QCD Corrections to Toponium Production at Hadron Colliders

17 Pages·0.17 MB·English

by  J. H. K"uhn

TTP92–32
Nov. 1992
QCD CORRECTIONS TO
TOPONIUM PRODUCTION
AT HADRON COLLIDERS1
J.H. Ku¨hn and E. Mirkes
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Institut fu¨r Theoretische Teilchenphysik
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Universit¨at Karlsruhe

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a Kaiserstr. 12, Postfach 6980
J

4 7500 Karlsruhe 1, Germany

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Abstract
e
h
:
v Toponium production at future hadron colliders is investigated. Per-
i
X turbative QCD corrections to the production cross section for gluon
r fusionarecalculatedaswellasthecontributionsfromgluon-quarkand
a
quark-antiquark collisions to the total cross section. The dependence
on the renormalization and factorization scales and on the choice of
the parton distribution functions is explored. QCD corrections to the
branching ratio of η into γγ are included and the two-loop QCD po-
t
tential is used to predict the wave function at the origin. The branch-
ing ratio of η into γZ, ZZ, HZ and WW is compared with the γγ
t
channel.
1Supported by BMFT Contract 055KA94P
1 Introduction
The production of heavy quarks at hadron colliders has received a lot of attention
from the experimental as well as from the theoretical side. Higher order QCD cal-
culations for open top quark production [1, 2, 3] provide the theoretical basis for
the current experimental limits [4] on the mass of the top quark. Given the cur-
rent range for the top mass of 90 to 180 GeV deduced from electroweak radiative
corrections [5] ongoing experimental studies at the TEVATRON should be able
to discover the top quark in the near future.
Planned experiments at LHC or SSC will lead to a huge number of tt¯events
and may pin down the top quark mass with a remaining uncertainty of perhaps
5 GeV [6]. Aside from the intrinsic interest in the precise top mass determination,
the aim to fix the input in one-loop electroweak corrections [7] constitutes one of
the important motivations for the drive for future increased precision in the top
mass.
In [8] it has been proposed to decrease the experimental error significantly
throughthestudyofη initsγγ decaychannel.Making useofthelargeluminosity
t
of future hadron colliders, one should be able to overcome the tiny cross section
multiplied by the small branching ratio into two photons. The excellent gamma
energy resolution, originally developed for the search for a light Higgs boson,
would in principle allow the measurement of the mass of the bound state at
about 100 MeV accuracy. This would result in a determination of the top quark
mass limited only by theoretical uncertainties. It was shown that the toponium
signal could be extracted from the γγ background with a signal to noise ratio
decreasing from 2/3 down to 1/8 for m between 90 and 120 GeV.
t
The estimates of [8] for the production and decay rate were based on the Born
approximation and on fairly crude assumptions on the bound state wave function
which was derived from a Coulomb potential, neglecting QCD corrections. The
signal to noise ratio and the statistical significance of the signal as estimated in
[8] is just sufficient for the discovery. Therefore a more precise evaluation of the
production rate is mandatory.
In lowest order the reaction is induced by gluon-gluon fusion (fig. 1a). QCD
corrections will be calculated in this work. They affect and indeed increase the
rate in this channel by about 50%. Quark-gluon scattering and quark-antiquark
annihilation into η ‘ enter at order α3. As we shall see the qg process increases
t s
the production cross section by about 8% whereas the effect of qq¯annihilation is
negligible.
The production cross section for η with nonvanishing transverse momentum
t
is infrared finite in the order considered here. The total cross section, however,
exhibitssingularitiesthathavetobeabsorbedinappropriatelychosenanddefined
1
parton distribution functions. The dependence on the factorization as well as
renormalization scales will be studied as well as the dependence on the choice on
differentparametrizationsofpartondistributionfunctions.AfullNLOcalculation
requires the corresponding corrections for the physically interesting branching
ratio. These are available and will be implemented.
The bound state production cross section and the branching ratio of η into
t
γγ depend (markedly!) on R(0)2, the square of the bound state wave function
at the origin and hence on the potential. The range of R(0) compatible with the
current knowledge of the perturbative two-loop QCD potential and the present
experimental results for Λ will be explored.
QCD
As we shall see in the following, fairly optimistic assumptions are necessary
to open even a narrow window for toponium discovery in the mass range of
m = 90 110 GeV. This is the consequence of the dominance single quark
t
−
decays (SQD) and the small branching ratio into two photons. The situation
improves dramatically if new hypothetical quarks are considered with suppressed
single quark decay. Examples are b′ or isosinglet quarks with small mixing with
ordinary d,s or b quarks. These would dominantly decay into gauge and Higgs
bosons, leading to spectacular signatures.
