Factorization and its Breaking in Dijet Single Transverse Spin Asymmetries in $pp$ Collisions

There has been a strong interest in correlated dijet production in various hadronic collisions Abazov:2004hm ; Abelev:2007ii ; Khachatryan:2011zj ; daCosta:2011ni ; Aad:2010bu ; Chatrchyan:2011sx ; Adamczyk:2013jei ; Adamczyk:2017yhe ; Aaboud:2019oop , where the two jets are produced mainly in the back-to-back configuration in the transverse plane,

Here $A$ and $B$ represent the two incoming hadrons with momenta $P_{A}$ and $P_{B}$, respectively. The azimuthal angle between the two jets is defined as $\phi=\phi_{1}-\phi_{2}$ with $\phi_{1,2}$ being the azimuthal angles of the two jets. In the leading order naive parton picture, the Born diagram yields a delta function at $\phi=\pi$. One-loop gluon radiation will lead to a singular distribution around $\phi=\pi$. This divergence arises when the total transverse momentum of the dijet (imbalance) is much smaller than the individual jet momentum, $q_{\perp}=|\vec{k}_{1\perp}+\vec{k}_{2\perp}|\ll|k_{1\perp}|\sim|k_{2\perp}|\sim P_{T}$, where large logarithms appear at every order of the perturbative calculation. In the kinematic region $q_{\perp}\ll P_{T}$, the appropriate resummation method that needs to be applied is the so-called transverse momentum dependent (TMD) resummation or the Collins-Soper-Sterman (CSS) resummation Collins:1984kg . There have been several theoretical efforts to resum the large logarithms for this process Banfi:2003jj ; Banfi:2008qs ; Hautmann:2008vd ; Chiu:2012ir ; Mueller:2013wwa ; Sun:2014gfa ; Sun:2015doa ; Chien:2020hzh . The differential cross section can be written as,

where $d\Omega=dy_{1}dy_{2}dP_{T}^{2}$ represents the phase space of dijet production. Here $y_{1}$ and $y_{2}$ are the rapidities of the two jets, $P_{T}$ is the leading jet transverse momentum, and $q_{\perp}$ the imbalance transverse momentum between the two jets as defined above. Moreover, $\sigma_{0}$ is the overall normalization of the differential cross section. The first term on the right hand side, $W_{ab\to cd}$, contains the all order resummation and the second term, $Y_{ab\to cd}$, takes into account fixed order corrections. At next-to-leading logarithmic (NLL) order, the resummation for $W$ was conjectured to take the following form Sun:2014gfa ; Sun:2015doa

for each partonic channel $ab\to cd$, where $Q^{2}=\hat{s}=x_{1}x_{2}S$, representing the hard momentum scale. In addition, we have $b_{0}=2e^{-\gamma_{E}}$, with $\gamma_{E}$ being the Euler constant. The $f_{a,b}(x,\mu)$ are the parton distributions for the incoming partons $a,b$, and $x_{1,2}=P_{T}\left(e^{\pm y_{1}}+e^{\pm y_{2}}\right)/\sqrt{S}$ are the fractions of the incoming hadrons’ momenta carried by the partons. In the above equation, the hard and soft factors $\mathbf{H}$ and $\mathbf{S}$ are expressed as matrices in the color space of the partonic channel $ab\to cd$, and $\gamma_{ab\to cd}^{s}$ are the associated anomalous dimensions for the soft factor. The Sudakov form factor ${\cal S}_{\textrm{Sud}}$ resums the leading double logarithms and the universal sub-leading logarithms,

