

 10 Apr 1995

UM-P-95/37 RCHEP-95/11

Quark Fragmentation into 3PJ Quarkonium

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

Abstract: We calculate the functions of parton fragmentation into 3PJ quarkonium at order ff2s, where the parton can be a heavy or light quark. The obtained functions explicitly satisfy the Altarelli-Parisi equation and they are divergent, behaving as z\Gamma 1 near z = 0. However, if one choses the renormalization scale as twice of the heavy quark mass, the fragmentation functions are regular over the whole range of z.

1. Introduction Fragmentation functions are in general nonperturbative objects in the context of the factorization theorem in QCD[1]. This makes it hard to study them by starting directly from QCD. However, if partons, i.e., quarks and glouns, undergo fragmentation into a quarkonium, fragmentation functions can be factorized--they are sums of products of constants and coefficient functions. The constants represent the nonperturbative effects and may be defined as matrix elements in nonrelativestic QCD(NRQCD) or are related to a wavefunction, while the coefficients functions can be calculated with perturbative theory. With this factorization, fragmenation functions for various quarkonia were studied in [2-9].

A quarkonium contains a heavy quark Q and its antiquark _Q, which move with a small velocity v in the quarkonium rest frame. Because of the small velocity the bound state effect, i.e., the nonperturbative effect in the quarkonium, can be well described by employing NRQCD. Recently, such an approach has been established[10]. The approach is basically distinct than early treatment within the color-singlet model(for a nice review see [11]). In the color-singlet model one treats a quarkonium system simply as a bound state of Q and _Q, where the Q _Q pair is in a color-singlet state, and the nonperturbative effect is contained in the wavefunction of the bound state. Then an expansion in the small parameter v is made and to leading order only the wavefunction at the origin or its derivative at the origin is involved. This model has serious problems. First, it does not tell us how to handle the Coulomb singularities. Since an expansion in v is made, this type of singularity must appear because of massless photons and massless gluons. The effect related to the Coulomb singularities is nonperturbative. In the color-singlet model these singularities were absorbed into the wavefunction without solide reasons from theories. Second, infrared(I.R.) singularities appear when a P -wave quarkonium is involved and appear even at the leading order of ffs. In the model the I.R. singularities were regularized as divergences in the limit of the zero-binding energy. Such I.R. singularities clearly indicate that the wavefunction of a P -wave quarkonium can not contain all the nonperturbative effects. In the approach[10], quarkonium systems are analysed in the framework of QCD. A systematice expansion in v can be made and the nonperturbative effect is represented by matrix elements in NRQCD. With these matrix elements one can show that Coloumb singularities are factorized into these matrix elements. From the point of view of a relativistice quantum theory a quarkonium system consists not only of a Q _Q color-singlet but also many other components like jQ _QG ? \Delta \Delta \Delta etc.. In the approach of [10] the effect of these other components is also included in the systmatic expansion in v. It should be emphasized [10,12] that a Q _Q color-octet state is as important as a

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Q _Q color-singlet for a P -wave quarkonium. Therefore one should take both states into account. It is already shown that results at the one-loop level for gluon fragmenation into 3PJ quarkonium[9] are free from I.R. singularities and from Coloumb singularities.

In this work we will study the quark fragmentation into a 3PJ Q _Q quarkonium at leading order ffs. We will show that because of the contribution of the color-octet Q _Q pair, a light quark q can also fragment into the quarkonium at the same order of ffs as a heavy quark Q. The heavy quark fragmentation into P -wave quarkonium was originally studied in[8]. As pointed out in [9] the results for charmonium can not be correct, because the fragmenation functions obtained there do not satisfy the Altarelli-Parisi equation.

The work is organized as follows: in Sect. 2 we introduce the definition[13] of renormalized quark fragmentation functions and the factorzation[10] forms for quark fragmentation into 3PJ quarkonium. Further details may be found in [10,13]. In Sect.3 we start from the definiton to calculate the heavy quark fragmentation functions. In Sect.4 we calculate the light quark fragmentation function. Finally we discuss and summarise the results of our work in Sect.5. Throughout this work we always assume that the polarization of the quarkonium is not observed. We will use dimensional regularization and work at leading order in the expansion of v.

