content_image/418555.jpg content_image/418560.jpg content_image/418556.jpg content_image/418558.jpg content_image/418554.jpg content_image/418553.jpg content_image/418557.jpg
from Eq. (17), regardless of the color factor, since it is imaginary. 

A symmetric analysis applies if the extra gluon is in- stead radiated from hadron H$_{2}$ as in Figs. 4(c,d). In that case n$_{1}$ is replaced by n$_{2}$ and l + is replaced by l $^{−}$. Since p$_{2}$ is unpolarized, it is the real parts of the eikonal fac- tors that contribute. Thus, since the real parts cancel between graphs, the eikonal factors from Fig. 4(c,d) do not contribute to the unpolarized TMD PDF of hadron H$_{2}$ . 

The uncanceled terms like Eq. (17) would ordinarily signal a breakdown of standard TMD-factorization be- cause they are not consistent with the standard Wil- son line structure in a TMD PDF. This is exactly what is observed for the Abelian gauge theory calculation in Ref. [17]. In our non-Abelian example, however, these single extra gluon contributions to a “factorization anomaly” are exactly zero because they happen to in- clude the trivial color factor: 

So, in our specific non-Abelian calculation, Eqs. (8, 10) are the only eikonal propagators that contribute to an SSA at the level of just one extra gluon. 

The non-vanishing eikonal factors (coming from Figs. 3(a,b)) are exactly what is obtained from an order g expansion of the Wilson lines in the standard definition of the TMD PDFs: 

<img src='content_image/418555.jpg'>

The standard Wilson line operator U [ n$_{1}$ ] jk [0 , w ] is inserted between the two scalar quark fields. The triplet color indices j, k are shown explicitly to emphasize the flow of color. The full Wilson line insertion is, 

<img src='content_image/418560.jpg'>

where 

<img src='content_image/418556.jpg'>

with P denoting a path-ordering operator. The super- script [ n$_{1}$ ] in Eq. (19) refers to the direction of the main leg of the Wilson line, starting from point 0. The ex- tra gluon attachment in Fig. 3(a) on the left side of the cut contributes to [ V$_{0}$ ( n$_{1}$ )]$_{k}$ ′ k while the extra gluon in Fig. 3(b) contributes to [ V † $_{w}$( n$_{1}$ ) ] jj $_{′}$. To close the Wil- son line and ensure that it is exactly gauge invariant, the path needs a transverse link at light-cone infinity [27], 

<img src='content_image/418558.jpg'>

<img src='content_image/418554.jpg'>

where C is a path in the transverse direction connect- ing the points, (0 , ∞ , w$_{t}$ ) and (0 , ∞ , 0$_{t}$ ). In a deriva- tion of TMD-factorization in Feynman gauge, contribu- tions to the link at infinity do not arise explicitly. (See, e.g., Refs. [3, 8, 22] and references therein for a review of the steps for resumming collinear gluon attachments and identifying the resulting gauge links.) 

Note that the Wilson line insertion in the standard TMD PDF of Eq. (19) contracts the color indices of the quark fields, so Φ [ n$_{1}$ ] H$_{1}$ ( x$_{1}$, k$_{1}$$_{T}$ ) has no leftover color indices — it is a real-valued function as is appropriate for a gauge invariant probability density. 

Similar steps result in the single-gluon contribution to the Wilson line in the standard TMD PDF for H$_{2}$ when the extra gluon is radiated from the upper spectator as in Figs. 3(c,d). There the Wilson line insertion instead points in the direction n$_{2}$ = (1 , 0 , 0$_{t}$ ): 

<img src='content_image/418553.jpg'>

In the Abelian case considered in Ref. [17], breaking of standard TMD-factorization was observed for an SSA because the factorization anomaly terms analogous to Eq. (17) did not vanish. There, the “color factor” was the Abelian charge g$_{2}$ for Abelian “gluons” rather than the non-Abelian color factor Tr$_{C}$ [ t $^{a}$] = 0. That the anoma- lous terms vanish at the level of one extra gluon in our non-Abelian example is only due to the highly simple color structure in the particular hard process that we have considered. Generally, when color is exchanged in the hard part, standard TMD-factorization breaking will already appear at the level of one extra gluon. In our example, violations of standard TMD-factorization only appear at the level of two extra gluons or higher. 

Schematically, the standard TMD-factorization for- mula suggested by the sum of one-extra-gluon graphs in Figs. 3(a-d) is 

<img src='content_image/418557.jpg'>

Here H is the same hard factor that appeared at zeroth order in Sect. II. The “!” on the equal sign is to emphasize that this formula is ultimately incorrect, as explained in Refs. [17, 18]. We will see this explicitly at the level two extra gluons in the next section. 

In this section we review the steps for illustrating a violation of standard TMD-factorization at the level of two extra gluons radiated from one of the hadrons. The structure of the resulting TMD-factorization anomaly 

## IV. TWO EXTRA GLUONS 