where $R_{1,2}$ are the jet radii of the two jets, respectively. In practice the jets are of course reconstructed with the same radius $R$ but to clarify the structure of our calculation we use two different radii $R_{1,2}$ to differentiate between the dijets. At one-loop order, $A=C_{A}\frac{\alpha_{s}}{\pi}$, $B=-2C_{A}\beta_{0}\frac{\alpha_{s}}{\pi}$ for a gluon-gluon initial state, $A=C_{F}\frac{\alpha_{s}}{\pi}$, $B=\frac{-3C_{F}}{2}\frac{\alpha_{s}}{\pi}$ for a quark-quark initial state, and $A=\frac{(C_{F}+C_{A})}{2}\frac{\alpha_{s}}{\pi}$, $B=(\frac{-3C_{F}}{4}-C_{A}\beta_{0})\frac{\alpha_{s}}{\pi}$ for a gluon-quark initial state. In addition, $D_{1,2}=C_{A}\frac{\alpha_{s}}{2\pi}$ for a gluon jet and $D_{1,2}=C_{F}\frac{\alpha_{s}}{2\pi}$ for a quark jet, respectively. Here, $\beta_{0}=(11-2N_{f}/3)/12$, with $N_{f}$ being the number of effective light quarks.

The resummation formula in Eq. (3) was obtained in Refs. Sun:2014gfa ; Sun:2015doa by a detailed analysis of the soft gluon radiation at one-loop order. The leading contributions from soft gluon radiation can be factorized into the associated TMD parton distributions and can be resummed by solving the relevant evolution equations. At NLL, the soft gluon radiation is factorized into the soft factor ${\bf S}$ which is given by a matrix in the color space of the partonic channels. The matrix form of the factorization is the same as was found for threshold resummation for the dijet production in proton-proton collisions Kidonakis:1998bk ; Kidonakis:1998nf ; Kidonakis:2000gi ; Kelley:2010fn ; Catani:2013vaa ; Hinderer:2014qta .

It is known that TMD factorization in dijet production in hadronic collisions is highly nontrivial and that there are potential factorization breaking effects Boer:2003tx ; Qiu:2007ey ; Collins:2007nk ; Rogers:2010dm ; Bacchetta:2005rm ; Vogelsang:2007jk ; Bomhof:2007su ; Catani:2011st ; Mitov:2012gt ; Schwartz:2017nmr ; Schwartz:2018obd . First, non-global logarithms (NGLs) Dasgupta:2001sh ; Dasgupta:2002bw start to contribute to the cross section at two-loop order. It has been shown that they cannot easily be included into a factorization formula, although numerical simulations can be made and their contribution can be taken into account Banfi:2003jj ; Banfi:2008qs . In addition, TMD factorization will be explicitly broken at three-loop order for the unpolarized cross section. This leads to a modification of the coefficient $A^{(3)}$ in the above Sudakov form factor Collins:2007nk ; Becher:2010tm ; Catani:2011st ; Mitov:2012gt ; Rothstein:2016bsq ; Schwartz:2017nmr ; Schwartz:2018obd ; Sun:2015doa .

Factorization breaking effects are particularly evident for the single transverse spin asymmetry (SSA) in dijet production Collins:2007nk , $\Delta\sigma(S_{\perp})=(\sigma(S_{\perp})-\sigma(S_{\perp}))/2$, where $S_{\perp}$ represents the transverse polarization vector for one of the incoming nucleons. The SSA for this process is expressed as $\Delta\sigma(S_{\perp})\propto\epsilon^{\alpha\beta}S_{\perp}^{\alpha}q_{\perp}^{\beta}$, i.e., the total transverse momentum of the two jets $q_{\perp}$ will have a preferred direction Boer:2003tx . This asymmetry is sensitive to the parton’s Sivers function where the transverse momentum distribution is correlated with the transverse polarization vector Sivers:1989cc . In Refs. Bacchetta:2005rm ; Bomhof:2007su , all initial/final state interaction contributions to the SSA were factorized into a complicated gauge link structure associated with the quark Sivers function for the polarized nucleon. However, for the double spin asymmetries involving two Sivers functions, it was shown explicitly that the generalized gauge-link approach to TMD factorization does not apply Rogers:2010dm .