Theoutlayofthispaperisasfollows:Insection2theQCDcorrectionstogluon
gluonfusion into η andthe cross section forthe qq¯andqG initiated processes will
t
be derived. Insection 3the formulae forthe decay ratesofη will belisted andthe
t
dependence of the wave function on the choice of the potential studied. Section 4
contains the numerical evaluation of the production cross section and the study
of scale dependencies. Section 5 contains our conclusions. A brief account of this
work has been presented in [9].
2 QCD corrections to hadronic η production
t
For the complete evaluation of NLO corrections for η production in gluon gluon
t
fusion virtual corrections (fig. 1b) are required as well as corrections from real
gluon radiation (fig. 1c,d). In addition one needs the cross section for quark-
antiquark annihilation into η +gluon (fig. 2a) and the (anti-)quark-gluon (fig. 2b)
t
initiated reaction into η +quark. Technically it is quite convenient to employ
t
dimensional regularization for both ultraviolet and infrared singularities. The
calculation of amplitudes for bound state production (or decay) can be performed
in two different ways. One may either evaluate directly the amplitude for the
¯
production of a (QQ) bound state with the desired spin and orbital angular
momentum configuration and arrive at an amplitude proportional to the wave
function at the origin or its derivative [10, 11]; or alternatively one may evaluate
the production rate for open QQ¯ and subsequently identify the rate at threshold
2
with the rate for bound state production, substituting
R(0)2
dPS (k +k ;k ,...,k ,k ,k ) dPS (k +k ;k ,...,k ,q) (1)
n 1 2 3 n Q Q¯ n−1 1 2 3 n
−→ 2πM
where M = 2m + E and k = k = q/2. For the one particle final state
t Bind Q Q¯
relevant for resonant η production in gluon gluon fusion this implies (in 4 2ǫ
t
−
dimensions):
R(0)2
dPS (k +k ;k ,k ) 2πδ(s M2) (2)
2 1 2 Q Q¯ ⇒ 2πM −
and for the η + parton configuration that will be of main concern in the following
t
R(0)2 1 4πs ǫ 1
dPS (k +k ;k ,k ,k ) dt (3)
3 1 2 3 Q Q¯
⇒ 2πM 8πs ut Γ(1 ǫ)
(cid:18) (cid:19) −
where the Mandelstam variables are defined by
s = (k +k )2 t = (k q)2 u = (k q)2 = M2 t s (4)
1 2 1 2
− − − −
Inthelatterapproachitisimportanttomakesurethat—inthelimitofvanishing
relativevelocity—thecrosssectionreceivessolelycontributionsfromtheQQ¯ spin
parity configuration corresponding to the bound state. It is therefore mandatory
to restrict the evaluation of the amplitude to QQ¯ color singlet configurations and
in the reaction under consideration to those three gluon states that are totally
antisymmetric with respect to color as well as to their Lorentz structure.
The former approach leads typically to more compact expressions2. However,
for states with JPC = 0−+ the formula for the bound state amplitude involves
the γ5 matrix whose formulation in the context of dimensional regularization
introduces complications. This purely formal difficulty is absent in the second
approach, which for this reason has been adopted in this calculation.
2.1 The hadronic cross section
In this section we define the general structure of the cross sections for η produc-
t
tion in hadronic collisions
h (P )+h (P ) η +X γ +γ +X (5)
1 1 2 2 t
→ →
within the framework of perturbative QCD. Here h ,h are unpolarized hadrons
1 2
with momenta P . The hadronic cross section in NLO is thus given by
i
σH(S) = dx dx fh1(x ,Q2)fh2(x ,Q2)σˆab(s = x x S,α (µ2),µ2,Q2) (6)
1 2 a 1 F b 2 F 1 2 s F
Z
2
See for instance [13] for particularly impressive examples.
3
where one sums over a,b = q,q¯,g. fh(x,Q2) is the probability density to find
a F
parton a with fraction x in hadron h if it is probed at scale Q2 and σˆab denotes
F
the parton cross section for the process
a(k = x P )+b(k = x P ) η +X (7)
1 1 1 2 2 2 t
→
from which collinear initial state singularities have been factorized out at a scale
Q2 and implicitly included in the scale-dependent partondensities fh(x,Q2). We
F a F
next specify the possible partonic subprocesses that contribute to η production
t
in LO and NLO.
