# Calculating Fragmentation Functions from Definitions

UM–P-94/01 OZ-94/01

Calculating Fragmentation Functions from De?nitions

arXiv:hep-ph/9401249v1 14 Jan 1994

J.P. Ma Recearch Center for High Energy Physics School of Physics University of Melbourne Parkville, Victoria 3052 Australia

Abstract: Fragmentation functions for hadrons composed of heavy quarks are calculated directly from the de?nitions given by Collins and Soper and are compared with those calculated in another way. A new fragmentaion function for a P-wave meson is also obtained and the singularity arising at the leading order is discussed.

Recently it has been shown[1,2,3,4] that the fragmentation functions for a hadron as a bound-state of two heavy quarks can be calculated perturbatively. The reason such calculations are possible is that the bound-states can be described well by the nonrelativstic wavefunctions and hence the e?ect of long distance in the fragmentaion can be factorized into the wavefunction, expressed through the radial wavefunction at the origin. Although it is realized[2,4] that the fragmentaion functions are universal, the fragmentation functions calculated in these works are extracted from speci?c processes. It should be noted that in QCD the de?nitions of the fragmentaion functions (or decay functions) are already given by Collins and Soper[5] and these functions should be universal and independent of pocesses. The so called factorization thorem is based on such de?nitions(for the factorization theorem see [6] and references cited there). It is interesting to calculate fragmentation functions directly from these de?ntions and to compare those obtained previously. On the other hand, it is also convenient to work with the de?ntions for calculating the high order corrections. In this letter we will calculate fragmentation functions from the de?nitions. To give the de?nitions for a fragmentaion function it is convenient to work in the light-cone coordinate system. In this coordinate system a 4-vector p is expressed as p? = √ √ (p+ , p? , pT ), with p+ = (p0 + p3 )/ 2, p? = (p0 ? p3 )/ 2. Introducing a vector n with n? = (0, 1, 0T ), the fragmentation functions for a spinless hadron H are de?ned as[5]: DH/Q (z) = z 4π dx? e?iP ? P exp{?igs
0 ∞
+ ?

x /z 1 ∞

1 Trcolor TrDirac {n · γ < 0|Q(0) 3 2

dλn · GT (λn? )}a? (P + , 0T )aH (P + , 0T ) H (1)

P exp{igs
x?

? dλn · GT (λn? )}Q(0, x? , 0T )|0 > < 0|G
b,+ν

?z DH/G (z) = 32πk +

dx e

? ?iP + x? /z ∞ 0 ∞

(0)

? {P exp{?igs {P exp{igs
a

dλn · G(λn? )}}bc a? (P + , 0T )aH (P + , 0T ) H

x?

dλn · G(λn? )}}cd Gd,+ (0, x? , 0T )|0 > ν

Where G? (x) = Ga (x) λ , Ga (x) is the gluon ?eld and the λa (a = 1, . . . , 8) are the Gell? ? 2 1

Mann matrices. The subscription T denotes the transpose. Ga,?ν is the gluon ?eld strength and a? (P) is the creation operator for the hadron H. The function DH/Q (z) or DH/G (z) H are interpreted as the probablity of a quark Q or a gluon G with momentum k to decay into the hadron H with momentum component P + = zk + , both are gauge invariant from the de?nitions. The de?ntions in (1) are the unrenormalized versions, i.e. all quantities are bar quantities(for renormalization see [5]). When one has a parton, i.e. quark or gluon, instead of the hadron in (1), the perturbative expansion in gs , the strong coupling constant, can also be expressed with Feymann diagrams, the Feymann rules can be found in [5,6]. The de?nitions can easily be generalized to hadrons with nonzero spin. From the de?ntions a direct calculation is possible if one can express the hadron operators with the parton operators. This is possible for a hadron composed of two heavy quarks, where, as mentioned, the nonrelativstic approach works. To see how to express the hadron operator as a parton one, let us consider a spinless hadron to be a bound-state ? of a heavy quark Q1 and an anti-quark Q2 and to have the quantum number 1 S0 . In its rest frame the state can be expressed as: |1 S0 >= A0 with 1 χs1 s2 = √ 2 0 ?1 1 0 (3)
s1 s2

1 d3 q f0 (|q|)Y00 (θ, φ)χs1 s2 √ (2π)3 3

a?1 (q)b?2 (?q)|0 > s s
color

(2)

