Table Of ContentΥ radiative decays to light quark jets and color octet mechanism
Ying-Jia Gao (a),Yu-Jie Zhang (a), and Kuang-Ta Chao (b,a)
(a) Department of Physics, Peking University, Beijing 100871, People’s Republic of China
(b) China Center of Advanced Science and Technology (World Laboratory), Beijing 100080, People’s Republic of China
WestudyradiativedecaysofΥtolightquarkjetsinnonrelativisticQCDbytakingboththecolor
singletandcoloroctetb¯boperatorsintoconsideration. Thecutforquarkjetenergyandcutforthe
angle between two quark jets are introduced. The sensitivity to the soft and collinear singularities
in the loop integrals are greatly reduced by these cuts. With the jet energy cut of about 1 GeV,
andthejet angle cut of about 36◦,thebranchingratio for Υ→γqq¯isfound to be8.2×10−4 from
color singlet contributions. The color octet contributions could be much larger than that of color
singlet, depending on the estimate of the color octet matrix elements. This process may provide a
7
new test for thecolor octet mechanism in nonrelativistic QCD.
0
0
PACSnumbers: 12.38.Bx;13.25.Hw;14.40.Gx
2
n
a I. INTRODUCTION ences).
J Based on the above considerations, we will study the
2 three body decay Υ γqq¯, where q = u,d,s. We use
2 ggTghheaosbsbeerevnativoinewoefdΥadsecaaycriunctioaltherveeen-gtlufoonr jveetrsifΥyin→g tthheebr¯besbuoltusntdhsrtoautgehdbes→octrhipctoiolonr-fsoirnΥgleitnaNnRdQcoClDor-aoncdtegtivb¯eb
v quantum chromodynamics (QCD). The radiative decay
operators. In NRQCD, with the Fock state expansion
70 ΥQC→D.γgTghehsaesparlsoocebseseens aarneimcopmoprtuatnatblteesitningpegrrtouurbnadtifvoer in terms of v, the relative velocity between b and ¯b, the
decaywidthforΥ(1S)canbewrittenasasumofcontri-
1
6 QCD due to the large mass of the b quark. Recently butions fromvariousb¯bchannelswith differentcolorand
the CLEO Collaboration has accumulated a large num-
0 angular momentum:
ber data sample for the low lying Υ resonances [1, 2].
6
0 This encourages us to study the radiative decay process dΓ[Υ(1S)] = dΓ[b¯b (3S )] Υ (3S )Υ +dΓ[b¯b (3S )]
1 1 1 1 8 1
h/ Υligh→t qγuqaq¯r,kwjhetesr.e q and q¯(q = u,d,s) are referred to as × hΥ|O8(3S1)|Υh i|+OdΓ[b¯b8|(1Si0)]
p
Υ (1S )Υ + (2J +1)
- Jets are consideredto be footprints of quarksand glu- × h |O8 0 | i
p ons produced atshortdistances. While the gluonjet, its (cid:0)XJ
he production angular distribution, and the determination × dΓ[b¯b8(3PJ)] hΥ|O8(3P0)|Υi, (1)
: ofthegluonspinhaverecentlybeenstudied[3],lessatten- (cid:1)
v where the long distance matrix elements with different
tion has been paid to quark jets, say, e.g. in Υ decays.
i color (denoted by the subscript 1(8) for color-singlet
X Since the gluon jet is expected to be ”fatter” than the
(octet)) are involved. In Section 2 we will describe our
r quark jet[4] or the mean multiplicity of the gluon jet is
a expected to be larger than the quark jet (see e.g. [5]), it method and give a calculation for the color singlet con-
tribution. In section 3, we will study color octet contri-
will be interesting to measure the quark jets production
butions. Then we will give a summary in Section 4.
in comparison with the gluon jets in Υ radiative decays.
In heavy quarkoniumproductionand decay animpor-
tant issue is about the color octet mechanism in nonrel-
II. COLOR SINGLET CONTRIBUTIONS
ativistic QCD (NRQCD)[6], which is introduced to ex-
plain the large cross sections of inclusive charmonium
Theqandq¯dijetproductionthroughΥradiativedecay
hadroproductionatlargep in pp¯collisionsmeasuredat
T at leading order is described by the process
the Fermilab Tevatron[7], and other decay and produc-
tion processes. Though the color octet mechanism has Υ(P) γg∗g∗ γ(k )q(k )q¯(k ) (2)
1 2 3
gained a series of successes, some difficulties still remain → →
and more tests are needed (for a recent comprehensive as shown in Fig.1 (another three diagrams with reverse
review, see [8]). In this regard, the radiative decay pro- light quark lines are also taken into account).