2. The Definition of Quark Fragmentation Function

and the Factorization Form for 3PJ quarkonium

As we will use dimensional regularization we give the definiton of the quark fragmentation function in d dimensions. To give the definitions for a fragmentaion function it is convenient to work in the light-cone coordinate system. In this coordinate system a d-vector p is expressed as p_ = (p+; p\Gamma ; pT), with p+ = (p0 + pd\Gamma 1)=p2; p\Gamma = (p0 \Gamma pd\Gamma 1)=p2. Introducing a vector n with n_ = (0; 1; 0; \Delta \Delta \Delta ; 0) = (0; 1; 0T), the fragmentation function for a spinless hadron H or for a hadron without observing its polarization is defined as[13]:

D(0)H=q(z) = z

d\Gamma 3

4ss Z dx

\Gamma e\Gamma iP +x

\Gamma =z 1

3 Trcolor

1 2 TrDiracfn \Delta fl ! 0jq(0)

_P expf\Gamma igs Z

1

0 d*n \Delta G

T (*n_)gayH (P +; 0

T)aH (P +; 0T)

P expfigs Z

1

x

\Gamma d*n \Delta G

T (*n_)g_q(0; x\Gamma ; 0

T)j0 ?;

(2:1)

where G_(x) = Ga_(x)T a, Ga_(x) is the gluon field and T a(a = 1; \Delta \Delta \Delta ; 8) are the SU (3)-color matrices. The subscript T denotes the transpose. ayH (P) is the creation operator for the

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hadron H. For hadrons with nonzero spin the summation over the spin is understood. The definition is a unrenormalized version. Ultraviolet divergences will appear in D(0)H=q(z) and call for renormaization. Following [13] the renormalized gluon fragmenation function can be defined as:

DH=Q(z) = D(0)H=Q(z) + X

a=G;q

Z 1

z

dy

y La(

y z )D

(0) H=a(y): (2:2)

Here the summation over all possible partons is understood. The function La(z) is chosen so as to cancel the U.V. divergences. In MS scheme La(z) takes the form:

La(z) = X

N

1 fflN L

(N)a (z); (2:3)

where ffl = 4 \Gamma d. From Eq.(2.2) one can derive the Altarelli-Parisi type evolution equation for the fragmentation function. We will use the modified MS scheme, where La(z) is chosen to cancel the terms with Nffl = 2ffl \Gamma fl + ln(4ss) . The function DH=q(z) is interpreted as the probability of a quark q with momentum k to decay into the hadron H with momentum component P + = zk+, it is gauge invariant by definiton. Further, it is also invariant under a Lorentz boost along the moving direction of the hadron and under a rotation with the direction as the rotate axis.

If the hadron is a 3PJ quarkonium, a factorized form for the fragmentation function can be taken. We will use the notation O/J for the 3PJ quarkonium. At the leading order of v DH=q(z) can be written according to [10] as:

DO/J =q(z) = ^D1(z; J )M 5 ! 0jOO/J1 (3PJ )j0 ? + ^D8(z)M 3 ! 0jOO/J8 (3S1)j0 ? : (2:4) where D1(z; J ) and ^D8(z) are dimensionless and ^D8(z) is same for all J . The operators OO/J1 (3PJ ) and OO/J8 (3P1) are given by:

OH8 (3S1) = O/yoeiT a\Gamma ayH aH \Delta yoeiT aO/ OH1 (3P0) = 13 O/y\Gamma \Gamma i2

$D \Delta oe\Delta \Gamma ay

H aH \Delta y\Gamma \Gamma i2

$D \Delta oe\Delta O/

OH1 (3P1) = 12 O/y\Gamma \Gamma i2

$D \Theta oe\Delta

i\Gamma a

y H aH \Delta y\Gamma \Gamma i2

$D \Theta oe\Delta

iO/

OH1 (3P2) = O/y\Gamma \Gamma i2

$D

fi oejg\Delta \Gamma ayH aH \Delta y\Gamma \Gamma i2

$D

fi oejg\Delta O/;

(2:5)

where D is the space part of the covariant derivative D_ and oei(i = 1; 2; 3) is the Pauli matrix. The notation fijg means only the symmetric and traceless part of a tensor taken.

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In Eq. (2.5) and O/y are fields with two components for the heavy quark Q and its antiquark _Q in NRQCD. M is the mass of the heavy quark. a+H is the creation operator for the hadron in its rest frame. The matrix elements in Eq.(2.4) are defined in NRQCD. In Eq.(2.4) the part with ^D8 is the contribution from a color-octet Q _Q pair in a 3S1 state and the part with ^D1(z; J ) is the contribution from a color-singlet Q _Q pair in a 3PJ state. We will call them as the color-octet and color-singlet components respectively. The matrix elements represent the nonperturbative effect, while ^D1 and ^D8 can be calculated perturbatively and they should be free from I.R. singularities.