? In Eq(2), a?1 (b?2 ) is the creation operator for Q1 (Q2 ), the indices s1 and s2 are spins s indices, and A0 is a suitable normalization constant. The function ψ0 (q) = f0 (|q|)Y00 (θ, φ) is obtained through Fourier transformation of the nonrelativstic wavefunction. f0 (|q|) goes rapidly to zero, when |q| → ∞. This means, the region with |q| ≈ 0 is dominant in the integral of Eq.(2). In this work we will always take nonzero leading order in |q| and the hadron mass M = m1 + m2 . Through a Lorenz boost the state can be transformed into a P = 0 state: |1 S0 , P >= A0 1 1 d3 q f (|q|) √ χs1 s2 √ 3 0 (2π) 3 4π 2 a?1 (p1 )b?2 (p2 )|0 > s s
color

(4)

where p1 = m1 m2 P + qp , p2 = P ? qp m1 + m2 m1 + m2 (5)

qp is related through the Lorenz boost to q in the rest frame. Now it is easy to show that the bound-state satis?es the normalization condition given in [5] as required for the de?nitions in (1), if one takes the following form for the hadron creation operator: a? (P ) =A0 H A?1 = 0 1 1 d3 q f (|q|) √ χs1 s2 √ 3 0 (2π) 4π 3 a?1 (p1 )b?2 (p2 ) s s
color

(6)

2m1 m2 P 0 + P 3 · m1 + m2 P 0 + |P| 1 d3 q 1 f (|q|) √ (?i (λ) · σi χ)s1 s2 √ 3 0 (2π) 4π 3 a?1 (p1 )b?2 (p2 ) s s
color

Similarly, one can also obtain the operator for the 3 S1 hadronic state: a? (P, λ) = A0 H (7)

Here σi (i = 1, 2, 3) are the Pauli matrices and ?i (λ)(i = 1, 2, 3) is the component of the polarization vector of the hadron in its rest frame. λ labels the heilicity with respect to the P direction. With Eq.(6) and Eq.(7) we can now calculate the fragmentaion function for hadrons with quantum number 1 S0 and 3 S1 from the de?nions in Eq.(1). In the leading order of gs there are four Feymann diagrams for a quark Q1 to decay to those hadrons. The diagrams are given in Fig.1, where the double lines represent the Wilson line operators in Eq.(1). One can work in the light cone gauge n · G = 0. In this gauge the Wilson line operators in the de?ntions Eq. (1) disappear and only one of the four diagrams contributes. We will take Feymann gauge. With the Feymann rule in [5,6] one obtains the contribution from each diagram to the fragmentaion function DH/Q1 (z). It is straightforward to calculate, and in the calculation one meets the so called spin-sums, which are: u(p1 , s1 )?(p2 , s2 )χs1 s2 v √ m1 m2 = √ γ5 (γ · P ? M ), 2M for H = 1 S0 state √ m1 m2 (γ · P + M )?p (λ) · γ, =? √ 2M for H = 3
3

(8)

u(p1 , s1 )?(p2 , s2 )(?i (λ)σi χ)s1 s2 v

S1 state

Here ?p (λ) is the 4-vector for the polarization of H in the P = 0 frame. Similar results for the spin sums in Eq.(8) and below can also be found in [7]. After a straightforward calculation we obtain for the H = DH/Q1 (z) =
1

S0 bound-state:

2 |R0 (0)|2 2 y2 z(1 ? z)2 αs 81π m3 (1 ? y1 z)6 2

2 · {6 + 18(1 ? 2y1 )z + (68y1 ? 62y1 + 15)z 2 2 2 2 + 2y1 (?18y1 + 17y1 ? 5)z 3 + 3y1 (1 ? 2y1 + 2y1 )z 4 }

(9)

and for H =

3

S1 bound-state:

2 |R0 (0)|2 2 y2 z(1 ? z)2 DH/Q1 (z, λ = 0) = αs 81π m3 (1 ? y1 z)6 2

2 · {2 ? 2(1 + 2y1 )z + (15 ? 22y1 + 16y1 )z 2 2 3 2 + 2y1 (?5 + 7y1 ? 6y1 )z 3 + 3y1 (1 ? 2y1 + 2y1 )z 4 }

2 |R0 (0)|2 2 y2 z(1 ? z)2 αs DH/Q1 (z, λ = ±1) = 81π m3 (1 ? y1 z)6 2
2 + 2y1 (y1 ? 5)z 3 + 3y1 z 4 }

(10)

2 · {2 ? 2(1 + 2y1 )z + (15 ? 16y1 + 10y1 )z 2

Here yi = mi /M for i = 1, 2 and R0 (0) is the radial wavefunction at the origin. Comparing the fragmentation functions calculated in previous works[3,4] we ?nd agreement. Exchang? ? ing y1 → y2 one can get DH/Q2 . If we take Q2 = Q1 , then we obtain the fragmentation ? ? functions for the quarkonia (Q1 Q1 ). ? ? Now we turn to gluon fragmentaion function. If a hadron has Q2 = Q1 , then a gluon
4 can decay into that hadron at the order of gs . In this leading order there are also four