Ref. Qiu:2007ey provided an understanding of the SSA from the twist-three framework where the Qiu-Sterman-Efremov-Tereyav matrix elements are the basic ingredients Efremov:1981sh ; Efremov:1984ip ; Qiu:1991pp ; Qiu:1991wg ; Qiu:1998ia , and the high momentum Sivers function is generated by collinear gluon radiation. In particular, it was shown in Refs. Qiu:2007ey ; Efremov:1984ip that the collinear gluon radiations parallel to the incoming hadrons can be factorized into the associated TMD parton distribution functions. It was also suggested that a factorization formula similar to that in the unpolarized case may hold for the single spin dependent differential cross section Qiu:2007ey ,

Here $f_{1Tb}^{\perp(\rm SIDIS)}$ and $f_{a}^{(\rm SIDIS)}$ denote the transverse-spin dependent TMD quark Sivers function and the unpolarized TMD parton distribution, respectively. These TMD parton distribution functions were chosen following their definitions in semi-inclusive deep-inelastic scattering (SIDIS) with future pointing gauge links. Although it was not explicitly shown in Ref. Qiu:2007ey , a matrix form of the factorization was suggested, where $H_{ab\to cd}^{\rm Sivers}$ and $S_{ab\to cd}$ are partonic hard and soft factors and the $[\quad]_{c}$ represents a trace in color space between the hard and soft factors, similar to the unpolarized case in Eq. (3).

In order to check the factorization formula of Eq. (I), it is important to carry out the calculation of soft gluon radiation. Soft gluon emissions contribute in a nontrivial way to the factorization formula. In particular, it will be crucial to show that these contributions can be included in the soft factor in the matrix form of the factorization formula. The goal of the current paper is to derive the soft gluon radiation contribution at one-loop order. As mentioned above, in Ref. Qiu:2007ey it was shown that collinear gluon radiation associated with the incoming nucleons can be treated following the general factorization arguments. This indicates that factorization holds in the leading logarithmic approximation (LLA). However, in order to obtain also all the subleading logarithmic contributions, we need to consider the soft gluon radiation as well. After including soft gluon radiation, we will obtain the complete double logarithmic result.

Our calculations presented in this work show that the factorization and resummation is expected to be valid at LLA. However, factorization breaking effects will emerge at NLL accuracy, in the sense that the contributions from soft gluon radiation cannot be factorized into the same soft factor as for the unpolarized case. This implies that beyond the LLA, we do not have a factorization formula for $\Delta\sigma$ as in Eq. (3), at least not in the standard way with a spin independent soft factor.

In the LLA, we can express the spin dependent differential cross section in terms of the Fourier transform $b_{\perp}$ variable,

Here, we neglect the $Y$-term contribution compared to the unpolarized case above. In this work we show that the leading logarithmic factorization of $W^{T\beta}$ takes the form

where $S_{\rm Sud}^{T}(Q^{2},b_{\perp})$ can be written in analogy to Eq. (4),

Here $A$, $B$, $D_{1,2}$ are the same as in Eq. (4). In the above equation, $T_{F}$ is the Qiu-Sterman matrix element which is also related to the transverse momentum-moment of the quark Sivers function. It is defined as follows

where $F^{\mu\nu}$ represents the gluon field strength tensor. From the leading order derivation, we have $\frac{1}{M_{P}}\int d^{2}k_{\perp}\,\vec{k}^{2}_{\perp}\,f_{1T}^{\perp(\rm SIDIS)}(x,k_{\perp})=-T_{F}(x,x)$. For our soft gluon calculation at leading power, we take the correlation limit, i.e., we neglect all power corrections of the form $q_{\perp}/P_{T}$. In this limit, the leading double logarithm is proportional $1/q_{\perp}^{2}\times\ln(P_{T}^{2}/q_{\perp}^{2})$. We will show that these contributions will be consistent with the resummation formula of Eq. (7). Some sub-leading logarithmic terms can be factorized in this form as well. These include collinear gluon radiation associated with the incoming hadrons and the final state jets. The former can be resummed by including the scale evolution of the integrated parton distribution and the Qiu-Sterman matrix elements, e.g., by evaluating these distributions at the scale $\mu_{b}=b_{0}/b_{\perp}$. The latter are taken into account by the $\ln(1/R^{2})$ terms in the Sudakov form factor $S_{\rm Sud}$ of Eq. (4). Therefore, Eq. (7) is an improvement of the leading logarithmic approximation, to which we will refer as $\rm LLA^{\prime}$ in the following.