2.2 Gluon fusion
In Born approximation the cross section forgg η (fig. 1a) is given by (N = 3)
t C
→
1 1 1 R(0)
σˆgg = 2 2πδ(s M2) (8)
Born 2s644(1 ǫ)2 |MBorn| 2πM −
− X
with
N2 1
2 = C − g416(1 2ǫ)(1 ǫ) (9)
|MBorn| 4N s − −
C
TheQQ¯ statehasbeenprojectedontothecolorsingletconfiguration.Thedivision
by 64 4 (1 ǫ)2 is implied by the average over 82 gluon colors and (n 2)2 gluon
· · − −
helicities in n dimensions. g will be replaced by α through g2 = 4πα µ2ǫ. The
s s s s
Born cross section can therefore be cast into the form
π2α2R(0)2 (1 2ǫ)
σˆgg = s µ4ǫ − δ(1 z) (10)
Born 3s M3 (1 ǫ) −
−
where z = M2/s. Virtual corrections to the rate for QQ¯ production close to
threshold (fig. 1b) can be combined to yield3 [12]
4πµ2 ǫ
2 = 2 Γ(1+ǫ)
virtual Born Born
|M ∗M | |M | M2 !
X X
α π2 1 1 1
s
C +b N b +A (11)
× π F v 0ǫUV − Cǫ2IR − 0ǫIR !
where
4 π2 5
A T ln2+C 5 +N π2 +1
F F C
≡ − 3 4 − ! (cid:18)12 (cid:19)
11 2
b N n T (12)
0 C f F
≡ 6 − 3
3
Here CF =4/3; NC =3; and TF=1/2.
4
1/ǫ and 1/ǫ represent poles arising from ultraviolet and infrared divergent
UV IR
integrals respectively. The mass singularity 2ln(m /M) in [12] has been replaced
i
by 1/ǫ . Since we are considering bound state production the 1/v term (v de-
IR
notes the relative velocity of the two heavy quarks) in eq. (11) is absorbed in
the instantaneous potential and thus effectively dropped. The QCD coupling is
renormalized in the MS scheme and is given by
α α α 1
s = MS(µ2) 1 MSb +ln4π γ (13)
0 E
π π − π ǫ −
(cid:20) (cid:18) UV (cid:19)(cid:21)
Combining Born result and virtual corrections one thus obtains for resonant pro-
duction
α 4πµ2 ǫ µ2
σˆgg = σ (ǫ) 1+ s Γ(1+ǫ) b ln
Born+virtual 0 " π M2 ! 0 M2
1 1
N b +A δ(1 z) (14)
− Cǫ2IR − 0ǫIR !# −
where the normalization factor
1π2R2(0) (1 2ǫ)
σ (ǫ) = α2 (µ2)µ4ǫ − (15)
0 s 3 M3 MS (1 ǫ)
−
has been split off for convenience.
In a second step the cross section for
gg gQQ¯ (16)
→
has to be calculated (fig. 1c,d). Only the kinematical situation will be treated
¯
where the relative momentum of Q and Q vanishes. In an intermediate step the
matrix element for
ggg QQ¯ (17)
→
will be considered (fig. 1e,f) and reaction (16) can be subsequently obtained
through crossing. The QQ¯ state will be projected onto a color singlet configura-
tion through δ /√3. The overall wave function of the gluons is of course totally
ij
symmetric. Requiring a totally antisymmetric configuration in color space im-
plies, in turn, a totally antisymmetric wave function in Minkowski space. This
latter condition secures JPC = 0−+ for the QQ¯ bound state. (Conversely, requir-
ing symmetric wave functions in both color and Minkowski space would lead to
JPC = 1−−.)
