Charmonium Production via Fragmentation at Higher Orders in alpha_s

18 Pages·0.16 MB·English

by  P. Ernstrom

NORDITA-96/3 P
hep-ph/9601333
January 25, 1996
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Charmonium Production via Fragmentation
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at Higher Orders in α
n s
a
J

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P. Ernstr¨om, P. Hoyer, and M. V¨anttinen
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NORDITA
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3 Blegdamsvej 17, DK-2100 Copenhagen Ø
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h Abstract
p
- Quarkonium production at a given large p is dominated by parton
p T
e fragmentation: a parton which is produced with transverse momen-
h tum p /z fragments into a quarkonium state which carries a fraction
: T
v z of the parton momentum. Since parton production cross sections
i
X fall steeply with p , high z fragmentation is favored. However, quan-
T
r tum number constraints may require the emission of gluons in the
a
fragmentation process, and this softens the z dependence of the frag-
mentation function. We discuss the possibility that higher-order pro-
cesses may enhance the large z part of fragmentation functions and
thus contribute significantly to the quarkonium cross section. An ex-
plicit calculation of light quark fragmentation into η shows that the
c
higher-order process q qη in fact dominates the lowest-order pro-
c
→
cess q qgη .
c
→
1. Introduction
Heavy quark production is a hard QCD process, and the perturbative ex-
pansion of production amplitudes in α is expected to apply. However, the
s
perturbation series is not always dominated by its lowest-order term. For
example, when the transverse momentum p of a collinear heavy quark pair
T
is much larger than its invariant mass M, light parton fragmentation dom-
inates the lowest-order production processes by a power of p2/M2 [1]. The
T
data on charmonium production indeed shows a 1/p4 behavior of the cross
T
section [2, 3], compatible with the prediction for a fragmentation process.
QCD calculations based on the color singlet model [4] nevertheless dis-
agree with data on the relative production rates of S and P wave quarkonia
and on the absolute normalization of the cross sections at both low [5, 6]
and high [2, 3, 7] values of p . Order-of-magnitude discrepancies have been
T
observed for both charmonium and bottomonium.
In QCD, the fragmentation of a virtual gluon into a 3S quarkonium state
1
(J/ψ, Υ) requires the emission of at least two extra gluons, g 3S +gg,
∗ 1
→
whereas fragmentation into a 3P state can proceed with the emission of a
J
single gluon, g 3P +g. The emission of the extra gluon suppresses the
∗ J
→
calculated 3S cross section considerably compared to the 3P cross section,
1 J
due to the extra power of α and also because the emitted gluons carry away
s
part of the transverse momentum of the fragmenting gluon. The experimen-
tal 3S /3P cross section ratio is much larger than the calculated one.
1 2
The discrepancy could be related to the bound state dynamics of the
quarkonia. This is the solution proposed by the color octet model [8, 9]. The
gluonisassumedtofirst fragmentintoaQQ¯ pairinacoloroctetstate. Later,
after a formation time characteristic of the bound state, the pair couples to
1
thephysicalquarkoniumthroughtheabsorptionoremissionofoneortwosoft
gluons. The momenta of the emitted gluons are typical of the quarkonium
bound state dynamics, and thus they carry away only a minor part of the
transverse momentum.
Although the probabilities of these nonperturbative transitions are essen-
tially free parameters, the octet model makes specific predictions about the
polarization of the produced quarkonium [10]. The 3S quarkonia produced
1
by the fragmentation of a nearly real gluon are expected to be transversely
polarized. The datato test thisprediction is notyet available. Notethat the-
oretical calculations of quarkonium polarization can be compared with data
in fixed-target experiments. In this case, neither the color singlet model nor
the color octet model can explain the experimentally observed unpolarized
production [6, 11].
Here we wish to draw attention to the possibility that higher-order per-
turbative contributions to fragmentation mechanisms could be enhanced by
the so-called trigger bias effect. When the transverse momentum p of the
T
