Heavy flavor azimuthal correlations in cold nuclear matter

Background: It has been proposed that the azimuthal distributions of heavy flavor quark-antiquark pairs may be modified in the medium of a heavy-ion collision.

Purpose: This work tests this proposition through next-to-leading order (NLO) calculations of the azimuthal distribution $d\sigma /d\varphi$, including transverse momentum broadening, employing $\langle k_T^2\rangle$ and fragmentation in exclusive $Q\overline{Q}$ pair production. While these studies were done for $p+p,p+\overline{p}$, and $p+\mathrm{Pb}$ collisions, understanding azimuthal angle correlations between heavy quarks in these smaller, colder systems is important for their interpretation in heavy-ion collisions.

Methods: First, single inclusive $p_T$ distributions calculated with the exclusive hvqmnr code are compared to those calculated in the fixed-order next-to-leading logarithm approach. Next the azimuthal distributions are calculated and sensitivities to $\langle k_T^2\rangle ,p_T$ cut, and rapidity are studied at $\sqrt{s}=7$ TeV. Finally, calculations are compared to $Q\overline{Q}$ data in elementary $p+p$ and $p+\overline{p}$ collisions at $\sqrt{s}=7$ and 1.96 TeV as well as to the nuclear modification factor $R_{p\mathrm{Pb}}\left(p_T\right)$ in $p+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=5.02$ TeV measured by ALICE.

Results: The low $p_T$ ($p_T<10$ GeV) azimuthal distributions are very sensitive to the $k_T$ broadening and rather insensitive to the fragmentation function. The NLO contributions can result in an enhancement at $\varphi \sim 0$ absent any other effects. Agreement with the data was found to be good.

Conclusions: The NLO calculations, assuming collinear factorization and introducing $k_T$ broadening, result in significant modifications of the azimuthal distribution at low $p_T$ which must be taken into account in calculations of these distributions in heavy-ion collisions.

Figure 1

Leading order $Q\overline{Q}$ production diagrams with (a) the $q\overline{q}$ initial state and (b)–(d) the $gg$ contributions.

Figure 2

Examples of real contributions to next-to-leading order $Q\overline{Q}$ production. Diagrams (a)–(c) illustrate contributions to $gg\to Q\overline{Q}g$ while (d) shows an example of $qg\to qQ\overline{Q}$ production.

Figure 3

The fragmentation functions used in the hvqmnr code and fonll for (a) charm and (b) bottom are compared. The red curves show the standard Peterson function parameter while the black curves are calculated with the values of ${\epsilon }_P$ used in this paper. The fonll results are shown by the dashed blue curves. For charm quarks, the total fonll contribution to $D^0$ fragmentation includes the vector (V) and pseudoscalar (PS) contributions, shown separately.

Figure 4

The single inclusive (a) charm and (b) bottom quark distributions in $\sqrt{s}=7$ TeV $p+p$ collisions at next-to-leading order using the hvqmnr code. The charm distributions are given at forward rapidity, $2.5<y<5$, while the bottom quark distributions are given at midrapidity, $\left|y\right|<2.4$. Results are shown for various combinations of $\langle k_T^2\rangle$ [Eq. (3)] and ${\epsilon }_P$ [Eq. (4)].

Figure 5

The single inclusive (a) $D^0$ and (b) $b$-quark hadron, $H_b$, distributions in $\sqrt{s}=7$ TeV $p+p$ collisions are compared to data from LHCb [22] at $2.5<y<3$ and ATLAS [43] at $\left|\eta \right|<2.5$ respectively. The curves show the extent of the uncertainty bands calculated employing Eqs. (7) and (8). The hvqmnr code (blue dashed curves) utilizes $(\langle k_T^2\rangle \left({\mathrm{GeV}}^2\right),{\epsilon }_P)=(1.5,0.008)$ for charm and (3,0.0004) for bottom. The corresponding fonll uncertainty band (red curves) is also shown. The same quark mass and scale parameters are used in both calculations.

Figure 6

The NLO azimuthal distribution between two heavy quarks, $d\sigma /d\varphi$ in $p+p$ collisions at $\sqrt{s}=7$ TeV using the hvqmnr code for (a) $c\overline{c}$ pairs at forward rapidity, $2.5<y<5$, and (b) $b\overline{b}$ pairs at midrapidity, $\left|y\right|<2.4$. The results are shown for the same combinations of $\langle k_T^2\rangle$ in Eq. (3) and ${\epsilon }_P$ in Eq. (4) as in Fig. 4.