The remainder of this paper is organized as follows. In Sec. II, we will briefly review the soft gluon contribution to unpolarized dijet production. The basic formalism, including the Eikonal approximation, the phase space integrals to obtain the leading contributions, and the subtraction method to derive the soft gluon radiation associated with the final state jets will be introduced. Sec. III contains the main new derivations of this work. We will carry out the calculation of the soft gluon radiation for the spin dependent differential cross sections. We introduce the general framework, the twist-three collinear expansion, and derive the soft gluon radiation amplitude in this formalism. We apply these techniques to different partonic channels and demonstrate that the leading double logarithmic contributions factorize, and we verify our resummation formula at LLA^′. In Sec. IV, we consider an example and show that factorization breaking effects appear at NLL, in particular, from soft gluon radiation that does not belong to the incoming hadrons and final state jets. In Sec. V, we will present phenomenological results for the single spin asymmetries in dijet production at RHIC, and compare to recent STAR data Abelev:2007ii ; starpreliminary . Finally, we will summarize our paper in Sec. VI.

Dijet production at the leading order can be calculated from partonic $2\to 2$ processes,

where $p_{1,2}$ and $k_{1,2}$ are the momenta of the incoming and outgoing two partons, respectively. Their contributions to the cross section can be written as

where the overall normalization of the differential cross section is $\sigma_{0}=\frac{\alpha_{s}^{2}\pi}{s^{2}}$. The partonic cross sections $h^{(0)}$ for all the production channels depend on the kinematic variables $\hat{s}=(p_{1}+p_{2})^{2}$, $\hat{t}=(p_{1}-k_{1})^{2}$ and $\hat{u}=(p_{1}-k_{2})^{2}$. As mentioned above, at the leading order, they contribute to a delta function setting $q_{\perp}=0$, which corresponds to the back-to-back configuration of the two jets in the transverse plane. For soft gluon emissions, we can apply the leading power expansion and derive the dominant contribution from the Eikonal approximation Mueller:2013wwa ; Sun:2015doa . For example, for the outgoing quark, antiquark and gluon lines, we obtain the following factors in the Eikonal approximation:

respectively, at one-loop order. Here $g$ is the strong coupling and the $k_{i}$ represent the momenta of the outgoing particles. For incoming quark, antiquark and gluon lines, we have,

respectively, where $p_{1}$ corresponds to the momentum of the incoming particle.

Following Ref. Mueller:2013wwa ; Sun:2015doa , we choose physical polarizations of the soft gluon along the incoming particle with momentum $p_{2}$, so that the polarization tensor of the radiated gluon takes the following form:

This choice will simplify the derivation since there is no soft gluon radiation from the incoming parton line $p_{2}$. Therefore, as shown in Fig. 1, the leading contributions come from the initial state radiation from the line with momentum $p_{1}$, and the final state emissions from the lines $k_{1}$ and $k_{2}$. The contributions by these diagrams can be evaluated by taking the amplitudes squared of the Eikonal vertex with the polarization vector of the radiated gluon contracted with the above tensor. This leads to the following expressions for the soft gluon radiation contributions,

Here $S_{g}(p,q)$ is a short-hand notation for

When we integrate out the phase space of the radiated gluon to obtain the finite transverse momentum for the dijet imbalance, we have to exclude the contributions that belong to the jets. Therefore, only gluon radiation outside of the jets with radius $R_{1,2}$ contributes. These diagrams have been calculated in Refs. Sun:2015doa , where an offshellness was considered to regulate the collinear divergence associated with the jet within the narrow jet approximation Aversa:1988vb ; Mukherjee:2012uz . In Ref. Liu:2018trl ; Liu:2020dct , a subtraction method was employed to derive the soft gluon radiation contribution. For completeness, we show details of the derivation in the Appendix. Here, we list the final results:

Compared to the results in Ref. Sun:2015doa , the above results differ by a term proportional to $\epsilon\,\pi^{2}/6$. This is a result of the approximation made in Ref. Sun:2015doa .

In this section, we will investigate the soft gluon radiation contribution to the SSA in dijet production. The leading order analysis and collinear gluon radiation contributions have been studied in Ref. Qiu:2007ey . In the following, we will first review the leading order results and then derive the soft gluon radiation contribution.