Effective-Field-Theory Approach to Top-Quark Production and Decay

0.51 MB·English

by  Cen Zhang

Effective-Field-Theory Approach to
Top-Quark Production and Decay
Cen Zhang and Scott Willenbrock
1
1 Department of Physics, University of Illinois at Urbana-Champaign
0 1110 West Green Street, Urbana, IL 61801
2

n
a January 12, 2011
J

0
1
Abstract

]
h Wediscussnewphysicsintop-quarkinteractions,usinganeffectivefieldtheoryapproach. Weconsider
p top-quark decay, single top production, and top-quark pair production. We identify 15 dimension-six
- operators that contribute to these processes, and we compute the deviation from the Standard Model
p
induced by these operators. The results provide a systematic way of searching for (or obtaining bounds
e
on) physicsbeyond theStandard Model.
h
[

3
1 Introduction
v
9
6
8 Thestandardmodel(SM)ofthestrongandelectroweakinteractionshasbeenverysuccessfulphenomenolog-
3 ically. However,therearereasonsto believethatthe SMis notacomplete descriptionofnature. Itcontains
8. many arbitrary parameters with no connections, and provides no explanation for the symmetry-breaking
0 mechanism that gives rise to the masses of gauge bosons and fermions. In addition, the existence of dark
0 matter and darkenergy arenot accommodatedby the SM. It alsoprovides no explanationofthe strong CP
1
problem. Consequently,there couldbe further particles and interactions as one probes higher energy scales.
:
v These energy scales could be the Planck scale or an intermediate scale Λ.
i
X
AcompletedescriptionofanynewphysicsbeyondtheSMrequiresafundamentaltheory. Therearemany
r models proposedfor new physics. For example, the new particles could be those of supersymmetric theories
a
or fermions that transform under different representations of the SM gauge group. The new interactions
may also originate from quark substructure or preon exchange. Because of the diversity of these models, it
is useful to introduce a model-independent approach.
Ifthephysicsbeyondthe SMliesatanenergyscaleΛlessthan1TeV, thenweshouldbe abletoobserve
it directly at high-energy colliders. If it lies at a scale much greater than 1 TeV, then we can parameterize
its effects via higher-dimensionoperators,suppressed by inverse powers of the scale Λ. If the new physics is
too heavy to appear directly in low energy processes, then we can integrate it out from the Lagrangian. In
this way, the effective Lagrangianis:
1 1
L =L + L + L + (1)
eff 0 Λ 1 Λ2 2 ···
where L is the SM Lagrangian of dimension four, L is the new interaction of dimension five, L is of
0 1 2
dimension six, etc. The procedure is quite general and independent of the new interactions at scale Λ. The
only constraint is that all L are SU(3) SU(2) U(1) invariant.
i C × L× Y
1
At dimension five, the only operator allowed by gauge invariance is [1]
cij
L = (LiTǫφ)C(φTǫLj)+h.c. (2)
eff
Λ
where Li is the lepton doublet field of the ith generation and φ is the Higgs doublet field. When the Higgs
doublet acquires a vacuum-expectation value, this term givesrise to a Majoranamass for neutrinos. Due to
the tinyneutrinomasses,the scaleΛ isprobablyaround1015 GeV.Incontrasttothisunique dimension-five
operator, there are many independent dimension-six operators [2, 3].
Because the top quark is heavy relative to all the other observed SM fermions, we expect that the new
physics at higher energy scales may reveal itself at lower energies through the effective interactions of the
top quark, and deviations with respect to the SM predictions might be detectable. In this paper we will
study the effect of these dimension-six operators on top quark interactions at hadroncolliders. We focus on
three different processes: top quark decay, single top production, and top pair production. If no deviation
is observed experimentally, then one can place bounds on the coefficients of the dimension-six operators.
The effects of non-standardinteractions on top-quarkphysics at linear colliders and photon colliders can be
found in Refs. [4, 5, 6, 7].
We choose the effective Lagrangian to realize the weak symmetry linearly, as the precision electroweak
data favors a light Higgs boson. The situation where the weak symmetry is realized nonlinearly is studied
in Ref. [8, 9, 10, 11, 12]. We will use the operator set introduced by Buchmuller and Wyler [3]. In their
paper they categorize all possible gauge-invariantdimension-six operators, and use the equations of motion
(EOM)tosimplify theminto80independentoperators(foronegeneration). Subsequentlyitwasfoundthat
severalof these operators are actually not independent [13, 14, 15]. We focus on the operators that have an
influence on the top quark.