cess Υ γqq¯ may provide an interesting test for the In our work, all outgoing particles are taken to be on-
color-oc→tet mechanism, since the color-octet b¯b compo- mass-shell, k2 = k2 = k2 = 0 and P2 = M2. Here
1 2 3 Υ
nent(withsoftgluons)intheΥwavefunctioncouldgive the light quark q (with momentum k ) and antiquark q¯
2
substantial contributions to this process (for discussions (withmomentumk )aretakentobe massless,asforthe
3
on color-octet contributions in Υ hadronic and radiative photon (with momentum k ). We can see explicitly that
1
decays, see, e.g. [9, 10, 11], see also[8] for more refer- thereexistbothcollinearandsoftsingularitiesintheloop
2
γ
q q
Υ Υ
q¯ q¯
γ
(a) (b)
q
Υ γ
q¯
(c)
FIG. 1: Feynman diagrams for Υ→γ+q+q¯with thecolor singlet b¯b
integralswithintheregionwherethe lightquarkandan- fermionflowcontributeafactorC togetherwithacolor
F
tiquark interact with two gluons and are connected with unit matrix. Gauge invariance ensures that we may
a massless quark propagator. In general, these soft and sum over the photon spin by employing the substitu-
collinear mass singularities may be separated by artifi- tions ε ε∗ = g . To sum over the Υ spin, we use
µ ν − µν
ciallyintroducingaquarkmassandagluonmass. Inthe ǫρǫP(∗)σ = gρσ+PρPσ/M2.
present case of dijet production, in order to define two PInΥoΥur calc−ulation, we choose two parameters to de-
jets, the jet energy has to be higher than certain energy scribethejets: oneistheminimumenergycutofthe jet,
scale, and the two jets have to be well separated by in- which is, say, about 1 GeV, and the other is the angle
troducing certain value for the angle between q and q¯. between the two jet axes,which is, say,about 36◦ (these
With these two infrared cuts, the sensitivity of soft and values are just for illustration). We choose the following
collinear singularities will become much weaker. region:
WeusetheNRQCDdescriptionfortheb¯bboundstate.
Inthestaticapproximation,inwhichtherelativemotion x s , z z , (4)
0 0
ofthebottomquarksintheboundstateisneglected,the ≥ | |≤
decay amplitude factorizes into a short distance ampli- where x is the energy fraction of the jet (k0 = mx),
tude (b¯b γqq¯),withb¯binthecolor-singletstatewith and z=cosθ, θ is angle between two jet axes.
M →
zerorelative momentum, and a long distance matrix ele- Sincewehavetakensomestepstoseparatethesoftand
mentwhich is relatedto (0), the wavefunction ofΥ at collinear divergences, we can complete the phase space
R
the origin. integralin4-dimension. Thestandardformofthreebody
Within the static approximation, we write down the phase space in 4-dimension is:
amplitude of Fig.1(a) (for other related processes, see,
e.g., Ref.[12]) 3 d3~k
dΦ = i (2π)4δ4(P k ). (5)
3 2k0(2π)3 − i
= CFgs4eQ R(0) ε(∗) d4q u¯ γ q/γ v (3) iY=1 i Xi
M 8√N πM 2p k λ,k1Z (2π)4 i µ ν j
c · 1 With the parameters
Tr Π (P,M)γν(q/+k/ p/+m)γµ(p/ k/ +m)γλ
1Sz 3− − 1 .
(cid:2) q2 (q k )2 (q+k )2 ((q+k p)2 m2) (cid:3) 2Pki
2 3 3 x = , x =2, (6)
− − − i M2 i
X
where p = P/2, m = M/2, i and j are color indexes, g
s
we can get
and e are the strong and electromagnetic couplings re-
spectively, and Q is the magnitude of the bottom quark
charge in units of e. Charge conjugation invariance im- M2 3
dΦ = dx δ(2 x ). (7)
plies that the Feynman graphs are symmetric under re- 3 2(4π)3 i − i
Y X
i=1
version of the fermion flow. All six amplitudes con-
tributing to the leading-order cross section are propor- With the following constrains:
tional to the same color factor 1/(2√N ) δ , and the
c ab
δ combined with the color matrix of the right hand s x 1, s x 1, cosθ z , (8)
ab 0 2 0 3 0
≤ ≤ ≤ ≤ | |≤
3
we can get the integration region of phase space: z wave function of Υ contains high Fock states with color-
0
cosθ z fors x (2 2s +r)/(2+s z s )−; z≤ octet b¯b pair and associated soft gluons . We can see
0 0 2 0 0 0 0 0
cos≤θ (2 2≤s +r≤)/(s−x )+1 2/s for−(2 2s−+ that the color-octet b¯b operators (3S ), (1S ), and
0 0 1 0 0 8 1 8 0
≤r)/(2+≤s z −s ) x (2 2s −+r)/(2 s z− s ), (3P ) will contribute to the pOrocess Υ O γqq¯. The
0 0 0 2 0 0 0 0 8 J
− ≤ ≤ − − − O →
wherer=m2/m2. Herefornumericalestimatewechoose corresponding diagrams are shown in Fig.2.