quarkonium is fixed, processes which allow the fragmenting parton to be pro-
duced with the lowest possible transverse momentum p /z are favored. The
T
large p cross section is a convolution of the production cross section σ of a
T i
parton i and its fragmentation function D to the quarkonium state ,
i
→O O
dσ (s,p ) 1 dσ
T i
O = dz (s,pT/z,µ)Di (z,µ), (1)
dpT i Z0 dpiT →O
X
where µ is the factorization scale. The parton production cross section
dσ /dp falls approximately as (p /z) 4, which implies the enhancement
i iT T −
of high z fragmentation by a factor z4. As a rough measure of the impor-
tance of a fragmentation function D (z,µ) we can therefore use its fifth
i
→O
2
(5)
moment D (µ), defined by
i
→O
1
D(n) (µ) dzzn 1D (z,µ). (2)
i→O ≡ Z0 − i→O
We have studied the importance of higher-order fragmentation contribu-
tions in the case of light quark fragmention into 1S quarkonium (η ) within
0 c
the color singlet model. The relevant Feynman diagrams are shown in Fig.
1. In the nonrelativistic limit the fragmentation function is a product of the
fragmentation probability into a collinear, on-shell cc¯pair in a 1S state and
0
the square of the η wave function at r = 0,
c
R (0) 2
S
Di→ηc(z,µ) = Di→cc¯(1S0)(z,µ)| 4π | . (3)
h h
c c
q q
g
Figure1: (a)Alowest-order diagramcontributing totheprocessq η +X.
c
→
(b) A higher-order diagram with no gluon emission. In each case there is
another diagram with the c-quark–gluon vertices interchanged.
At lowest order, q η +X fragmentation is due to the process q qη g
c c
→ →
of Fig. 1a. The emission of the gluon suggests that this process may have a
softer fragmentation function than the higher-order process q qη shown
c
→
in Fig. 1b. Due to the trigger bias effect, the higher-order process could be
enhanced.
3
2. Results
Our calculation of the q η +X fragmentation processes shown in Fig. 1
c
→
is described in the Appendix. The contribution of the lowest-order process
(Fig. 1a) to the fragmentation function is of the form
µ2 4m2
D(a) (z,µ) = f(z)ln +g(z)+ c . (4)
q→ηc 4m2! O µ2 !
c
The coefficient functions are
α3C R (0) 2 π2
f(z) = s F | S | 6 (2+z) (z) 3z ln(z)
48π2m3cNc ( " 6 −L2 #−
2 6
+ 18+12z +4z2 + 6+ 12z ln(1 z) , (5)
z − (cid:18) z − (cid:19) − )
α3C R (0) 2 2
g(z) = s F | S | π2 (2+z) ln(z)+ 18+ +18z +4z2 ln(z)
48π2m3cNc ( (cid:18)− z (cid:19)
6 1
+ 6+ 12z ln(1 z) ln(z) 3zln(z)2 +34+π2 2
z − − − z −
(cid:18) (cid:19) (cid:18) (cid:19)
71 53z 13z2 5
+ + − +9z 4z2 ln(1 z)
− 6z − 2 3 z − −
(cid:18) (cid:19)
+(18 12z) (z)+6 (2+z) [ (z) ln(z) (z) ζ(3)] , (6)
2 3 2
− L L − L − )
where
z ln(1 t)
(z) = − dt (7)
2
L − t
Z0
is the dilogarithmic function,
z (t)
2
(z) = L dt, (8)
3
L t
Z0
and ζ(3) 1.202. The logarithmic term f(z)ln[µ2/(4m2)] arises from the
≈ c
two-stepprocesswhereq qg splittingisfollowedbyg η g fragmentation;
c
→ →
the function f(z) can be written as
1 dy
f(z) = P (z/y)D (y), (9)
y q→g g→ηc
Zz
4
where P (z/y) is thestandard q qg splitting function [14] andD (y) is
q→g → g→ηc
the g η g fragmentation function at lowest order [1]. A similar result has
c
→
been obtained in the case of J/ψ production by light quark fragmentation
[12, 13].
A lower limit of the loop contribution (see Fig. 1b) is obtained by consid-
ering only the imaginary part of the loop amplitude. There is no logarithmic
term in this case:
4m2
D(b) (z,µ) h(z)+ c , (10)
q→ηc ≥ O µ2
(cid:16) (cid:17)
where
α4 R (0) 2C2 π2
h(z) = s| S | F 14(1 z) (1 z)
96πm3cNc ( − " 6 −L2 − #
2z z(7z2 18z +12)
+z + ln(z)+ − ln2(z) . (11)
1 z (1 z)2 )
− −
The functions f(z), g(z) and h(z) are plotted in Fig. 2, using α = 0.26,
s
−
R (0) 2 = (0.8 GeV)3, and m = 1.5 GeV. The loop contribution dominates
S c
| |
over the lower-order Born contribution for z > 0.3, even though the real part
∼
of theloopwas neglected. More quantitatively, thefifthmoments oftheBorn
and loop contributions have the numerical values
1 µ2
D(5,a) (µ) = dzz4D (z,µ) 2.4ln 5.1 10 7, (12)
q→ηc Z0 q→ηc ≈ " 4m2c!− #× −
1
D(5,b) (µ) dzz4h(z) 1.1 10 6. (13)
q ηc ≥ ≈ × −
→ Z0
Depending on the fragmentation scale µ, the contribution from the loop
diagram is thus up to an order of magnitude larger than the lowest-order
Borncontribution. Neglecting thehigher-orderprocesswouldleadtoamajor
underestimate of the fragmentation cross section.
It is possible to further simplify the calculation of the fragmentationfunc-
tionsbytakingadvantageofthefactthatonlythelargez regionisimportant,
5
25 -g(z)
20
6
- 15
0
h(z)
1