We expect the leading modification to SM processes at order 1 . In this paper we don’t consider higher
Λ2
order contributions. We expect the scale Λ to be large (at least largerthan the scale we can probe directly)
so 1 contributionswouldbesmallcomparedtotheuncertaintyontopquarkmeasurements. Thusweignore
Λ4
all dimension-eight (and higher) operators, as well as effects involving two dimension-six operators.
For any physical observable, the 1 contribution comes from the interference between dimension-six
Λ2
operatorsandtheSMLagrangian. Thiscontributionmightbesuppressedforavarietyreasons. Forexample,
since all quark and lepton masses are negligible compared to the top quark mass, a new interaction that
involves a right-handed quark or lepton (except for the top quark) has a very small interference with the
SM charged-current weak interactions, which only involve left-handed fermions. It turns out that although
there are a large number of dimension-six operators,only a few of them have significant effects at order 1 .
Λ2
We list these operators in Tables 1 and 2.
In Table 1, only one of the four-quark operators, O(1,3) = (q¯iγ τIqj)(q¯γµτIq), is listed explicitly. Here
qq µ
the superscripts i,j denote the first two quark generations, while q without superscript denotes the third
generation. In single top production, this is the only (independent) four-quark operator that contributes.
However,there are many other four-quarkoperatorswith differentisospinand colorstructures [2, 3]. In the
toppairproductionprocessqq¯ tt¯,sevensuchoperatorscontribute. Thedetails arediscussedinSection4.
→
In Table 2, the CP-odd operators are listed. These interactions interfere with the SM only if the spin of
the top quark is taken into account. The reasonis that the SM conserves CP to a good approximation(the
only CP violation is in the CKM matrix), and the inteference between a CP-odd operator and a CP-even
operator is a CP violation effect. It was shown in Ref. [16] that, in the absence of final-state interactions,
any CP violation observable can assume non-zero value only if it is T -odd, where T is the “naive” time
N N
reversal, which means to apply time reversal without interchanging the initial and final states. Thus an
observable is T -odd if it is proportional to a term of the form ǫ vµvνvρvσ. If we don’t consider the
N µνρσ
top quark spin, v must be the momentum of the particles, and such a term will not be present because
the reactions we consider here involve at most three independent momenta. Therefore top polarimetry is
essentialforthe studyofCPviolation. Since thetopquarkrapidlyundergoestwo-bodyweakdecayt Wb
→
with a time much shorter than the time scale necessary to depolarize the spin, information on the top spin
2
operator process
O(3) =i(φ+τID φ)(q¯γµτIq) top decay, single top
φq µ
O =(q¯σµντIt)φ˜WI (with real coefficient) top decay, single top
tW µν
O(1,3) =(q¯iγ τIqj)(q¯γµτIq) single top
qq µ
O =(q¯σµνλAt)φ˜GA (with real coefficient) single top, qq¯,gg tt¯
tG µν →
O =f GAνGBρGCµ gg tt¯
G ABC µ ν ρ →
O = 1(φ+φ)GA GAµν gg tt¯
φG 2 µν →
7 four-quark operators qq¯ tt¯
→
Table1: CP-evenoperatorsthathaveeffectsontop-quarkprocessesatorder1/Λ2. Hereq istheleft-handed
quark doublet, while t is the right-handed top quark. The field φ (φ˜ = ǫφ∗) is the Higgs boson doublet.
D =∂ ig 1λAGA ig1τIWI ig′YB isthecovariantderivative. WI =∂ WI ∂ WI+gǫ WJWK
µ µ− s2 µ− 2 µ− µ µν µ ν− ν µ IJK µ ν
istheW bosonfieldstrength,andGA =∂ GA ∂ GA+g fABCGBGC isthegluonfieldstrength. Because
µν µ ν − ν µ s µ ν
of the Hermiticity of the Lagrangian, the coefficients of these operators are real, except for O and O .
tW tG
The operator O(3) with an imaginary coefficient can be removed using the EOM.