q
s = 0.2 and z = 0.8, corresponding to the jet energy
0 0 The color octet operators e.g. (3S ) and (1S )
cut 0.95 GeV and the angle 36◦ between q and q¯. O8 1 O8 0
are given by
In our calculation, we define Υ(1S)(3S ) 9
hO1 1 i ≃ 2Nc
(0)2[13], and take m M/2=4.73 GeV, and
|R | ≈ (3S ) = ψ†σTaχ χ†σTaψ,
8 1
O ·
αs =0.184, |R(0)|2 =7.12 GeV3, (9) O8(1S0) = ψ†Taχχ†Taψ. (11)
which are extracted from the observed decay widths of
Υ 3g and Υ e+e− with theoretical expressions for Differing fromthe colorsingletcase,the colorprojection
→ →
these decay widths including next to leading order QCD of the octet is
radiative corrections. We set the gluon mass as 10−6–
10−8 GeV, together with the light quark mass m as
10−2–10−3GeV,andfindthenumericalresultsarestqable h3i;3j| 8ai = √2 Tiaj. (12)
asthese massestendtozero. Finallywefindthe branch-
ing ratio for the dijet productionin the color-singletsec-
The spin projection operator e.g. Π (P,M) is [14]:
tor to be 00
Br(Υ γqq¯)=8.2 10−4. (10)
→ ×
1
Π (P,M) = (P/ +M)γ . (13)
00 5
−√4 M
III. COLOR OCTET CONTRIBUTIONS
Next we take the color octet contributions into con- We can write down the decay amplitudes of the color
sideration. With the velocity expansion in NRQCD the octet b¯b as shown in Fig.2(a),(b), and (c) respectively:
Tr[Π (P,M)γµ] 2p/ k/
= i√2 g2e Q Tr[Ta Tb]ε(∗)(k ) 1Sz qγλ − 3 Taγ q,
Ma − s q λ 1 4m2 (2p k )2 µ
3
−
Tr[Π (P,M)γµ(p/ k/ )γλ]
= i√2 g2e Q Tr[Ta Tb]ε(∗)(k ) 00 − 1 qTaγ q,
Mb s b λ 1 (k +k )2[(p k )2 m2] µ
2 3 1
− −
qTaγ q
= i√2 g2e Q Tr[Ta Tb]ε(∗)(k ) µ Tr[Πα (P,M)γµ(p/ k/ )γλ]
Mc − s b λ 1 (k +k )2[(p k )2 m2] 1Sz − 1
2 3 − 1 − (cid:2)
2 kα
+ Tr[Π (P,M)γµγαγλ]+ 1 Tr[Π (P,M)γµ(p/ k/ )γλ] , (14)
1Sz (p k )2 m2 1Sz − 1
− 1 − (cid:3)
whereQ is the lightquarkcharge. Thenwecanexpress can write down the branching ratio as
q
the decay width as functions of the jet energy cut s
0
and the jet angle cut z0. With s0=0.2 (corresponding to Br(Υ→γqq¯) = 2.4×10−4hΥ|O1(3S1)|Υi
the jet energy cut 0.95 GeV) fixed, the decay width (in + 0.061 Υ (3S )Υ
8 1
arbitrary units) as functions of z are shown in Figs.4- ×h |O | i
0 + 0.084 Υ (1S )Υ
6. We see that the decay width increases nearly linearly ×h |O8 0 | i
+ 0.043 Υ (3P )Υ (15)
as z increases when z < 0.8, whereas when z > 0.8 8 0
0 0 0 ×h |O | i
the decay width will increase rapidly. This may imply
that taking the jet angle cut to be z < 0.8 (θ > 36◦)
0
Fornumericalresultsofthe decaybranchingratio,the
is a reasonable choice to avoid the sensitivity of infrared
main uncertainty comes from the estimates of the color-
singularities.
octetmatrixelements. Themostdirectwayis tousethe
Choosing α (M )=0.184 and α=1/137, and includ- velocity scaling rules. However, it is shown in Refs.[16,
s Υ
ingcolor-singletandvariouscolor-octetcontributions,we 17]thatbyusingrenormalizationgroupequationstheob-
4
q
γ
(3S )8
1
q¯
(a)
γ q γ q
(1S )8 (3P )8
0 J
q¯ q¯
(b) (c)
FIG. 2: Feynman diagrams with thecolor octet b¯b .
