Ⅱ-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 51 (2018)

Covariant calculation of quark distribution functions in ρ+ -meson†
Y. Ninomiya,∗1 W. Bentz,∗1,∗2 and I. C. Cloët∗3(λ)(λ)

where q (λ) (x) = q↑ + q↓ (x) is the unpolarized valence quark distribution in a longitudinally polarized
ρ+ meson with spin projection λ = 0 and λ = ±1. Accordingly, the function fLL (x) satisﬁes the sum rules∫ 1∫ 1dx fLL (x) =x dx fLL (x) = 0 ,00

which simply mean that the valence quark number andmomentum are independent of the hadron’s spin state.
Our result for the spin sum at the model scale is∫ 1dx g(x) = 0.56.
0This means that the total valence quark and antiquark contribution to the spin of the ρ meson is 56%,
†
∗1
∗2
∗3

Condensed from an article by Yu Ninomiya et al., Phys. Rev.
C 96, 045206 (2017).
Department of Physics, Tokai University
RIKEN Nishina Center
Physics Division, Argonne National Laboratory- 119 -f (x)1.5

fLL (x)Rho PDFsg(x)
h(x)

1.0

0.5

000.2

0.4x0.60.81Fig. 1. The unpolarized (f (x)), helicity (g(x)), transversity(h(x)) and tensor polarized (fLL (x)) PDFs of the ρ+
meson at the model scale Q20 = 0.16 GeV2 .

Rho PDFs

Spin-1 hadrons are characterized by three elastic
form factors (FFs) and four leading-twist parton distribution functions (PDFs). As compared to the familiar
spin-1/2 case, the new functions for the spin-1 case are
the quadrupole FF and the tensor polarized PDF, bothcarrying important information on the orbital motionof the constituents. Recent calculations1) have shownthat the quadrupole form factor of the ρ+ meson isof opposite sign compared to the deuteron. It is thenof interest to compare their tensor polarized PDFs aswell.
In this work we use the Nambu-Jona-Lasinio (NJL)
model as an eﬀective quark theory of QCD to calculate the nine T -even transverse momentum-dependent
quark distribution functions (TMDs) of the ρ+ meson in a completely covariant way. Important eﬀectsof quark conﬁnement are included via the use of theproper-time regularization scheme, which avoids unphysical thresholds for the decay of hadrons into freequarks. Integration of the TMDs over the transverse
quark momentum components leaves the four PDFs,
namely the unpolarized f (x), helicity g(x), transversity h(x), and tensor polarized fLL (x) distributionfunctions.2) Our results for these four PDFs are shownin Fig. 1 at the model scale of Q20 = 0.16 GeV2 , and inFig. 2 at the scale Q2 = 5 GeV2 .
Our function fLL (x) is related to the function b1 (x)of Ref. 4) by b1 (x) = 12 fLL (x), and its partonic interpretation at the model scale can be expressed as)
1 ( (1)fLL (x) = q (0) (x) −q (x) + q (−1) (x) ,20.35

x f (x)

0.30

x fLL (x)0.25

x g(x)x h(x)

0.20
2Q = 5 GeV2

0.15
0.10
0.05
0−0.05
00.2

0.4

x

0.6

0.8

1Fig. 2. The four PDFs of the ρ+ meson, multiplied by x, atthe scale Q2 = 5 GeV2 . The NLO non-singlet DGLAP
evolution was performed3) to evolve the PDFs of Fig. 1from 0.16 GeV2 to 5 GeV2 .

which implies a substantial contribution of 44% fromthe quark orbital angular∫ 1 momentum. Our result for
the tensor charge is 0 dx h(x) = 0.94, which is close
to the naive quark model expectation of unity.
In a qualitative comparison between our tensor polarized PDF of the ρ+ meson and the HERMES Collaboration data for the deuteron5) we ﬁnd a similar
behavior as functions of x, but the overall sign is opposite and the magnitude is larger for the case of theρ+ meson by roughly one order.
References
1) M. E. Carrillo-Serrano, W. Bentz, I. C. Cloët,A. W. Thomas, Phys. Lett. B 759, 178 (2016).
2) A. Bacchetta, P. J. Mulders, Phys. Lett. B 518, 85(2001).
3) M. Hirai, S. Kumano, M. Miyama, Comput. Phys. Commun. 111, 150 (1998).
4) P. Hoodbhoy, R. L. Jaﬀe, A. Manohar, Nucl. Phys. 312,571 (1989).
5) A. Airapetian et al., (HERMES), Phys. Rev. Lett. 95,242001 (2005).