φq
operator process
O =(q¯σµντIt)φ˜WI (with imaginary coefficient) top decay, single top
tW µν
O =(q¯σµνλAt)φ˜GA (with imaginary coefficient) single top, qq¯,gg tt¯
tG µν →
O =f G˜AνGBρGCµ gg tt¯
G˜ ABC µ ν ρ →
O = 1(φ+φ)G˜A GAµν gg tt¯
φG˜ 2 µν →
Table 2: CP-odd operators that have effects on top-quark processes at order 1/Λ2. Notations are the same
as in Table 1, and G˜ =ǫ Gρσ.
µν µνρσ
can be obtained from its decay products. CP violation will be discussed in Section 5.
Thereisanargumentthatcanbeusedtoneglectsomeofthenewoperators[17]. Somenewoperatorscan
be generatedattreelevelfromanunderlyinggaugetheory,whileothersmustbe generatedatlooporder. In
general the loop generated operators are suppressed by a factor of 1/16π2. However, the underlying theory
may not be a weakly coupled gauge theory, or the loop diagrams could be enhanced due to the index of a
fermion in a large representation. Furthermore, the underlying theory may not be a gauge theory at all.
Fortunately, the effective field theory approachdoes not depend on the underlying theory. We will consider
all dimension-six operators, without making any assumptions about the nature of the underlying theory.
We do not make any assumptions about the flavorstructure ofthe dimension-six operators,althoughwe
don’t consider any flavor-changing neutral currents in this paper. The charged-current weak interaction of
the top quark is proportionalto V , so the SM rate for top decay and single top production is proportional
tb
to V2. We write all dimension-six operators in terms of mass-eigenstate fields, so no diagonalization of the
tb
new interactions is necessary. Hence, in charged-currentweak interactions,the interference between the SM
amplitudeandthenewinteractionisproportionaltoV C ,whereC isthe(real)coefficientofthedimension-
tb i i
six Hermitian operator O (also recall that V itself is purely real in the standard parameterization [18]).
i tb
If the operator is not Hermitian, the coefficient C is complex; CP-conserving processes are proportional to
i
V ReC , while CP-violating processes are instead proportional to V ImC .
tb i tb i
Deviations oftop-quarkprocessesfromSMpredictions haveoften beendiscussedusinga vertex-function
approach, where the Wtb vertex is parameterized in terms of four unknown form factors [19]. Given our
precision knowledge of the electroweak interaction, this approach is too crude. The effective field theory
approach is well motivated; it takes into consideration the unbroken SU(3) SU(2) U(1) gauge sym-
C × L× Y
metry; it includes contact interactions as well as vertex corrections;it is valid for both on-shell and off-shell
quarks; and it can be used for loop processes [20]. None of these virtues are shared by the vertex function
approach [21].
3
2 Top Quark Decay
Whenthe fermionmasses(exceptforthe topquark)areignored,thereareonlytwoindependentdimension-
six operators in [3] that contribute to top-quark decay at leading order:1
O(3) = i(φ+τID φ)(q¯γµτIq) (3)
φq µ
O = (q¯σµντIt)φ˜WI (4)
tW µν
The operators O(3) and O modify the SM Wtb interaction. Upon symmetry breaking, they generate the
φq tW
following terms in the Lagrangian:
C(3)gv2
L = φq ¯bγµP tW−+h.c. (5)
eff Λ2 √2 L µ
C
L = 2 tWv¯bσµνP t∂ W−+h.c. (6)
eff − Λ2 R ν µ
(3)
where v = 246 GeV is the vacuum expectation value (VEV) of φ. The operator O simply leads to a
φq
C(3)v2
rescaling of the SM Wtb vertex by a factor of (1+ φq ), so it does not affect any distributions, and is
Λ2Vtb
therefore impossible to detect in angular distributions of top-quark decays. The vertex-function approach
to top-quark decay is pursued in Refs. [22, 23].
TheseoperatorsinterferewiththeSMamplitude,asisshowninFigure1. Wecancomputetheircorrection
to the SM amplitude. The t be+ν squared amplitude is:
→
1ΣM 2 = Vt2bg4u(m2t −u) + Cφ(3q)Vtbv2g4u(m2t −u) + 4√2ReCtWVtbmtmW g2su (7)
2 | | 2(s m2 )2 Λ2 (s m2 )2 Λ2 (s m2 )2
− W − W − W
where C is the coefficient of operator O , and s,t,u are generalizations of the usual Mandelstam variables
i i
(s=(p p )2, t=(p p )2, u=(p p )2). C(3) is real.
t− b t− ν t− e+ φq
Figure1: TheFeynmandiagramsfort be+ν. (a)istheSMamplitude;(b)representsthevertexcorrection
→
induced by the operator O(3) and O .