· 10
f(z)
5
0
0.2 0.4 0.6 0.8 1
z
Figure 2: The functions f(z), g(z) and h(z) as defined in the text.
−
6
due to the trigger bias effect. We have verified that using only the leading
part of an expansion of D(z,µ) around z = 1 changes the fifth moments of
the loop and Born contributions by less than 10%.
3. Discussion
The trigger bias effect in large p quarkonium production favors fragmenta-
T
tion processes where the quarkonium takes a large fraction z of the momen-
tum of the fragmenting parton. When estimating the relative importance
of different fragmentation processes, the shape of their z dependence must
therefore be considered.
In particular, some higher-order perturbative contributions may be en-
hancedrelativetothelowest-ordercontributionsduetothetriggerbiaseffect.
In this paper, we analyzed the process q η +X, where such an enhance-
c
→
ment can be expected because gluon emission is not required in higher-order
processes. Wefoundthatthereisaloopcontributionwhichindeeddominates
the Born contribution by a large factor.
It is likely that an analogous result is obtained in the case of q J/ψ
→
fragmentation. Some relevant Born and loop diagrams are shown in Fig.
3. At higher orders, all the gluons coupling to the heavy quark line can be
attached to the light quark line instead of being emitted, which suggests a
hard z dependence of the fragmentation function.
These higher-order contributions are part of the standard perturbation
series and thus do not bring in any new parameters. Their relative impor-
tance should depend only weakly on the quark mass (through the decrease of
α (m ) with m ). This is in qualitative agreement with total cross section
s Q Q
7
g
y
J/ y
J/
q q
g
a b
Figure 3: Light quark fragmentation into a J/ψ.
data [2, 3, 5, 6], which shows a disagreement with Born term calculations
(within the colour singlet model) of similar magnitude for bottomonium and
for charmonium.
The calculation presented here is not, however, immediately applicable
to the present data on quarkonium production. The primary production
mechanism for quarkonia at large p in hadron collisions is expected to be
T
gluon fragmentation. Even at higher orders, a minimum of two extra gluons
needtoaccompany aproducedJ/ψ,duetochargeconjugationinvariance(cf.
Fig. 4). In this case, loop diagrams like the one in Fig. 4b simply represent
radiative corrections to the lowest-order process. Whether they enhance the
kinematic region where the emitted gluons carry little momentum (the large
z region) can only be determined by an explicit calculation.
On the other hand, processes such as the one in Fig. 3b could be signifi-
cant in collisions where light quarks are more copiously produced relative to
gluons, such as at HERA. There, however, also charm quark fragmentation
becomes important as a charmonium production mechanism at large p [15].
T
8
In summary, we have pointed out that the trigger bias enhancement of
large z fragmentation is crucial in quarkonium production at large p . As
T
a specific example, we considered the q η fragmentation process and
c
→
calculated a higher-order perturbative correction whose contribution to the
cross section exceeds the lowest-order fragmentation contribution by a large
factor.
Acknowledgement. We are grateful for discussions with Stan Brodsky.
g g g
g
y y
J/ J/
g
g
a b
Figure 4: Gluon fragmentation into a J/ψ. (a) A lowest-order diagram. (b)
A higher order diagram.
Appendix
We describe here our calculation of the q η fragmentation functions. As
c
→
shown in Figs. 5 and 6a, we denote by p the momentum of the quarkonium
state; Q denotes the momentum of the fragmenting light quark, and s = Q2
is its virtuality.
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