Non-Abelian bremsstrahlung and azimuthal asymmetries in high energy $p+A$ reactions

We apply the GLV reaction operator solution to the Vitev-Gunion-Bertsch (VGB) boundary conditions to compute to all orders in nuclear opacity the non-Abelian gluon bremsstrahlung of event-by-event fluctuating beam jets in nuclear collisions. We evaluate analytically azimuthal Fourier moments of single gluon, $v_n^M\{1\}$, and even numbered $2\ell$ gluon distribution, $v_n^M\{2\ell \}$, inclusive distributions in high-energy $p+A$ reactions as a function of harmonic $n$, target recoil cluster number, $M$, and gluon number, $2\ell$, at the RHIC and LHC. Multiple resolved clusters of recoiling target beam jets together with the projectile beam jet form color scintillation antenna (CSA) arrays that lead to characteristic boost-noninvariant trapezoidal rapidity distributions in asymmetric $B+A$ nuclear collisions. The scaling of the intrinsically azimuthally anisotropic and long range in $\eta$ nature of the non-Abelian bremsstrahlung leads to $v_n$ moments that are similar to results from hydrodynamic models, but due entirely to non-Abelian wave interference phenomena sourced by the fluctuating CSA. Our analytic nonflow solutions are similar to recent numerical saturation model predictions but differ by predicting a simple power-law hierarchy of both even and odd $v_n$ without invoking $k_T$ factorization. A test of the CSA mechanism is the predicted nearly linear $\eta$ rapidity dependence of the $v_n(k_T,\eta )$. Non-Abelian beam jet bremsstrahlung may, thus, provide a simple analytic solution to the beam energy scan puzzle of the near $\sqrt{s}$ independence of $v_n(p_T)$ moments observed down to 10 AGeV, where large-$x$ valence-quark beam jets dominate inelastic dynamics. Recoil bremsstrahlung from multiple independent CSA clusters could also provide a partial explanation for the unexpected similarity of $v_n$ in $p(D)+A$ and noncentral $A+A$ at the same $dN/d\eta$ multiplicity as observed at the RHIC and LHC.

Figure 1

$v_n(p_T)$ with $n=2$ to 5 obtained for $|\mathrm{\Delta }\eta |>2$ and the $p_T$ range of 1–3 GeV. An overlay sketch of preliminary rapidity-even $v_1$ data shown at QM14 [14] is also indicated by a dark green curve. The error bars and shaded boxes represent the statistical and systematic uncertainties, respectively. ATLAS $v_2(v_3)$ data in the $220<N_{ch}^{rec}<260$ range are compared to the CMS data [2] obtained by subtracting the peripheral events (the number of offline tracks $N_{off}^{trk}<20$), shown by the dashed (solid) curves. Reproduced from ATLAS Ref. [13] $p+\mathrm{Pb}$ Fig. 24.

Figure 2

Single GB beam jet bremsstrahlung azimuthal Fourier moments, $v_n^{\mathrm{GB}}(k,q)$ from Eq. (11), are shown versus $k/\mu$ for $n=1-5$ for $q/\mu =1(3)$ as solid (dashed) curves.

Figure 3

Single GB beam jet bremsstrahlung azimuthal Fourier moments, $⟨v_n^{\mathrm{GB}}(k,q){⟩}_q$ averaged over $q$ with $M^2/(q^2+M^2{)}^2$, are shown versus $k/\mu$ for $n=1-5$ for $(M/\mu {)}^2=1(10)$ as solid (dashed) curves.

Figure 4

The ideal $1/n$ power scaling of $q$ averaged $⟨v_n^{\mathrm{GB}}(k,q){⟩}^{1/n}$ with $k\lesssim M$ [see Eq. (11)] breaks down at higher $k$, because in the $\mu =0$ limit of non-Abelian bremsstrahlung limits, $k\le q$ [see Eq. (12)].

Figure 5

Schematic diagram corresponding to coherent bremsstrahlung from the projectile dipole from Eqs. (19) and (20). At opacity order $n$, the azimuthal distribution is enhanced for transverse momenta $\mathbf{k}$ near the total accumulated momentum transfer ${\mathbf{Q}}_0\equiv {\mathbf{Q}}_{1n}=\sum _a{\mathbf{Q}}_a$, where $a=1,\dots ,M$ groups of recoiling target dipoles.

Figure 6

(Schematic diagram corresponding to partial coherent backward $\eta <0$ gluon bremsstrahlung from Eq. (23). At opacity order $n$, the azimuthal distribution is enhanced for transverse momenta $\mathbf{k}$ near the recoil momentum transfers $-{\mathbf{Q}}_a$, where $a=1,\dots ,M$ labels incoherent target groups of color dipoles fragmenting into the negative rapidity region.

Figure 7

Example illustrating apparent perfect “triangular flow” but arising entirely from non-Abelian bremsstrahlung sourced by color scintillation antenna (CSA) arrays. In this case, $M=2$ target beam jet clusters recoil off a projectile beam jet with ${\mathbf{Q}}_0=-\sum _{a=1}^M{\mathbf{Q}}_a$, and all $Q_a$ are assumed to have same magnitude but spaced in azimuth by $2\pi /3$. A $Z_3$ CSA radiates only $n=3$ harmonics: $v_n\{2\}(k_1,k_2)={\delta }_{n,3}{v}_3^{\mathrm{GB}}(k_1,Q_0)v_3^{\mathrm{GB}}(k_2,Q_0)$. Part (a) shows an extreme case with $v_3=0.45$, while (b) shows a more realistic $v_3=0.07$ case. An arbitrary isotropic soft nonperturbative background is assumed to be subtracted out.

Figure 8

As in Fig. 7, but for a $Z_5$ symmetric CSA that radiates an apparent “perfect pentatonic flow” pattern with $v_n\{2\}(k_1,k_2)={\delta }_{n,5}{v}_5^{\mathrm{GB}}(k_1,Q_0)v_5^{\mathrm{GB}}(k_2,Q_0)$. Part (a) shows an extreme $v_5=0.45$ case, while part (b) shows a more realistic $v_5=0.03$ case.