φq tW
Using the narrow width approximation for the W boson, the differential decay rate is
dΓ = V2 + 2Cφ(3q)Vtbv2 g4 (m2 m2 )2[m2+m2 +(m2 m2 )cosθ](1 cosθ)
dcosθ tb Λ2 !4096π2m3tmWΓW t − W t W t − W −
ReC V g2
+ tW tb m2 (m2 m2 )2(1 cosθ) (8)
128√2π2Λ2m2Γ W t − W −
t W
1TheoperatorODt=(q¯Dµt)Dµφ˜listedinRef.[3]canberemovedusingtheEOM[14].
4
(3)
SM and O
φq
O
tW
θ
s
o
c
d
/
Γ
d
0 π/2 π
θ
Figure 2: The differential decay rate induced by different operators. The curves are normalized so that the
area is the same.
Here θ is the angle between the momenta oftop quark andthe neutrino in the W restframe, and Γ is the
W
width of the W boson. In the SM, at tree level Γ is given by:
W
3α
W
Γ = m . (9)
W W
4
The angular dependence is shown in Figure 2. The curves are normalized to have equal areas. The
contributionfromO(3) isthesameastheSMcontribution,becauseO(3) simplyrescalesthe SMWtbvertex.
φq φq
It therefore does not affect angular distributions. The angular dependence of the contribution from O is
tW
not dramatically different from the SM.
The partial width is given by
Γ= V2+ 2Cφ(3q)Vtbv2 g4(m6t −3m4Wm2t +2m6W) +ReC V g2m2W(m2t −m2W)2 . (10)
tb Λ2 ! 3072π2ΓWm3tmW tW tb 64√2π2Λ2ΓWm2t
Both dimension-six operators affect the partial width. The total width is given by the above expression
times afactorofnine. Unfortunately,itisnotknownhowtomeasurethe partialortotalwidths inahadron
collider environment.
Wealsoconsiderthe energydependenceofthe leptonsinthetopquarkrestframe. TheSMcomputation
can be found in [24, 25]. The correctionfrom dimension-six operators at leading order is:
dΓ = V2+ 2Cφ(3q)Vtbv2 g4Ee+(mt−2Ee+) + ReCtWVtbg2m2W(mt−2Ee+)
dEe+ tb Λ2 ! 128π2mWΓW 16√2π2Λ2ΓW
dΓ = V2+ 2Cφ(3q)Vtbv2 g4(−4Eν2m2t +2Eν(m3t +2m2Wmt)−m2W(m2t +m2W))
dE tb Λ2 256π2m2m Γ
ν ! t W W
ReC V g2m2 (2E m m2 )
+ tW tb W ν t− W (11)
16√2π2Λ2m Γ
t W
where m2 /2m < E ,E < m /2 are the energies of the electron and neutrino, respectively. We don’t
W t e+ ν t
list the energy dependence of the bottom quark, because the narrow width approximation for the W boson
is used and the energy of the bottom quark is given by E = (m2 m2 )/2m . These results are shown
b t − W t
in Figure 3 and 4. Again the curves are normalized so that the areas are the same. Compared to Figure
5
(3) (3)
SM and O SM and O
φq φq
O O
tW tW
e ν
E E
d d
Γ/ Γ/
d d
20 40 60 80 20 40 60 80
E (GeV) E (GeV)
e ν
Figure 3: The energy dependence of the electron. Figure 4: The energy dependence of the neutrino.
2, the two curves are more distinct, which implies the effect of O would be more apparent in the energy
tW
distribution of the leptons.