Figure 7

The ratio of azimuthal angle distributions for the same $\langle k_T^2\rangle$ and ${\epsilon }_P$ combinations as in Fig. 6 relative to $\langle k_T^2\rangle =0$, ${\epsilon }_P=0$ for (a) $c\overline{c}$ pairs at $2.5<y<5$ and (b) $b\overline{b}$ pairs at $\left|y\right|<2.4$.

Figure 8

The mass and scale uncertainty band on the NLO azimuthal distributions in $p+p$ collisions at $\sqrt{s}=7$ TeV using the hvqmnr code for (a) $c\overline{c}$ pairs at forward rapidity, $2.5<y<5$, and (b) $b\overline{b}$ pairs at midrapidity, $\left|y\right|<2.4$. The calculations utilize $(\langle k_T^2\rangle \left({\mathrm{GeV}}^2\right),{\epsilon }_P)=(1.5,0.008)$ for charm and (3,0.0004) for bottom. The solid curves show the central results; the dashed show the edges of the uncertainty band; and the light dot-dashed curves are the individual mass and scale combinations.

Figure 9

The effect of changing the $p_T$ cut on the azimuthal angle distribution between the two heavy quarks. The calculations utilize $(\langle k_T^2\rangle \left({\mathrm{GeV}}^2\right),{\epsilon }_P)=(1.5,0.008)$ for charm (a) and (3,0.0004) for bottom (b). The $c\overline{c}$ distributions are shown at forward rapidity, $2.5<y<5$, while the $b\overline{b}$ distributions are at midrapidity, $\left|y\right|<2.4$. From top to bottom the results are $p_T<10$ GeV, $p_T>10$ GeV, $p_T>20$ GeV, $p_T>30$ GeV, $p_T>40$ GeV, $p_T>50$ GeV, $p_T>75$ GeV. The $c\overline{c}$ curves for all but the lowest $p_T$ cut are scaled up by ${10}^3$. The $b\overline{b}$ curves, which do not include a scale factor, also include a calculation with $p_T>100$ GeV.

Figure 10

The azimuthal angle distributions for (a) $c\overline{c}$ and (b) $b\overline{b}$ pairs in the central rapidity range $\left|y\right|<2.4$ with $p_T<10$ GeV. Calculations are shown with $\langle k_T^2\rangle =0$ and for values of $\mathrm{\Delta }$ from $-3/2$ to 1 for charm and $\mathrm{\Delta }=-1/2$ to 1 for bottom, as in Eq. (9).

Figure 11

The ratio of azimuthal angle distributions relative to that of $\langle k_T^2\rangle =0$ for (a) $c\overline{c}$ and (b) $b\overline{b}$ pairs in the central rapidity range $\left|y\right|<2.4$ with $p_T<10$ GeV. Calculations are shown for values of $\mathrm{\Delta }$ from $-3/2$ to 1 for charm and $\mathrm{\Delta }=-1/2$ to 1 for bottom, as in Eq. (9).

Figure 12

The azimuthal angle distributions for (a) $c\overline{c}$ and (b) $b\overline{b}$ pairs in the central rapidity range $\left|y\right|<2.4$ with $p_T>10$ GeV. Calculations are shown with $\langle k_T^2\rangle =0$ and for values of $\mathrm{\Delta }$ from $-3/2$ to 1 for charm and $\mathrm{\Delta }=-1/2$ to 1 for bottom, as in Eq. (9).

Figure 13

The ratio of azimuthal angle distributions relative to that of $\langle k_T^2\rangle =0$ for (a) $c\overline{c}$ and (b) $b\overline{b}$ pairs in the central rapidity range $\left|y\right|<2.4$ with $p_T>10$ GeV. Calculations are shown for values of $\mathrm{\Delta }$ from $-3/2$ to 1 for charm and $\mathrm{\Delta }=-1/2$ to 1 for bottom, as in Eq. (9).

Figure 14

The azimuthal angle distributions for $D^0{\overline{D}}^0$ (red), $D^0{D}^{-}$ (blue), and $D^{+}{D}^{-}$ (magenta) pairs measured in $p+p$ collisions at $\sqrt{s}=7$ TeV by LHCb [48]. The data are compared to calculations in the same acceptance with $\langle k_T^2\rangle =0$ and for values of $\mathrm{\Delta }$ from $-3/2$ to 1 in Eq. (9).