A good method to calculate ^D1 and ^D8 is to use wavefunctions to project out different states from a general Q _Q pair. At the leading order of v the projection can easily be worked out, details may be found in [14]. We will use a radial wavefunction R1(r) to project the 3PJ color-singlet Q _Q state and an octet radial wavefunction R(a)8 (r) to project the 3S1

color-octet Q _Q state. Calculating with these wavefunctions the l.h.s of Eq.(2.4) and the matrix elements in the r.h.s of Eq.(2.4) we can extract the functions ^D1(z; J ) and ^D8(z). The results for ^D1(z; J ) and ^D8(z) are independent on these wavefunctions. At the order of ffs we consider, only the tree-level results for the matrix elements are needed, they are:

! 0jOO/J1 (3PJ )j0 ? = 9(2J + 1)2ss jR01(0)j2;

! 0jOO/J8 (3S1)j0 ? = 38ss X

c j

R(c)8 (0)j2;

(2:6)

where R01(0) is the first derivative of R1(r) at the origin.

3. The Heavy Quark Fragmentation Function From the defintion in Eq.(2.1) we can always decompose the fragmentation function by sandwiching the operator PX jX ?! Xj between ayH and aH as:

D(0)H=Q(z) , X

X

Tr\Phi n \Delta flT yH TH \Psi ; (3:1)

where TH may be called the fragmentation amplitude for Q ! H + X. Here we have the conservation of total momentum only in the +-direction.

3.1 The Color-Singlet Component The color-singlet component receives nonzero contributions at order ff2s. The Feynman diagrams for TH are given in Fig.1. Because the Q _Q pair is in a color-singlet state,

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there are two gluon lines attached to the quark line. Here, there are no divergences, so renormalization is not required. The calculation is complicated. Because of the summation over intermediate states we encountered integrals of the type:Z \Gamma

dqT

2ss \Delta

d\Gamma 2\Gamma q2

T + (2 \Gamma z)

2M 2

z2 \Delta

\Gamma n; for n = 2; 3; 4; 5: (3:2)

Here qT is the transversel momentum of the quark as the intermediate state in Fig.1. The integrals are finite and after performing the integrations we can extract:

^D1(z; J = 0) = 16729 ff2s(_) z(1 \Gamma z)

2

(2 \Gamma z)8\Delta (192 + 384z + 528z2 \Gamma 1376z3 + 1060z4 \Gamma \Gamma 376z5 + 59z6);

^D1(z; J = 1) = 64729 ff2s(_) z(1 \Gamma z)

2

(2 \Gamma z)8\Delta (96 \Gamma 288z + 496z2 \Gamma 408z3 + 202z4 \Gamma 54z5 + 7z6);

^D1(z; J = 2) = 1283645 ff2s(_) z(1 \Gamma z)

2

(2 \Gamma z)8\Delta (48 \Gamma 192z + 480z2 \Gamma 668z3 + 541z4 \Gamma 184z5 + 23z6):

(3:3)

These results agree with those in [8]. It is interesting to note that there is a common factor z(1 \Gamma z)2(2 \Gamma z)\Gamma 8 for all J , whereas there is a common factor z(1 \Gamma z)2(2 \Gamma z)\Gamma 6 for heavy quark fragmentation into S-wave quarkonium[6]. Note that the same diagrams in Fig.1 contribute to heavy quark fragmentation into S-wave quarkonium. The difference between these two factors are because for P -wave quarkonium the derivative of the fragmentation amplitude with the relative momentum between Q and _Q is involved whereas for S-wave quarkonium only the amplitude itself is involved. The appearance of these common factors can be roughly understood by counting the denominators due to the quark- and gluon propagators in the amplitudes and factors from phase-space. A successful model for heavy quark fragmentation was obtained through this way[15]. However, we will see in the next subsection that such a counting rule will be violated due to renormalization.

3.2 The Color-Octet Component For the color-octet componenet, not only the diagrams in Fig.1, but also the diagrams in Fig.2 will contribute The two diagrams in Fig.2 were missing in [8] and they lead to divergences. Instead of integrals in Eq.(3.2), we have:Z \Gamma

dqT

2ss \Delta

d\Gamma 2\Gamma q2

T + (2 \Gamma z)Mz \Delta

\Gamma n; for n = 1; 2; 3; 4: (3:4)

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The integral with n = 1 is ultraviolet divergent, requiring renormalization. For the renormalization we note that the function of gluon fragmentation into O/J quarkonium is nonzero at order ffs[9,6,7]:

DO/J =G(z) = ss24 ffsffi(1 \Gamma z) \Delta 1M 3 ! 0jOO/J8 (3S1)j0 ? +O(ff2s): (3:5) Substituting this into Eq.(2.2) for H = O/J we can easily chose the function LG(y) to cancel the divergence. Finally, we obtain the renormalized function ^D8(z):