The angulardistribution and the energydistribution are not independent. The energy ofthe leptons are
fixedinthe W restframe. Thereforetheirenergyinthe topquarkrestframeisgivenbyaboost,whichonly
depends on the angle θ:
1
E = (E+ q cosθ) (12)
ν
2 | |
1
E = (E q cosθ) (13)
e+ 2 −| |
whereE =(m2+m2 )/2m and q =(m2 m2 )/2m aretheenergyandmomentumoftheW bosoninthe
t W t | | t− W t
top quark rest frame. Furthermore, both the angular distribution and energy distribution can be expressed
using the W helicity fractions [22]:
1 dΓ 3 3 3
= (1+cosθ)2F + (1 cosθ)2F + sin2θF (14)
R L 0
Γdcosθ 8 8 − 4
and
1 dΓ 1
= 3(E E )2F +3(E E )2F +6(E E )(E E )F
ΓdE (E E )3 e+ − min R max− e+ L max− e+ e+ − min 0
e+ max min
−
(cid:0) (1(cid:1)5)
where E =m /2 and E =m2 /2m , F =Γ /Γ are the W boson helicity fractions, corresponding to
max t min W t i i
positive (R), negative (L), or zero (0) helicity. The helicity fraction is affected by the operator O :
tW
m2 4√2ReC v2m m (m2 m2 )
F = t tW t W t − W
0 m2+2m2 − Λ2V (m2+2m2 )2
t W tb t W
2m2 4√2ReC v2m m (m2 m2 )
F = W + tW t W t − W
L m2+2m2 Λ2V (m2+2m2 )2
t W tb t W
F = 0 (16)
R
These equationsmake manifestthe earlierobservationthatthe operatorO(3), whichsimply rescalesthe SM
φq
vertex,cannotaffectanydistributions. Thustop-quarkdecayis sensitiveonlyto theoperatorO ,andcan
tW
be used to measure (or bound) its coefficient.
Finally,weinvestigatethepolarizeddifferentialdecayrate. Intherestframeofthetopquark,theangular
distribution of any top quark decay product is given by [24, 25]
1 dΓ 1+α cosθ
i i
= (17)
Γdcosθ 2
i
6
where θ =θ ,θ ,θ is the angle between the spin axis of the top quark and the momentum of the bottom
i b v e+
quark, neutrino or positron. The “analyzing power” α measures the degree to which the direction of the
i
decay product i is correlated with the top spin. If dimension-six operators are added, the relation still
holds, but the coefficient α will be affected by the new operators. Since O(3) is just a rescaling of the SM
i φq
interaction,the only correctionis fromO . This could be anindependent way to determine the coefficient
tW
ReC . At leading order, the correction is given by:
tW
m2 2m2 ReC v28√2m m (m2 m2 )
α = t − W + tW t W t − W
b −m2+2m2 Λ2V (m2+2m2 )2
t W tb t W
m6 12m4m2 +3m2m4 (3+8ln(m /m ))+2m6
α = t − t W t W t W W
v m6 3m2m4 +2m6
t − t W W
ReC v212√2m m (m6 6m4m2 +3m2m4 (1+4ln(m /m ))+2m6 )
tW t W t − t W t W t W W
− Λ2V (m2+2m2 )2(m2 m2 )2
tb t W t − W
α = 1 (18)
e+
Thesameequationsholdforhadronictopdecay,withαu =αν,αd¯=αe+. Thecoefficientαe+ isnotaffected
by dimension-six operators. This is consistent with the results in Ref. [26].
The measurementof these coefficients requiresa source of polarizedtop quarks. This is addressedin the
next section.
3 Single Top Production
Single top quarks are produced through the electroweak interaction. There are three separate processes:
s-channel [27], t-channel [28, 29, 30], and Wt production [31]. An effective field theory approach to the s-
and t-channel processes was advocated in Ref. [32]. We update that analysis by including an additional
operator,whichwasneglectedinthatstudybecause itisloop-suppressedifthe underlyingtheoryisagauge
theory. We also perform an effective field theory analysis of the Wt process. The vertex-function approach
to single-top production is pursued in Refs. [33, 34, 35].
Single top production contains four distinct channels: the s-channel process ud¯ t¯b, the t-channel
processes ub dt and d¯b u¯t, and the Wt associated production channel gb Wt. W→e first consider the
→ → →
s and t channels. The following operators contribute [32]:
O(3) = i(φ+τID φ)(q¯γµτIq) (19)
φq µ
O = (q¯σµντIt)φ˜WI (20)
tW µν
O(1,3) = (q¯iγ τIqj)(q¯γµτIq) (21)
qq µ
For the four-quark operator O(1,3), the superscripts i,j denote the first two quark generations. Another
qq
four-quark operator that could contribute is (q¯iγ q)(q¯γµqj). However, using the Fierz identity, this can
µ
be turned into a linear combination of O(1,3) and some other four-quark operators with different isospin
qq
and color structures which do not contribute to this process. Four-quark operators are neglected in the
vertex-function approachto the Wtb vertex.