Feymann diagrams depicted in Fig.2. Since Hadrons are colorless, there are always two gluons attached to the fermion lines to bulid a color singlet. Because charge-conjungation invariance a gluon can not decay at this order into a hadron with the quantum number
3

S1 (i.e. J P C = 1?? ). Computing the contributions from the diagrams we obtain the

fragmentaion function for a 1 S0 hadron: DH/G (z) = α2 |R0 (0)|2 s {(3 ? 2z)z + 2(1 ? z)ln(1 ? z)} 24π m3 1 4 (11)

This is also in agreement with that obtained in [2]. With our results calculated directly from Eq.(1) one can start with the de?ntions to calculate the high order corrections. It should be pointed out that the fragmentaion functions calculated here are at the scale ? = M . For the functions at an arbitrary scale ? one needs to solve the correponding renormalization group equations and use the results in Eq.(9,10,11) as initial conditions. With Eq.(9,10,11) we complete the calculations for the fragmentation functions of a S-wave meson. The long-distance e?ect in these functions are included in the nonrelativstic approach through the radial wave functions at the origin, which can be calculated from potential models or can be directly measured in leptonic decays. For a P -wave meson the calculation is lengthy but still straightforward. Considering a meson to be a J P C = 1++ ? bound state of (Q1 Q1 ), the creation operator for this hadron is: a? (P, λ) = A0 3 8π d3 q qi f (|q|) 1 ?i2 (λ)(σi3 χ)s1 s2 εi1 i2 i3 a?1 (p1 )b?2 (p2 ) s s 3 1 (2π) |q| (12)

and the spin sum used in the calculation: u(p1 , s1 )?(p2 , s2 )qi1 ?i2 (λ)(σi3 χ)s1 s2 εi1 i2 i3 = v (13) 1 1 σ √ (γ · p1 + m1 )ε?νσρ P ? ?ν (λ)qp γ ρ (γ · P ? M )(γ · p2 ? m1 ) p 2 4M 3 At order α2 a gluon can decay into the J P C = 1++ meson, the Feymann diagrams are s same as in Fig.2. Keeping the leading order at |q| we obtain:
′ z α2 |R1 (0)|2 s 2 DH/G (z, λ = 0) = 5 { (1 ? z) (1 ? z + 2z )} 108 4πm1 ′ α2 |R1 (0)|2 z DH/G (z, λ = ±1) = s (1 ? z + z 2 )} 5 { 108 4πm1 (1 ? z)

(14)

′ The R1 (0) is the derivative of the radial wavefunction at the origin. The results in Eq.(14)

are divergent when z → 1. The reason is that the gluon exchanged between the quarks in Fig.2 can move collinearly along the quarks, if we take the nonzero leading order at |q|. Such singularity is actually well known as the singularity in the zero-binding limit[8], appearing in the calculations of the hadronic decay widths of a P -wave meson. The situation is similar, if we transpose the diagrams in Fig.2 and calculate the gluon distribution 5

of the hadron. The singularity at z = 1 means that the long-distance e?ect is present not
′ only in the R1 (0) but also in the remaining parts which we calculated perturbativly and

hence it prevents the naive use of perturbation thoery for z near to 1. Recent progress[9] on the P -wave hadron decay shows that the long-distance e?ect can be clearly seperated from the short-distance e?ect by a new factorization theorem[9] for the decay, where one realizes that any meson is a superpostion of many components:

? ? |M >= ΨQQ |QQ > +ΨQQG |QQG > + · · · ? ?

(15)

? and the con?guration of |QQG > plays an important roll in the decay. In the hadronic ? decay of a P -wave meson one should also take account on the |QQG > con?guration and the decay width can be written in the new factorization form, where instead of one, two parameters represent the long-distance e?ect and the remaining parts can be calculated perturbativly without the singularity. To apply this idea in our case, one needs to know the wavefunction ΨQQG in Eq.(15), which can not be obtained through the nonrelativstic ? approach. A detailed study is needed. Here we recall a phenomenological treatment for the singularity at z = 1. Similar treatment of the decay was also used in [8,10]. In calculating the contribution from the diagrams to DH/G we integrated the transverse monmentum |pT |, which is carried by the exchanged gluon, from zero to in?nity. However, the meson is not a point-like particle and it has a spatial extension. The extension in the transverse direction is of the order R ≈ 1/M . If |pT | is smaller than 1/R, one should treat the gluon as a part of the meson. From this point of view, the integral over |pT | should be from a nonzero |pT |min ≈ 1/R to ∞. We introduce a parameter β: |pT |min = βM (16)