Figure 15

The invariant mass distributions for $D^0{\overline{D}}^0$ (red), $D^0{D}^{-}$ (blue), and $D^{+}{D}^{-}$ (magenta) pairs measured in $p+p$ collisions at $\sqrt{s}=7$ TeV by LHCb [48]. The data are compared to calculations in the same acceptance with $\langle k_T^2\rangle =0$ and for values of $\mathrm{\Delta }$ from $-3/2$ to 1 in Eq. (9).

Figure 16

The $\left|\mathrm{\Delta }y\right|$ distributions for $D^0{\overline{D}}^0$ (red), $D^0{D}^{-}$ (blue), and $D^{+}{D}^{-}$ (magenta) pairs measured in $p+p$ collisions at $\sqrt{s}=7$ TeV by LHCb [48]. The data are compared to calculations in the same acceptance with $\langle k_T^2\rangle =0$ and for values of $\mathrm{\Delta }$ from $-3/2$ to 1 in Eq. (9).

Figure 17

The azimuthal angle distributions for $D^0{D}^{*-}$ (red) and $D^{+}{D}^{*-}$ (blue) pairs measured in $p+\overline{p}$ collisions at $\sqrt{s}=1.96$ TeV by CDF [50]. The data are compared to calculations in the same acceptance with $\langle k_T^2\rangle =0$ and for values of $\mathrm{\Delta }$ from $-3/2$ to 1 in Eq. (9).

Figure 18

The azimuthal angle distributions for $B\overline{B}$ pairs measured in $p+p$ collisions at $\sqrt{s}=7$ TeV by CMS [49]. The $p_T^{\mathrm{jet}}$ lower limits of 56 GeV (red), 84 GeV (blue), and 120 GeV (black) are scaled by factors of 4, 2, and 1 respectively. The data are compared to calculations in the same kinematics for $\langle k_T^2\rangle =0$ (blue curves) and for values of $\mathrm{\Delta }$ from $-1/2$ to 1 in Eq. (9).

Figure 19

The $\mathrm{\Delta }R$ distributions for $B\overline{B}$ pairs measured in $p+p$ collisions at $\sqrt{s}=7$ TeV by CMS [49]. The $p_T^{\mathrm{jet}}$ lower limits of 56 GeV (red), 84 GeV (blue), and 120 GeV (black) are scaled by factors of 4, 2, and 1 respectively. The data are compared to calculations in the same kinematics for $\langle k_T^2\rangle =0$ (blue curves) and for values of $\mathrm{\Delta }$ from $-1/2$ to 1 in Eq. (9), the green dotted to solid red curves respectively.

Figure 20

The nuclear modification factor $R_{p\mathrm{Pb}}\left(p_T\right)$ in 5.02-TeV $p+\mathrm{Pb}$ collisions for $-0.96<y_{\mathrm{c}.\mathrm{m}.}<0.04$. The ALICE data for the average of their $D^0,D^{+}$, and $D^{*+}$ measurements [52] at central rapidity are shown. Results are shown for no $k_T$ broadening and no fragmentation (solid blue); the standard result from Sec. 3, $\langle k_T^2\rangle =1.5$ ${\mathrm{GeV}}^2$ and ${\epsilon }_P=0.008$ (red dashed); and $\langle k_T^2\rangle =3$ ${\mathrm{GeV}}^2$ and ${\epsilon }_P=0.008$ (magenta dot-dashed). The EPS09 NLO uncertainty band, along with the central value, is shown. Note that the additional $k_T$ broadening is only applied to $p+\mathrm{Pb}$ collisions and not to $p+p$ collisions in the magenta curves.

Figure 21

The nuclear modification factor $R_{p\mathrm{Pb}}\left(\varphi \right)$ in 5.02 TeV $p+\mathrm{Pb}$ collisions. Results are shown for no $k_T$ broadening and no fragmentation (solid blue); the standard result from Sec. 3, $\langle k_T^2\rangle =1.5$ ${\mathrm{GeV}}^2$ and ${\epsilon }_P=0.008$ (red dashed); and $\langle k_T^2\rangle =3$ ${\mathrm{GeV}}^2$ and ${\epsilon }_P=0.008$ (magenta dot-dashed). The EPS09 NLO uncertainty band is shown along with the central value. No $p_T$ or rapidity cuts have been imposed. Note that the additional $k_T$ broadening in the magenta curves is only applied to $p+\mathrm{Pb}$ collisions and not to $p+p$ collisions.