The Feynman diagrams are shown in Figure 5. Since the operator O will be measured (or bounded)
tW
fromstudiesoftop-quarkdecay,thes-andt-channelproductionofsingletopquarkscanbeusedtomeasure
(or bound) the operators O(3) and O(1,3).
φq qq
Now we turn to consider the gb Wt process. The contributing operators are
→
O(3) = i(φ+τID φ)(q¯γµτIq) (22)
φq µ
O = (q¯σµντIt)φ˜WI (23)
tW µν
O = (q¯σµνλAt)φ˜GA (24)
tG µν
7
Figure5: Feynmandiagramsforthes-andt-channelsingletopproduction. (a-c)arethes-channeldiagrams,
while (d-f) are the t-channel diagrams. (a,d) are the SM amplitude, (b,e) are the correction from O(3) and
φq
O , and(c,f) are the four-fermioninteractionfromO(1,3). The diagramsfor the t-channelprocess d¯b u¯t
tW qq
→
can be obtained by interchanging u and d quarks in (d-f).
Again, the first two operators O(3) and O will affect the Wtb coupling. The “chromomagnetic moment”
φq tW
operator O modifies the gtt coupling:
tG
ReC
L = tGv t¯σµνλAt GA (25)
eff √2Λ2 µν
(cid:0) (cid:1)
This interaction is neglected in the vertex-function approachto the Wtb vertex.
The Feynman diagrams are shown in Figure 6. Since the operators O(3) and O will be measured
φq tW
(or bounded) from single-top production and top-quark decay, respectively, the Wt associated production
process can be used to measure (or bound) the operator O , which is also present in tt¯production (see
tG
Section 4).
Here we list all the corrections to the SM amplitudes and cross sections. The squared amplitude of the
three channels are:
s-channel:
41Σ|Mud¯→t¯b|2= Vt2b+ 2Cφ(3qΛ)V2tbv2!g44(us(−um−2Wm)2t2) − 2√2ReCtWΛ2VtbmtmW (s−g2msu2W)2 + 2Cq(Λq1,32)Vtbg2us(−um−2Wm(22t6))
t-channel:
14Σ|Mub→dt|2= Vt2b+ 2Cφ(3qΛ)V2tbv2!g44(ts(−s−m2Wm)2t2) − 2√2ReCtWΛ2VtbmtmW (t−g2mst2W)2 + 2Cq(Λq1,32)Vtbg2ts(−s−m2Wm(2t2)7)
14Σ|Md¯b→u¯t|2= Vt2b+ 2Cφ(3qΛ)V2tbv2!g44(ut(−um−2Wm)2t2) − 2√2ReCtWΛ2VtbmtmW (t−g2mu2Wt )2 + 2Cq(Λq1,32)Vtbg2ut(−um−2Wm(22t8))
8
Figure 6: The Feynman diagrams for Wt associated production process. (a,b) are the SM amplitude. (c,d)
(3)
are corrections due to the operator O and O . (e) is a modification on the gtt vertex.