If the picture given above is correct, β should be around 1. Here we take it as a free 6

parameter. With Eq. (16) we obtain:
′ α2 |R1 (0)|2 s ((1 ? z)2 + z 2 β 2 )?3 108 4πm5 1

DH/G (z, λ = 0) =

· {z(?2z 7 + 11z 6 ? 26z 5 + 35z 4 ? 30z 3 + 17z 2 ? 6z + 1) + 3β 2 z 4 (?2z 4 + 7z 3 ? 9z 2 + 5z ? 1) + 3β 4 z 5 (?2z 3 + 4z 2 ? 3z + 1)} DH/G (z, λ = ±1) =
′ α2 |R1 (0)|2 s ((1 ? z)2 + z 2 β 2 )?3 108 4πm5 1

· {z(?z 7 + 6z 6 ? 16z 5 + 25z 4 ? 25z 3 + 16z 2 ? 6z + 1) + 3β 2 z 3 (?z 5 + 4z 4 ? 7z 3 + 7z 2 ? 4z + 1)} (17)

The results in Eq. (17) are ?nite at z = 1 and for z → 0 the expressions approach those given in Eq. (14). In Eq.(17) there are two parameters which need to be determined, the
′ R1 (0) can be obtained from potential models or measured elsewhere, while β is unknown

in principle. However, in the fragmentation functions the dependence on the momentum fraction z is predicted by the perturbative calculations. It should be stressed that it is important to know the fragmentation functions for the P -wave meson or quarkonia. If we take Q1 as a c quark, the quarkonia then corresponds to a χc1 meson and χc1 can substantially decay into J/ψ via radiative decay. Hence, a theoretical prediction of χc1 production via fragmentation is very important for the prediction for the J/ψ production(for example see [11] and the references cited there). The de?ntions given in Eq.(1) are for unpolarized partons. One can generalize the de?nitions to polarized partons and then obtain fragmentation functions for a polarized parton. In this way one can study through heavy meson production the polarization properties of the quarks and gluons produced in high energy collisions, where one expects for large Pt processes that the heavy mesons are produced dominantly via fragmentation[2]. To summarize: We have shown in this letter that the fragmentation functions for a Swave meson composed of two heavy quarks can be calculated directly from the de?nitions given by Collins and Soper in [5] and the results obtained are same as those obtained by 7

other means. A new fragmentation function for a P -wave meson is also obtained, but in contrast to a S-wave meson, there is an extra parameter which needs to be estimated. From the naive picture given above this parameter should be around 1. Finally within the de?ntions one can conveniently study higher order corrections to the fragmentaion functions.

Acknowledgement: The author would like to thank Dr. M. Thomson for reading the manuscript carefully. This work is supported in part by the Australian Research Council.

8

References [1] Chao-Hsi Chang and Yu-Qi Chen, Phys. Rev. D46 (1992) 3845 C.R. Ji and F. Amiri, Phys. Rev. D35 (1987) 3318 [2] E. Braaten and T.C. YUan, Phys. Rev. Lett. 71 (1993) 1673 [3] A.F. Falk, M. Luke, M.J. Savage and M.B. Wise, Phys. Lett. B312 (1993) 486 [4] E. Braaten, K. Cheung and T.C. Yuan, Phys. Rev. D48 (1993) 4230 [5] J.C. Collins and D.E. Soper, Nucl. Phys. B194 (1982) 445, Nucl. Phys. B193 (1981) 381 [6] J.C. Collins, D.E. Soper and G. Sterman, in Perturbative Quantum Chromodynamics, edited by A.H. M¨lller, World Scienti?c, Singapore, 1989. u [7] B. Guberina, J.H. K¨n, R.D. Pecci and R. R¨ckl, Nucl. Phys. B174 (1980) 317 u u [8] R. Barbieri, R. Gatto and E. Remiddi, Phys. Lett. B61 (1976) 465 R. Barbieri, M. Ca?o and E. Remiddi, Nucl Phys. B162 (1980) 220 [9] G.T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. D46 (1992) R1914 [10] W. Kwong, P.B. Mackenzie and R. Rosenfeld and J.L. Rosner, Phys. Rev. D37 (1988) 3210 [11] M.L. Mangano, invited talk presented at the 9th Topical Workshop on Proton– Antiproton Collider Physics, 18–22 October 1993, Tsukuba, Japan, Preprint IFUP–TH 60/93

9

Figure Captions

Fig.1. The Feymann diagrams for a quark to decay into a hadron. Fig.2. The Feymann diagrams for a gluon to decay into a hadron.

10

Fig. 1

11

Fig. 2

12

This figure "fig1-1.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9401249v1

This figure "fig1-2.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9401249v1

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