Quark fragmentation functions in the NJL-jet model †

RIKEN Accel. Prog. Rep. 43 (2010)

Quark fragmentation functions in the NJL-jet model ^†

W. Bentz, ^∗1 T. Ito, ^∗1 I. C. Clo¨et, ^∗2 A. W. Thomas, ^∗3 and K. Yazaki ^∗4

QUARK FRAGMENTATION FUNCTIONS, Semi-inclusive pion production, Deep inelastic scattering

Quark distribution and fragmentation functions are the basic nonperturbative ingredients for a QCD-based analysis of hard scattering processes. In this paper we show the results of recent calculations of fragmentation functions in the NJL-jet model ¹⁾ .

The spin-independent fragmentation function for the process q → h is defined by

D ^h _q (z) = z 12

Z dω ⁻

2π e ^ip

^ω

^/z X ˆ

n

× h p(h), p n | ψ(0) | 0 i γ ⁺ h 0 | ψ(ω ⁻ ) | p(h), p n i . The field operators refer to a quark of flavour q, the symbol p(h) refers to a hadron h with momentum p, and p n labels the spectator state. The light-cone com- ponents of a 4-vector a ^µ are defined by a ^± = a ∓ = (a ⁰ ± a ³ )/ √

2. From this definition one can derive the expression

D ^h _q (z) dz = 1 6 dp −

Z d ² p ⊥

X

α

h k(α) | a ^† _h (p)a h (p) | k(α) i h k(α) | k(α) i , where the creation and annihilation operators refer to the hadron h, k(α) labels a quark state of flavour q with momentum k and spin-color α, and p − = zk − for some fixed k − > 0. The above result can be interpreted as the light-cone momentum distribution of the hadron h in the quark q.

The momentum and isospin sum rules obtained from the above formula are

X

h

Z 1

0

dz z D _q ^h (z) = 1 , X

h

Z 1

0

dz t h D _q ^h (z) = t q .

The condition which lies at the basis of these sum rules is that the initial quark state is an eigenstate of the momentum and isospin operators, expressed solely in terms of hadrons. Their physical content is that 100%

of the initial quark light-cone momentum (k − ) and isospin (t q ) are transferred to the hadrons. (Note that the definition of the fragmentation function implies an average over the isospin of the soft quark remainder of a fragmentation chain.)

In order to satisfy the momentum and isospin sum

†

Condensed from an article by T. Ito, W. Bentz, I.C. Clo¨ et, A.W. Thomas and K. Yazaki, Phys. Rev. D 80 (2009) 074008.

∗1

Department of Physics, Tokai University, Kanagawa, Japan

∗2

Department of Physics, University of Washington, Seattle, WA, U.S.A.

∗3

University of Adelaide, Adelaide, Australia

∗4

Radiation Laboratory, RIKEN, Saitama, Japan

rules, it is necessary to take into account the possibil- ity that the fragmenting quark produces a cascade of mesons. In order to describe these multi-fragmentation processes, we use the ideas of the quark jet model of Field and Feynman ²⁾ . Assuming that the fragmenting quark can produce a maximum of N mesons, we make a product ansatz to express the total fragmentation function as a product of N elementary splitting func- tions. Because only in the limit N → ∞ it becomes possible to transfer 100% of the initial quark momen- tum and isospin to the mesons, we take this limit in the final results. The details of this product ansatz can be found in Ref. ¹⁾ .

In the numerical calculations we take into account the fragmentation to pions only. The results for the

“favored” fragmentation process u → π ⁺ are shown in Fig. 1. This figure demonstrates the tremendous en- hancement of the fragmentation function arising from the cascade-type multi-fragmentation processes. In or- der to improve the agreement with the empirical frag- mentation function, one should perform the Q ² evolu- tion in next-to-leading order, and include the effects of fragmentation processes to other hadrons, mainly the nucleons, antinucleons and kaons.

LO

0 0.2 0.4 0.6 0.8 1.0 1.2

z D

0 0.2 0.4 0.6 0.8 1.0

z

NJL-elementary NJL-jet (Q

= 0.18 GeV

) NJL-jet (Q

= 4 GeV

) Empirical (Q

= 4 GeV

)

Fig. 1. Fragmentation function zD

(z). The dash-dotted line is the elementary fragmentation function, and the dotted line is the full fragmentation function in the NJL-jet model. The solid line is the result after LO evo- lution to Q

= 4 GeV

, and the dashed line is the em- pirical NLO result of Ref. ³⁾ , evolved to Q

= 4 GeV

.

References

1) T. Ito, W. Bentz, I.C. Clo¨et, A.W. Thomas, K. Yazaki, Phys. Rev. D 80 , 074008 (2009).

2) R. D. Field and R. P. Feynman, Phys. Rev. D 15 , 2590 (1977).

3) M. Hirai, S. Kumano, T. H. Nagai and K. Sudoh, Phys.

Rev. D 75 , 094009 (2007).

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