φq tW
Wt associated production:
916Σ|Mgb→Wt|2 = Vt2b+ 2Cφ(3qΛ)V2tbv2!24m2Wgs2(tgs2−m2t)2 m8t −(2s+t)m6t +((s+t)2−2tm2W −2m4W)m4t
(cid:0)
(t(s+t)2 2(s2 st+2t2)m2 +2tm4 4m6 )m2
− − − W W − W t
2tm2 (s2+t2 2(s+t)m2 +2m4 )
− W − W W
2ReC V g2m m
+ tW tb s t W 3m6 (2s+3t(cid:1)+6m2 )m4
3√2Λ2s(t m2)2 t − W t
− t
(s2+2st 3t2 6m4(cid:0))m2+t(s2 2st 3t2+6(s+t)m2 6m4 )
− − − W t − − W − W
ReC V2g2g m v
+ tG tb s t (m2+2s t) (cid:1) (29)
3√2Λ2(m2 t) t −
t −
As before, C is the coefficient of operator O and s,t,u are the usual Mandelstam variables. We have set
i i
V =1 for simplicity. The differential cross sections are as follows:
ud
dσud¯→t¯b = V2+ 2Cφ(3q)Vtbv2 g4(s−m2t)2 (1+cosθ) s+m2+(s m2)cosθ
dcosθ tb Λ2 !512πs2(s−m2W)2 t − t
(cid:0) (cid:1)
g2m m (s m2)2
+ReC V t W − t (1+cosθ)
tW tb
16√2πΛ2s(s m2 )2
− W
g2(s m2)2
+C(1,3)V − t (1+cosθ)(s+m2+(s m2)cosθ) (30)
qq tb64πΛ2s2(s m2 ) t − t
− W
with θ the angle between up quark and top quark momenta in the center of mass frame;
dσub→dt = V2 + 2Cφ(3q)Vtbv2 g4(s−m2t)2
dcosθ tb Λ2 32πs(2m2 +(s m2)(1 cosθ))2
! W − t −
g2m m (s m2)2(1 cosθ) g2(s m2)2
+ReC V t W − t − C(1,3)V − t (31)
tW tb4√2πΛ2s(2m2 +(s m2)(1 cosθ))2 − qq tb8πΛ2s(2m2 +(s m2)(1 cosθ))
W − t − W − t −
9
(3) (1,3) (3)
SM, O and O SM and O
φq qq φq
O O
tW tW
θ θ (1,3)
s s O
o o qq
c c
d d
σ/ σ/
d d
0 π/2 π 0 π/2 π
θ θ
Figure 7: The s-channel differential cross section at Figure 8: The t-channel (ub dt) differential cross
→
√s=2m . section at √s=2m .
t t
dσd¯b→u¯t = V2 + 2Cφ(3q)Vtbv2 g4(s−m2t)2(1+cosθ)(s+m2t +(s−m2t)cosθ)
dcosθ tb Λ2 128πΛ2s2(2m2 +(s m2)(1 cosθ))2
! W − t −
g2m m (s m2)3sin2θ
ReC V t W − t
− tW tb8√2πΛ2s2(2m2 +(s m2)(1 cosθ))2
W − t −
g2(s m2)2(1+cosθ)(s+m2+(s m2)cosθ)
C(1,3)V − t t − t (32)
− qq tb 32πΛ2s2(2m2 +(s m2)(1 cosθ))
W − t −
with θ the angle between bottom quark and top quark momenta in the center of mass frame;
dσgb→Wt = V2+ 2Cφ(3q)Vtbv2 g2gs2λ1/2 (m2+10m2 )s3
dcosθ tb Λ2 !1536πs3m2W(s+m2t −m2W −λ1/2cosθ)2 t W
(cid:0)
+(3m4+19m2m2 22m4 )s2 (9m6+8m4m2 +5m2m4 22m6 )s+5(m2 m2 )3(m2+2m2 )
t t W − W − t t W t W − W t − W t W
(m2+2m2 )λ3/2cos3θ+ (6m2 m2)s m4 m2m2 +2m4 λcos2θ
− t W W − t − t − t W W
(14m2 m2)s2 2(m4(cid:0) 7m2m2 +6m4 )s+3(m6 3m2m4(cid:1)+2m6 ) λ1/2cosθ
− W − t − t − t W W t − t W W
(cid:0) g2m m λ1/2 (cid:1) (cid:17)
ReC V s t W 5s3 9(m2 m2 )s2
− tW tb96√2πΛ2s3 s+m2 m2 λ1/2cosθ 2 − t − W
t − W − (cid:0)
+(19m4t +10m2tm2W −2(cid:0)9m4W)s−15(m2t −m2W)3+(cid:1)3λ3/2cos3θ−(5s−3m2t +3m2W)λcos2θ
3s2 10(m2 m2 )s 9(m2 m2 )2 λ1/2cosθ
− − t − W − t − W
(cid:0) g2g vm λ1/2 m2 m2 +5s (cid:1) λ1/2cosθ(cid:17)
+ReC V2 s t t − W − (33)
tG tb 96√2πΛ2s2 m2 m2 +s λ1/2cosθ
(cid:0) t − W − (cid:1)
with θ the angle between gluon and to(cid:0)p quark momenta in the ce(cid:1)nter of mass frame, and
λ = s2 +m4 +m4 2sm2 2sm2 2m2m2 . The angular dependence at √s = 2m (recall that the
t W − t − W − t W t
kinematic threshold is √s=m ) is shown in Figures 7-10 (areas are normalized). The total cross sections
t
are:
10