Finite-mass effects on inclusive $B$-meson hadroproduction

We calculate the transverse-momentum ($p_T$) distribution for the inclusive hadroproduction of $B$ mesons at intermediate values of $p_T$ at next-to-leading order (NLO) in a dedicated finite-mass scheme using realistic nonperturbative fragmentation functions that are obtained through a global fit to $e^{+}{e}^{-}$ data from CERN LEP1 and SLAC SLC exploiting their universality and scaling violations. We find that finite-mass effects moderately enhance the cross section, by about 20% at $p_T=2m_b$, and rapidly fade out with increasing value of $p_T$, so that the zero-mass prediction is reached. We also perform comparisons with recent $p\overline{p}$ data taken by the CDF Collaboration in run II at the Fermilab Tevatron and comment on the usefulness of the fixed-flavor-number scheme.

Figure 1

Comparisons of the ALEPH [22] (circles), OPAL [23] (squares), and SLD [24] (triangles) data with the NLO fits using (a) the Peterson ansatz (1) and (b) the power ansatz (2). The initial factorization scale for all partons is ${\mu }_0=m=4.5\mathrm{GeV}$. The different symbols can be better distinguished in the electronic edition, where the figures can be enlarged.

Figure 2

Examples of Feynman diagrams leading to contributions of (a) class (i), (b) class (ii), and (c) class (iii).

Figure 3

Transverse-momentum distribution $d\sigma /d{p}_T$ of $p\overline{p}\to B+X$ at c.m. energy $\sqrt{S}=1.96\mathrm{TeV}$ integrated over the rapidity range $|y|<1$. The contributions of class (i) evaluated at LO in the ZM-VFNS (dashed line) and the GM-VFNS (solid line), but with the NLO versions of ${\alpha }_s$, the PDFs, and the FFs, are compared.

Figure 4

Transverse-momentum distribution $d\sigma /d{p}_T$ of $p\overline{p}\to B+X$ at c.m. energy $\sqrt{S}=1.96\mathrm{TeV}$ integrated over the rapidity range $|y|<1$. The contributions of class (i) (solid lines) and their $gg$-initiated parts (dashed lines) evaluated at NLO in the ZM-VFNS (upper lines) and the GM-VFNS (lower lines) are compared.

Figure 5

Transverse-momentum distribution $d\sigma /d{p}_T$ of $p\overline{p}\to B+X$ at c.m. energy $\sqrt{S}=1.96\mathrm{TeV}$ integrated over the rapidity range $|y|<1$. The total NLO result in the GM-VFNS including classes (i)–(iii) (solid line) is broken up into the contributions from initial states consisting of (1) one gluon and one $b$ (anti)quark (upper dashed line); (2) one $q$ (anti)quark and one $b$ (anti)quark (middle dashed line); (3) two $b$ (anti)quarks (lower dashed line); (4) one gluon and one $q$ (anti)quark or two $q$ (anti)quarks; and (5) two gluons.

Figure 6

Transverse-momentum distribution $d\sigma /d{p}_T$ of $p\overline{p}\to B+X$ at c.m. energy $\sqrt{S}=1.96\mathrm{TeV}$ integrated over the rapidity range $|y|<1$. The central NLO prediction with ${\xi }_R={\xi }_F=1$ (solid line) of the GM-VFNS is compared with CDF data from Refs. 2 (open squares) and 3 (solid squares). The maximum and minimum values obtained by independently varying ${\xi }_R$ and ${\xi }_F$ in the range $1/2\le {\xi }_R$, ${\xi }_F\le 2$ with the constraint that $1/2\le {\xi }_R/{\xi }_F\le 2$ are also indicated (dashed lines).

Figure 7

Transverse-momentum distribution $d\sigma /d{p}_T$ of $p\overline{p}\to B+X$ at c.m. energy $\sqrt{S}=1.96\mathrm{TeV}$ integrated over the rapidity range $|y|<1$. The central NLO predictions in the FFNS with $n_f=4$ and without FFs (dot-dashed line), the ZM-VFNS (dashed line), and the GM-VFNS (solid line) are compared with CDF data from Refs. 2 (open squares) and 3 (solid squares). For reference, the historical FFNS prediction, evaluated with PDF set MRSD0 [33], a $b\to B$ FF of Peterson type [20] with $ϵ=0.006$, $m_b=4.75\mathrm{GeV}$, and ${\Lambda }_{\overline{\mathrm{MS}}}^{(4)}=215\mathrm{MeV}$, is also shown.

Figure 8

Transverse-momentum distribution $d\sigma /d{p}_T$ of $p\overline{p}\to B+X$ at c.m. energy $\sqrt{S}=1.96\mathrm{TeV}$ integrated over the rapidity range $|y|<0.6$. The central NLO predictions in the FFNS with $n_f=4$ and without FFs (dot-dashed line), the ZM-VFNS (dashed line), and the GM-VFNS (solid line) are compared in the large-$p_T$ range.