^D8(z) = 136 ff2s(_)\Phi 1z (1 + (1 \Gamma z)2)(ln \Gamma _

2

4M 2 \Delta \Gamma 2 ln(1 \Gamma

z 2 )) \Gamma z

+ 2(1 \Gamma z)9(2 \Gamma z)6 (192 \Gamma 1184z + 2016z2 \Gamma 1360z3 + 352z4 \Gamma 14z5 \Gamma 5z6)\Psi :

(3:6)

Here there is is no common factor like z(1 \Gamma z)2(1 \Gamma z)\Gamma 6. With the results in Eq.(3.6) and in Eq.(3.3) we complete the heavy quark fragmentation function at order ff2s. This function should in general satisfy its evolution equation:

_ @DH=Q(z; _)@_ =

NfX

q

Z 1

z

dy

y PQ!q(z=y; _)DH=q (y; _) + Z

1

z

dy

y PQ!G(z=y; _)DH=G(y; _):

(3:7) The spliting functions PQ!q(y; _) and PQ!G(y; _) are in the one-loop approximation the same as those for parton distributions. Using this fact and the result in Eq.(3.5) we obtain the evolution equation for the quark fragmenation at order of ff2s:

_ @DO/

J =Q(z; _)

@_ =

1 18 ff

2S (_) 1

M 3 ! 0jO

O/J 8 (3S1)j0 ? \Delta 1 + (1 \Gamma z)

2

z : (3:8)

Substituting our results into the l.h.s. in Eq.(3.8) one can check that our results are in agreement with this equation. From our result the fragmentation function is divergent as z\Gamma 1 when z ! 0. However, this singularity disappears if we chose the renormalization scale _ as twice of the mass M . The same was also found in the gluon fragmentation in [9]. This property is important for possible applications of our result. In practical applications one solves the evolution equation at _ numerically, where one needs the moments of our result for the fragmentation function at some initial scale _0 as the boundary condition. To insure that the perturbative result is a good approximation, one should chose _0 , M to avoid large logarithmc terms in higher order. Our result tells that one should chose _0 = 2M to avoid these terms in higher order and to safely calculate the moments in other hand.

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4. The Light Quark Fragmentation Function Since a color-octet Q _Q pair will lead to a contribution to P -wave quarkonium production at the leading order of v, a light quark q can undergo fragmentation into O/J by generating a color-octet Q _Q through emission of a virtual gluon. Such a process happens at the same order of ffs as the heavy quark fragmentation. The Feynman diagrams for TH are those in Fig.2, where the quark line attached by the double line is for the light quark q. At the leading order of ffs and v the light quark fragmentation function DO/J =q (z) can be written:

DO/J =q(z) = ^D8;q(z)M 3 ! 0jOO/J8 (3S1)j0 ? : (4:1)

The color-singlet component only becomes nonzero in higher order than ff2s. The calculation is similar to the previous section. We introduce the notation y = mqM , where mq is the mass of q. The result for ^D8;q(z) is:

^D8;q(z) = 136 ff2s(_)\Phi 1z (1 + (1 \Gamma z)2)\Theta ln \Gamma _

2

4M 2 \Delta \Gamma ln \Gamma (1 \Gamma z) +

1 4 y

2z2\Delta \Lambda

\Gamma z \Gamma 2(1 \Gamma z)(2 + y2) z4(1 \Gamma z) + y2z2 \Psi :

(4:2)

Again the light quark frgamentation function must satify its evolution equation. At order ff2s this equation is the same in Eq.(3.8). It is easy to check that the function in Eq.(4.1) and (4.2) satisfies the evolution equation. The light quark fragmenation function has the same property near z = 0 as the heavy one, i.e., it is divergent as z\Gamma 1 at any remormalizaion scale _ except when _ = 2M . The light quark mass mq can be safely neglected. With mq = 0 the function in (4.2) becomes:

^D8;q(z) = 136 ff2s(_)\Phi 1z (1 + (1 \Gamma z)2)\Theta ln \Gamma _

2

4M 2 \Delta \Gamma ln(1 \Gamma z)\Lambda \Gamma 2z\Psi : (4:3)

For the convenience of later discussions we introduce here some relations between the various matrix elements in Eq.(2.5). In principle these matrix elements have series expansions in v and the leading order is v2. Since we only work at the leading order, the higher order corrections can be neglected. In this case, the matrix elements in Eq.(2.5) are related to each other with a spin factor of O/J . We introduce two parameters H1 and H08 as in [10], and the relations can be expressed as:

! 0jOO/J1 (3PJ )j0 ? ss (2J + 1)M 4H1;

! 0jOO/J8 (3S1)j0 ? ss (2J + 1)M 2H08: (4:4)

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With these relations the whole set of the quark fragmention functions contains only two unknown parameters, which can only be computed nonperturbatively or extracted from experiments.

