Calculating hard probe radiative energy loss beyond the soft-gluon approximation: Examining the approximation validity

The soft-gluon approximation, which implies that radiated gluon carries away a small fraction of initial parton's energy, is a commonly used assumption in calculating radiative energy loss of high momentum partons traversing quark-gluon plasma created at the Relativistic Heavy Ion Collider and the Large Hadron Collider. While the soft-gluon approximation is convenient, different theoretical approaches have reported significant radiative energy loss of high-$p_{\perp }$ partons, thereby questioning its validity. To address this issue, we relaxed the soft-gluon approximation within Djordjevic–Gyulassy–Levai–Vitev (DGLV) formalism. The obtained analytical expressions are quite distinct from the soft-gluon case. However, numerical results for the first order in opacity fractional energy loss lead to small differences in predictions for the two cases. The difference in the predicted number of radiated gluons is also small. Moreover, the effect on these two variables has an opposite sign, which when combined results in almost overlapping suppression predictions. Therefore, our results imply that, contrary to the commonly held doubts, the soft-gluon approximation in practice works surprisingly well in DGLV formalism. Finally, we also discuss generalizing this relaxation in the dynamical QCD medium, which suggests a more general applicability of the conclusions obtained here.

Figure 1

The effect of relaxing the soft-gluon approximation on integrated variables to the first order in opacity of DGLV formalism, as a function of $p_{\perp }$. (a) Comparison of gluon's fractional radiative energy loss without (solid curve) and with (dashed curve) soft-gluon approximation. (b) The relative change of the radiative energy loss when the soft-gluon approximation is relaxed with respect to the soft-gluon limit. (c) Comparison of number of radiated gluons without (solid curve) and with (dashed curve) the soft-gluon approximation. (d) A percentage of radiated gluon number change when the soft-gluon approximation is relaxed.

Figure 2

The effect of relaxing the soft-gluon approximation on differential variables to the first order in opacity of DGLV formalism, as a function of $x$. The comparison of (i) fractional differential gluon radiative energy loss $((1/E)\times (d{E}^{\left(1\right)}/dx))$ and (ii) single-gluon radiation (spectrum) distribution in momentum fraction ($d{N}_g^{\left(1\right)}/dx$) between bsg (solid curve) and $sg$ (dashed curve) case, for different values of initial jet transverse momenta (5, 10, and 50 GeV, as indicated in panels) is shown in the first column ((a), (d), and (g)) and second column ((b), (e), and (h)), respectively. The relative change of the single-gluon radiation spectrum with respect to the soft-gluon limit is shown in panels (c), (f), and (i).

Figure 3

The effect of relaxing the soft-gluon approximation on $d{N}_g^{\left(1\right)}/dx$ for different $p_{\perp }$ values. The relative change of the single-gluon radiation spectrum with respect to soft-gluon case, calculated to the first order in opacity of DGLV formalism, for different values of initial $p_{\perp }$ (as indicated in the legend) is depicted as a function of $x$. The curves fade as transverse momentum increases.

Figure 4

The effect of relaxing the soft-gluon approximation on gluon nuclear modification factor $R_{AA}$ vs $p_{\perp }$. (a) The suppression of the gluon jet beyond the soft-gluon approximation (solid curve) is compared to the soft-gluon $R_{AA}$ (dashed curve) as a function of transverse momentum. (b) Quantification of the effect and its expression in percentage.

Figure 5

The role of initial gluon distribution in constraining the relevant $x$ region. The solid black curve represents initial gluon distribution as a function of $p_{\perp }$ at the LHC [34, 35]. The dot-dashed gray line marks the final gluon transverse momentum, while dotted arrows link parent gluons that lost momentum fraction equal to $x$ with their corresponding initial transverse momenta. The arrows fade as $x$ increases, as indicated in the legend.

Figure 6

Zeroth-order diagram that includes no interaction with the QCD medium and contributes to gluon radiation amplitude to the first order in opacity $L/\lambda$. The dashed circle represents the source $J$, which at longitudinal coordinate $z_0$ produces an off-shell gluon jet, propagating along the $z$ axis. $z$ denotes longitudinal coordinate at which the gluon is radiated. Latin indices denote color charges, while Greek indices denote components of 4-vectors. $k$ denotes 4-momentum of the radiated gluon carrying the color $c$, and $p$ denotes 4-momentum of the final gluon jet carrying the color $d$.