G G
35
12
30
10
25
8
20
6
15
10 4
5 2
0.2 0.4 0.6 0.8 z0 0.2 0.4 0.6 0.8 z0
FIG. 3: Decay width (in arbitrary units) as afunction of the FIG.5: Decay width (in arbitrary units) as afunction of the
angle cut from the color octet 3S1 operator. angle cut from thecolor octet 3PJ operator.
G hO8Υ(1S)(3S1)i 2.0±4.1−+00..56 3.0±3.8+−00..21
6 hO8Υ(1S)(1S0)i 13.6±6.8−+170.5.8 0
5 hOΥ(1S)(3P )i 0 13.9±7.1+11.4
5 m2 8 0 −8.0
b
4
TABLEI:ColoroctetmatrixelementsforΥ(inunitsof10−2
3 GeV3), taken from Table V in Ref.[13].
2
1 and we then get the color octet 3S contribution to be
1
0.2 0.4 0.6 0.8 z0 Br(Υ γqq¯) = 1.33 10−3, (17)
→ ×
FIG. 4: Decay width (in arbitrary units) as afunction of the which is larger than the color singlet contribution by a
angle cut from the color octet 1S0 operator. factor of 1.6.
Next, we use the results from the analysis of Υ pro-
duction at the Fermilab Tevatron, and we adopt the
tainedcoloroctetmatrixelementsaresmaller. Nonethe- assumption that the production matrix elements are 3
less, in our calculation we still choose velocity scaling times larger than the annihilation matrix elements just
ruleswithv2 =0.08forΥ[15]asaroughlyestimate. An- like the case of color singlet matrix elements. Note
other estimate can be made from the Υ production at that the color octet matrix element Υ(1S)(1S ) and
the Fermilab Tevatron using NRQCD [13], and the ex- 5 Υ(1S)(3P ) can not be determinhOed8indepen0diently,
tracted matrix elements are shown in Table.I (for more m2 hO8 0 i
and the values shown in Table.I from [13] are a sort of
discussions on the color octet matrix elements, see e.g.
simplification. We use the central values in the first col-
Refs.[18, 19]). umn (the 3P matrix element is zero), and in the second
0
Asthefirstestimate,wesetthecoloroctet1S0and3P0 column (the 1S0 matrix element is zero) respectively to
matrixelementstobezero,andusethefollowingvelocity calculate the decay branching ratio, and the results are:
scaling rule relation for the 3S matrix element[15]:
1
Br(Υ γqq¯)=4.21 10−3,
→ ×
Υ (3S )Υ = v4 Υ (3S )Υ . (16) Br(Υ γqq¯)=9.31 10−3. (18)
8 1 1 1
h |O | i h |O | i → ×
5
theresultverymildly,andchangeoftheanglecutbrings
36
34
32 28
30 26
-3 222210)2468 222024 133SS01
2(0 18 PJ
111B/dcos468 dcos 1146
1d2 dB/ 12
10 10
8 8
6 6
4
2 4
0 2
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0
cos
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
cos
FIG. 6: Decay branching ratio angular distributions. The
dotted line is for the color singlet contribution, dashed line FIG. 7: Angular distributions for different color octet opera-
for 3S1 and 3PJ color octet contributions, and solid line for tors.
3S and 1S color octet contributions.
1 0
a nearly linear effects in the region cosθ < 0.8. This
Thefirstbranchingratioiscontributedfrom (3S )and | |
8 1 implies that in this region those large logarithms which
(1S ), and the second is from (3S ) anOd (3P ).
8 0 8 1 8 J are introduced by regularizing the soft and collinear di-
O O O
These branching ratios are really big! To compare with
vergencesonly givesmallcontributionsand maythen be
the contribution from color singlet, we draw two group
ignored. Therefore, our numerical results should be rea-
curvesbasedonthelattercalculation. Fig.6showsangu-
sonable.
lardistributionsofthedecaybranchingratiofromdiffer-
Based on our calculation in NRQCD, we predict the
ent channel contributions. We see that both in the large branching ratio of Υ γqq¯to be 8.2 10−4 if contribu-
angleregionandthe smallangle region,coloroctet gives tions from the color s→inglet b¯b are con×sidered only .
greatercontribution than singlet does. If experimentally On the other hand, the color octet b¯b operators may
the jet angular distribution can be measured then we
play an important role in this process. When we take
might find useful hints of color octet contributions.