5. Discussion and Summary Some useful quantities of parton fragmentation functions are their first moments. These moments allow one to roughly estimate a single hadron production rate though fragmentation, where the rate may be taken as product of a parton production rate and the corresponding first moment, where the summation over different partons is understood. We will give results of the first moments of our fragmentation functions. We denote the first moment as M (qf ! O/J ), where qf stands for Q or q. Taking _ = 2M , we obtain:

M (q ! O/J ) ss 0:029(2J + 1)ff2s(2M ) H

08

M ; M (Q ! O/0) ss 0:024ff2s(2M ) H

08

M + 0:035ff

2s(2M ) H1

M ;

M (Q ! O/1) ss 0:072ff2s(2M ) H

08

M + 0:039ff

2s(2M ) H1

M ;

M (Q ! O/2) ss 0:12ff2s(2M ) H

08

M + 0:015ff

2s(2M ) H1

M :

(5:1)

Here we neglect the mass of the light quark. We take the c quark as an example to give some value for the moments. For the value of H1 and H08 we use the estimates in [12]. Taking M = mc = 1:5GeV and ffs(2mc) = 0:26, the value for M (q ! O/cJ ) is 3:8 \Theta 10\Gamma 6(2J + 1) and the value for M (Q ! O/cJ ) is 2:6 \Theta 10\Gamma 5, 3:4 \Theta 10\Gamma 5 and 2:5 \Theta 10\Gamma 5 for O/c0, O/c1 and O/c2 repectively. The contribution from the color-octet componennt is not negligible. For O/c2 the contribution from the color-octet component is roughly 70% of the heavy quark moment. From these values one can see that the moments for the light quark fragmentation are roughly one order of magnitude smaller than those for the heavy quark. But the contribution from light quarks should not be neglected, especially in a hadron reaction, since the production rate of light quarks as parton may be larger than the production rate of a heavy quark and hence a substantial contribution from light quark fragmentation to the O/J production is possible.

With the results here and those in [9] the functions of all possible parton fragmenation into 3PJ quarkonium are calculated at order ff2s. Only two parameters, which represent the nonperturbative effect at the leading order of v, are not known precisely. The functions have the general feature that they are divergent as z\Gamma 1 when z ! 0. But at _ = 2M

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they are regular distributions over the whole range of z. The functions also satify the Altarelli-Parisi equation, as expected.

Acknowledgment: The author would like to thank Dr. A. Rawlinson for reading the text carefully. This work is supported by Australain Research Council.

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edited by A.H. M"ulller, World Scientific, Singapore, 1989. [2] Chao-Hsi Chang and Yu-Qi Chen, Phys. Rev. D46 (1992) 3845

C.R. Ji and F. Amiri, Phys. Rev. D35 (1987) 3318 [3] E. Braaten and T.C. Yuan, Phys. Rev. Lett. 71 (1993) 1673 [4] A.F. Falk, M. Luke, M.J. Savage and M.B. Wise, Phys. Lett. B312 (1993) 486 [5] E. Braaten, K. Cheung and T.C. Yuan, Phys. Rev. D48 (1993) 4230, R5049 [6] J.P. Ma, Phys. Lett. B332 (1994) 398 [7] E. Braaten and T.C. Yuan, Phys. Rev. D50 (1994) 3176 [8] T.C. Yuan, Phys. Rev. D50 (1994) 5664 [9] J.P. Ma, Melbourne-Preprint, UM-P-95-23, RECHP-95-09,  [10] G.T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. D51 (1995) 1125 [11] E. Braaten, invited talk at the Tennessee International Symposium on Radiative

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G.T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. D46 (1992) R1914 [13] J.C. Collins and D.E. Soper, Nucl. Phys. B194 (1982) 445 [14] B. Guberina, J.H. K"uhn, R.D. Pecci and R. R"uckl, Nucl. Phys. B174 (1980) 317 [15] C. Peterson, D.Schlatter, I. Schmidt and P. Zerwas, Phys. Rev. D27 (1983) 105

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Figure Captions Fig.1: The Feynman diagrams for the color-singlet component of the heavy quark fragmentation. The line is for the heavy quark, the wavy line is for gluons. The double line represents the line operator in Eq.(2.1). Fig.2 The Feynman diagrams for the color-octet component of the heavy and light quark fragmentation.

11 Fig. 1 Fig. 2

