle H_{1T(d)}^{\perp h/g}\left(z,\bm{k}_{T}^{2}\right)$
	$\displaystyle=$	$\displaystyle\frac{\left(-2C^{2}_{A}C_{F}\right)g^{2}M_{h}}{32\pi^{5}k^{2}\left(1-z\right)^{2}z\bm{k}_{T}^{2}}\left[2\left(1-z\right)g_{1}g_{2}M_{h}\left[\mathcal{A}\left(z\left(1-z\right)k^{2}-2z^{2}\bm{k}_{T}^{2}\right)+z\bm{k}_{T}^{2}\left(\left(1-z\right)\mathcal{C}k_{-}+\left(1+z\right)I_{2}\right)\right.\right.$
		$\displaystyle+\left.\mathcal{B}M_{h}^{2}\left(1-z\right)\right]+g_{2}^{2}\left[\mathcal{A}M_{X}z\left(1-z\right)k^{2}+\mathcal{B}M_{h}\left(1-z\right)\left(M_{h}M_{X}-z^{2}\bm{k}_{T}^{2}\right)\right.$
		$\displaystyle+\left.\left.z\bm{k}_{T}^{2}\left(\mathcal{C}k_{-}+I_{2}\right)\left(M_{h}\left(1-z\right)^{2}+M_{X}\left(1-z^{2}\right)\right)\right]\right]\,,$		(33)
		$\displaystyle H_{1L(d)}^{\perp h/g}\left(z,\bm{k}_{T}^{2}\right)$
	$\displaystyle=$	$\displaystyle\frac{\left(-2C^{2}_{A}C_{F}\right)g^{2}g_{1}^{2}M_{h}^{2}z\left(\mathcal{A}-I_{2}\right)}{8\pi^{5}\left(1-z\right)k^{2}}-\frac{\left(-2C^{2}_{A}C_{F}\right)g^{2}}{128\pi^{5}z^{2}\left(1-z\right)k^{2}\bm{k}_{T}^{2}}\left[2g_{1}g_{2}M_{h}\left[2\left(M_{h}\left(z-1\right)+M_{X}z\right)\left(\mathcal{A}zk^{2}+\mathcal{B}M_{h}^{2}\right)\right.\right.$
		$\displaystyle+\left.z\bm{k}_{T}^{2}\left(2zM_{h}\left(4\mathcal{A}+\mathcal{B}\right)+\mathcal{C}k_{-}\left(5M_{h}\left(z-1\right)+8M_{X}z^{2}\right)+\mathcal{D}k_{-}M_{h}\left(z-1\right)z+I_{2}\left(8M_{X}z^{2}-M_{h}\left(3z+5\right)\right)\right)\right]$
		$\displaystyle-g_{2}^{2}\left[2\left(M_{h}-M_{X}\right)\left(M_{h}\left(z-1\right)+zM_{X}\right)\left(\mathcal{A}zk^{2}+\mathcal{B}M_{h}^{2}+2z\left(z-1\right)\bm{k}_{T}^{2}\left(\mathcal{C}k_{-}+I_{2}\right)\right)\right.$
		$\displaystyle+z\bm{k}_{T}^{2}\left(4zk^{2}\left(z\left(\mathcal{A}+\mathcal{B}\right)-I_{2}\right)-4\mathcal{C}k_{-}z\left(1-z\right)k^{2}+\left(\mathcal{C}k_{-}+I_{2}\right)M_{h}\left(9M_{X}-M_{h}\right)-2\mathcal{B}z\left(M_{h}^{2}\left(3-2z\right)\right.\right.$
		$\displaystyle+\left.\left.\left.\left.M_{h}M_{X}+2zM_{X}^{2}\right)-\mathcal{D}k_{-}zM_{h}\left(M_{h}-M_{X}\right)\right)\right]\right]\,.$		(34)

Here the functions $I_{i}$ are defined as

	$\displaystyle I_{1}=$	$\displaystyle\int d^{4}l~{}\delta(l^{2})\delta((k-l)^{2}-m^{2})=\frac{\pi}{2k^{2}}\left(k^{2}-m^{2}\right)\,,$		(35)
	$\displaystyle I_{2}=$	$\displaystyle\int d^{4}l~{}\frac{\delta(l^{2})\delta((k-l)^{2}-m^{2})}{\left(k-P_{h}-l\right)^{2}-M_{X}^{2}}=\frac{\pi}{2\sqrt{\lambda\left(M_{h},M_{X}\right)}}\ln\left(1-\frac{2\sqrt{\lambda\left(M_{h},M_{X}\right)}}{k^{2}-M_{h}^{2}+M_{X}^{2}+\sqrt{\lambda\left(M_{h},M_{X}\right)}}\right)\,,$		(36)
	$\displaystyle I_{3}=$	$\displaystyle\int d^{4}l~{}\frac{\delta(l^{2})\delta((k-l)^{2}-m^{2})}{-l\cdot n_{+}+i\varepsilon}\,,$		(37)
	$\displaystyle I_{4}=$	$\displaystyle\int d^{4}l~{}\frac{\delta(l^{2})\delta((k-l)^{2}-m^{2})}{\left(-l\cdot n_{+}+i\varepsilon\right)\left(\left(k-P_{h}-l\right)^{2}-M_{X}^{2}\right)}\,,$		(38)

with $\lambda\left(M_{h},M_{X}\right)=\left(k^{2}-\left(M_{h}+M_{X}\right)^{2}\right)\left(k^{2}-\left(M_{h}-M_{X}\right)^{2}\right)$, and $I_{34}$ is the linear combination of $I_{3}$ and $I_{4}$,

$\displaystyle I_{34}=k_{-}\left(I_{3}+(1-z)(k^{2}-m^{2})I_{4}\right)=\pi\ln\frac{\sqrt{k^{2}}(1-z)}{M_{X}}\,.$

(39)