them into consideration, the branching ratio could be
Fig.7 shows the branching ratio angular distributions
enhanced by about an order of magnitude [see Eq.(18)].
foreachindividualcoloroctetoperators. Herewechoose
This result is very encouraging. In this sense, the pro-
Υ(1S)(3S ) =2.0 10−2 GeV3. The values for other
hO8 1 i × cess Υ γqq¯ may provide an interesting test for the
two operators are from Table.I. These curves are also →
color octet mechanism in NRQCD. However, large un-
usefulindistinguishingbetweencontributionsofdifferent
certaintiesinthe estimate ofcolor-octetmatrixelements
color octet operators.
would prevent us from making definite predictions for
this process. A better understanding of the color octet
matrix elements in the future will certainly improve our
IV. SUMMARY
predictions.
WehopethatthemeasurementonΥ γqq¯willclarify
In this paper, we studied two mechanisms i.e. the →
the theoretical issues presented in this paper.
color-singletandcolor-octetmechanisms that contribute
to the process Υ γqq¯. With the help of FeynCalc
→
[20] and LoopTools[21], we calculated numerically the
Acknowledgments
radiative decay rates by introducing the energy cut for
the q and q¯ jets, and the angle cut between q and q¯
jets. These cuts correspond to a minimum jet energy We thank E. Berger for bringing their earlier work on
of about 1GeV and the angle between two jets larger inelastic photoproduction of Υ (Ref.[12]) to our atten-
thanθ =36.8◦. Experimentally,these cutsarenecessary tion. We would also like to thank Ce Meng for valu-
forthe two jets measurement. Theoretically,introducing able discussions. This work was supported in part by
these cuts greatly reduces the sensitivity to the soft and the National Natural Science Foundation of China (No
collinearsingularitiesinthe loopintegrals. Furthermore, 10421503), and the Key Grant Project of Chinese Min-
fromourstudieswefindthatchangeofenergycutaffects istry of Education (No 305001).
6
[1] G.Masek et al., CLEO Collaboration, Phys.Rev.D 65, [12] E.L.BergerandD.L.Jones,Phys.Rev.D23,1521(1981).
(2002)072002. [13] E.Braaten,S.Fleming,A.K.Leibovich, Phys.Rev.D63,
[2] D.Besson et al, CLEO Collaboration, hep-ex/0512003. 094006 (2001).
[3] TheOPALcollaboration, G.Abbiendietal., Phys.Rev. citeKuhn
D 69, 032002(2004). [14] J.H.Ku¨hn,Nucl.Phys.B157,125(1979);B.Guberina
[4] The CLEO collaboration, M. S. Alam et al., Phys. Rev. et al.,Nucl. Phys.B 174, 317 (1980).
D 46, 4822(1992). [15] S. Fleming, A. K. Leibovich , Phys. Rev. D 67, 074035
[5] S.J. Brodsky and J. Gunion, Phys. Rev. Lett. 37, 402 (2003).
(1976); K. Konishi, A. Ukawa, and G. Veneziano, Phys. [16] M. Gremm and A. Kapustin, Phys. Lett.
Lett. 78B, 243 (1978); J.W. Gary,Phys. Rev.D49, 4503 407B(1997)323.
(1994). [17] F. Maltoni and A. Petrelli, Phys. Rev. D59 (1999)
[6] G.T.Bodwin, E.Braaten andG.P.Lepage,Phys.Rev. 074006.
D 51 , 1125 (1995); 55, 5853(E) (1997). [18] N.Brambilla, D.Eiras, A.Pineda,J.SotoandA.Vairo,
[7] CDF Collaboration, F. Abe et al, Phys. Rev. Lett. 79, Phys. Rev.Lett. 88 (2002) 012003.
578 (1997). [19] G. T. Bodwin, J. Lee and D. K. Sinclair, Phys. Rev. D
[8] N.Brambilla et al.,arXiv:hep-ph/0412158. 72(2005) 014009.
[9] K. Cheung, W.Y. Keung, and T.C. Yuan, Phys. Rev. [20] R. Mertig, M. B¨ohm, A. Denner, Comput. Phys. Com-
D54, 929 (1996). mun. 64 (1991) 345.
[10] C.W.Bauer,C.W. Chiang,S.Fleming, A.K.Leibovich, [21] T. Hahn, M. Perez-Victoria, Comput. Phys. Commun.
I.Low, Phys. Rev.D 64, 114014 (2001). 118 (1999) 153.
[11] S. Fleming, C. Lee, and A. K. Leibovich, Phys. Rev. D
71(2005) 074002.