The amplitude arising from the diagram fig. 1e is given by:
ˆ(k ,k ,k ;ǫ ,ǫ ,ǫ ) =
1 2 3 1 2 3
A
i[( k + k + k )/2+m]
1 2 3
u¯((k +k +k )/2)( ig ǫ ) − 6 6 6 ( ig ǫ )
1 2 3 − 6 1 [( k +k +k )2/4 m2] − 6 2
1 2 3
− −
i[( k k + k )/2+m]
1 2 3
− 6 − 6 6 ( ig ǫ )v((k +k +k )/2) (18)
× [( k k +k )2/4 m2] − 6 3 1 2 3
1 2 3
− − −
5
to be multiplied the color factor
λa1 λa2 λa3 δ 1 1
ij
= (if +d ) (19)
2 2 2 ! √3 √34 a1a2a3 a1a2a3
ij
As discussed above, only the f term will be retained. The amplitude is obtained
from the six permutations of the diagram without triple boson coupling
i 1
ˆ
= f ǫ (k ,k ,k ;ǫ ,ǫ ,ǫ ) (20)
A √34 a1a2a3 ijmA i j m i j m
i,j,m
X
The second type of amplitues, represented by fig. 1f, involve the triple gluon
vertex. Their sum reads as follows
ˆ(k ,k ,k ;ǫ ,ǫ ,ǫ ) =
1 2 3 1 2 3
B
i[( k k + k )/2+m]
1 2 3
u¯((k +k +k )/2) ( igγ ) − 6 − 6 6 ( ig ǫ )
1 2 3 " − α [( k1 k2 +k3)2/4 m2] − 6 3
− − −
i[( k + k k )/2+m]
1 2 3
+( ig ǫ ) 6 6 − 6 ( igγ ) v((k +k +k )/2) (21)
− 6 3 [(k1 +k2 k3)2/4 m2] − α # 1 2 3
− −
i
− ( g) (k k )αǫ ǫ +(k +2k )ǫ ǫα +( 2k k )ǫαǫ
×(k +k )2 − 1 − 2 1 2 1 2 1 2 − 1 − 2 1 2
1 2
n o
with a color factor
λbλa3 δ 1 1
ij
f = f (22)
2 2 ! √3 a1a2b √32 a1a2a3
ij
Three cyclic permutations are combined in this class of diagrams
i 1 2
= f ˆ(k ,k ,k ;ǫ ,ǫ ,ǫ ) (23)
B √34 a1a2a3 iB i j m i j m
i,j,mX(cyclic)
The ghost amplitude is easily obtained from eq. (21) through the replace-
C
ment of the curly bracket ... kβ, where k ,k and k denote the momenta
{ } → 2 1 2 3
of the incoming ghost, antighost and gluon respectively.
The squared matrix element is finally obtained in n = 4 2ǫ dimensions
−
1
2 = + 2 2 2 = f2 g6128(1 ǫ) (24)
|M|R |A B| − |C| 48 a1a2a3 −
a1,a2,a3
X X X X
st+tu+us 2 M8 +s4 +t4 +u4
(1 2ǫ)+4ǫM2
× (s M2)(t M2)(u M2)! " stu − #
− − −
where
s = (k +k )2 t = (k +k )2 u = (k +k )2; M = 2m+E (25)
1 2 1 3 1 3 Bind
6
After crossing (k k ) this result is applicable to the reaction of interest
3 3
→ −
(16). In the QQ¯ threshold region (with the color restriction discussed above) one
obtains
1 1 1
dσ(gg gQQ¯) = 2 dPS (26)
→ 2s 644(1 ǫ)2 |M|R 3
− X
:=g6|M|2
R
| {z }
where gluon helicities and colors were averaged as usual. Replacing the three- by
the two-particle phase space as indicated in eqs. (1) and (3) one obtains
α 3M2
s
dσ(gg gη ) = σ (ǫ) (27)
t 0
→ π 2 s
st+tu+us 2 M8 +s4 +t4 +u4 4πsµ2 ǫ dt
× (s M2)(t M2)(u M2)! stu ut ! Γ(1 ǫ)
− − − −
The term proportional 4ǫM2 in eq. (24) without singular denominator will not
contribute in the limit ǫ 0 and has therefore been dropped. In the limit ǫ 0
→ →
the result coincides with eq. (8.45) of [13]. For the subsequent discussion it will be
convenient to follow closely the treatment of NLO corrections for the Drell-Yan
process presented in the classical paper by G.Altarelli et al. [14]. The invariants
s,t and u are expressed in terms of M2,y and z by
M2 M2 M2
s = ; t = (1 z)(1 y); u = (1 z)y (28)
z − z − − − z −
and one arrives at Gluon
α 4πµ2 ǫ
σ(gg g(QQ¯)) = σ (ǫ) sΓ(1+ǫ)
→ 0 π M2 !
6 1
fggzǫ(1 z)1−2ǫy−ǫ(1 y)−ǫdy (29)
×Γ(1+ǫ)Γ(1 ǫ) − −
− Z0
The normalization factors have been chosen to allow for easy combination with
the virtual corrections and fgg is given by
(1 ǫ)
fgg = − M2 2
(1 2ǫ) |M|R
−
2
z (1 z)(1 y)y 1
= − − −
4 (1 z)[1 y(1 z)][(1 z)y +z]!