Figure 7

Three diagrams, corresponding to interaction with one static scattering center, that contribute to gluon-jet radiation amplitude to the first order in opacity $L/\lambda$. $z_1$ denotes longitudinal coordinate of the interactions with one scattering center. Crossed circle represents scatterer that exchanges 3D momentum ${\vec{\mathbf{q}}}_1$ with the jet. Note that all three diagrams assume equivalently ordered Latin and Greek indices as indicated in panel (a). Remaining labeling is the same as in Fig. 6.

Figure 8

(a) Feynman diagram $M_{2,2,0}$ and its contribution to the first order in opacity gluon-jet radiative energy loss: (b) contact-limit $M_{2,2,0}^c$. $z_i$, where $i=1,2$, denotes longitudinal coordinate of the interactions with the consecutive scattering centers (or in the contact limit $z_1=z_2$). Crossed circles represent scatterers that exchange 3D momentum ${\vec{\mathbf{q}}}_i$ with the jet, which in the contact-limit case merge into one gridded ellipse. Note that all the following figures assume equivalently ordered Latin and Greek indices as in this figure. Remaining labeling is the same as in Figs. 6 and 7.

Figure 9

Feynman diagrams $M_{2,0,3}$ and $M_{2,0,0}$ in well-separated ((a) and (c)) and in contact-limit case ($z_1=z_2$), which contributes to the first order in opacity of gluon-jet radiative energy loss: $M_{2,0,3}^c$ and $M_{2,0,0}^c$ ((b) and (d)). Remaining labeling is the same as in Fig. 8.

Figure 10

Topologically indistinct Feynman diagrams: (a) $M_{2,0,1}^c$ and (b) $M_{2,0,2}^c$ in contact limit ($z_1=z_2$), which contribute to the first order in opacity of gluon-jet radiative energy loss. Remaining labeling is the same as in Fig. 8.

Figure 11

Illustration of calculating Cauchy principal value ($I_{PV}$) in the case when singularity $(-a)$ on the real (horizontal) axis arises.

Figure 12

Feynman diagrams: (a) $M_{2,1,0}^c$ and (b) $M_{2,1,1}^c$ in contact limit ($z_1=z_2$), which have negligible contribution to the first order in opacity gluon-jet radiative energy loss. Remaining labeling is the same as in Fig. 8.

Figure 13

The counterpart of Fig. 1, when uniform longitudinal distance distribution is considered. The effect of relaxing the soft-gluon approximation on integrated variables to the first order in opacity of DGLV formalism, as a function of $p_{\perp }$. (a) Comparison of gluon's fractional radiative energy loss in the bsg (solid curve) case with the $sg$ (dashed curve) case. (b) The percentage change of the radiative energy loss when the soft-gluon approximation is relaxed with respect to the $sg$ case. (c) Comparison of number of radiated gluons in bsg (solid curve) with $sg$ (dashed curve) case. (d) The relative change of radiated gluon number when the soft-gluon approximation is relaxed with respect to the $sg$ case.

Figure 14

The counterpart of Fig. 2, when uniform longitudinal distance distribution is considered. The effect of relaxing the soft-gluon approximation on differential variables to the first order in opacity of DGLV formalism, as a function of $x$. The comparison of $(1/E)\times (d{E}^{\left(1\right)}/dx)$ and $d{N}_g^{\left(1\right)}/dx$ between the bsg (solid curve) case and the $sg$ (dashed curve) case, for different values of initial jet $p_{\perp }$ (5, 10, and 50 GeV, as indicated in panels) is shown in the first ((a), (d), and (g)) and second ((b), (e), and (h)) columns, respectively. The quantification of the effect on the single-gluon radiation spectrum and its expression in percentage is shown in panels (c), (f) and (i).

Figure 15

The counterpart of Fig. 4, when uniform longitudinal distance distribution is considered. The effect of relaxing the soft-gluon approximation on gluon $R_{AA}$ as a function of final $p_{\perp }$. (a) Comparison of gluon-jet $R_{AA}$ between the bsg (solid curve) case and the $sg$ (dashed curve) case. (b) The quantification of the effect and its expression in percentage.

Figure 16

The two scenarios of relevant $x$ regions for the importance of the soft-gluon approximation. The effect of relaxing the soft-gluon approximation on $R_{AA}$ as a function of $p_{\perp }$. The suppression of gluon jet beyond the soft-gluon approximation (solid curve) is compared to the combined $R_{AA}$ (dot-dashed curve), obtained from (i) bsg expression for $x\le 0.4$ combined with $sg$ expression for $x>0.4$ in panel (a) ((ii) bsg expression for $x\le 0.3$ combined with $sg$ expression for $x>0.3$ in panel (c)). The quantification of the effect and its expression in percentage for these two scenarios are presented in panels (b) and (d), respectively.