− − − −
1 1
z4 +1+(1 z)4((1 y)4 +y4) + (30)
× − − y 1 y!
h i −
As a consequence of the symmetry of fgg with respect to y (1 y) it suffices
↔ −
to evaluate the 1/y term. The y integration can be performed with the help of
7
the distribution identities
1 1 lnx
x−1−ǫ = δ(x)+ ǫ
−ǫ (cid:18)x(cid:19)+ − x !+
1 1 lnz ln(1 z)
zǫ(1 z)−1−2ǫ = δ(1 z)+ +ǫ 2ǫ −
− −2ǫ − (cid:18)1−z(cid:19)+ 1−z − (1−z) !+
1 1 ln(1 y) lny
y−1−ǫ(1 y)−ǫ = δ(y)+ ǫ − ǫ (31)
− −ǫ y! − y − y !
+ +
and
1 1 π2
= 1 ǫ2 +... (32)
Γ(1 ǫ)Γ(1+ǫ) − 6
−
Combining the Born result with real radiation and virtual corrections one obtains
α 4πµ2 ǫ
s
σ(gg η +X) = σ (ǫ) δ(1 z)+ Γ(1+ǫ)
→ t 0 ( − π M2 !
µ2 π2 1
b ln N +A δ(1 z) P (z)+N F(z) (33)
× " 0 M2 − C 3 ! − − ǫ gg C #)
where b and A are given in eqs. (12) and
0
11z5 +11z4 +13z3 +19z2 +6z 12
F(z) = Θ(1 z) −
− " 6z(1+z)2
1 ln(1 z)
+ 4 +z(1 z) 2 ln(1 z)+ 4 − (34)
(cid:18)z − − (cid:19) − 1−z !+
2(z3 2z2 3z 2)(z3 z +2)z ln(z) 1
+ − − − − 3
(1+z)3(1 z) − ! 1 z #
− −
and
1 1
P = 2N + +z(1 z) 2 +b δ(1 z); (35)
gg C 0
z (cid:18)1−z(cid:19)+ − − ! −
The 1/ǫ term can be absorbed in the gluon densities. If these are defined at a
factorization scale Q2, one effectively subtracts from the above result the term
F
α 4πµ2 ǫ 1
s
σ (ǫ) Γ(1+ǫ) P (z)+C (z) (36)
0 2π Q2F ! (cid:18)−ǫ gg gg (cid:19)
In the MS scheme C = 0, in the DIS scheme
gg
1 z
C (z) = n z2 +(1 z)2 ln − +8z(1 z) 1 (37)
gg f
− − z − −
(cid:20)(cid:16) (cid:17) (cid:18) (cid:19) (cid:21)
8
The partonic cross section is then given by
1π2R(0)2
σˆgg = α2 (µ2) δ(1 z)
s 3 M3 MS ( −
α (µ2) µ2 π2 π2 4
+ MS δ(1 z) b ln +N 1+ +C 5 T ln(2)
0 C F f
π * − M2 12! 4 − !− 3 !
Q2
ln F P (z) C (z)+N F(z) . (38)
gg gg C
− M2 − +)
2.3 Gluon quark scattering
Gluonquarkscatteringg(k )+q(k ) q(k )+η (P)proceedsevidently inleading
1 2 3 t
→
order α3 through the diagrams in fig. 2b. Collinear singularities have to be ab-
s
sorbedinthe structure functions.The (dimensionally regularized) squared matrix
¯
element for the production of QQ at threshold is given by
s2 +u2 1 ǫ
gq 2(s,t,u) = 32 (1 2ǫ)+ (39)
|M | "−(s+u)2 t − t#
and has to be supplemented by the color factor
2
1 λaλb λb 1
δ = (40)
ij
(cid:12)(cid:12)√3 2 2 !ij 2 !kl(cid:12)(cid:12) 3
X(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
and the coupling constant g(cid:12)6. After spin and color av(cid:12)eraging the cross section can
s
then be cast into a form identical to eq. (29) with fgg replaced by
1 z 1+y2(1 z)2 ǫ
fgq = − (41)
9(1 z)(1 y) "(1 y(1 z))2 − 1 2ǫ#
− − − − −
The y integration is performed as before and only a single pole remains
α 4πµ2 ǫ 1
σgq = σ (ǫ) sΓ(1+ǫ) P (z)
0 2π M2 ! −ǫ gq
(cid:26)
(1 z)2 2(1 z)
+ ln − P (z)+C − (1 lnz)+C z (42)
gq F F
z ! z − )
with
1+(1 z)2
P (z) = C − (43)
gq F
z
The 1/ǫ term can again be absorbed by the (quark) structure function. This
amounts effecively to the subtraction of
α 4πµ2 ǫ 1
s
σ (ǫ) Γ(1+ǫ) P (z)+C (z) (44)
0 2π Q2F ! (cid:18)−ǫ gq gq (cid:19)
9