Andreev, V.

Shishkina, T. V.

English

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c© 2008 Proceedings of the ”Foundations & Advances in Nonlinear Science” Conference-School
Effects of new physics and gauge theory construction in
polarized lepton annihilation on ILC
Tatyana Shishkina∗
Faculty of Physics, Belarusian State University,
4, Nezalezhnasti av., 220030, Minsk, Belarus
Vasili Andreev†
Gomel State University, Sovetskay street, 102, Gomel, 246699, Belarus
The matrix element of the annihilation process was obtained in the lowest order of pertur-
bative theory of electroweak interaction. Additional diagram which corresponds to possible
Z ′- exchange was included. Matrix element was reduced to scalar functions using method of
basis spinors. Differential and total cross sections, polarization and other additional types
of observables were calculated and analyzed. Several non-minimal gauge models were con-
sidered and analyzed. Probable kinematic and energy regions of deviations from Standard
Model were investigated and possible directions of deviations were suggested.
Keywords: Z′-boson, additional types of observables, method of basis spinors
1. Introduction
The investigation of effects beyond Standard Model will hold a significant part of research pro-
gramme at future experiments on International Linear Collider (ILC). The analysis of non-minimal
gauge models can provide some important information useful for future experiments. This paper is
devoted to the consideration of gauge models with additional Z ′-boson. Therefore we analyze the ex-
pansions of Standard Model (SM) that include Z ′ as the only additional gauge vector boson. Various
gauge models which can be the expansion of SU(2) × U(1) were suggested [1]-[12]. We analyze sev-
eral most popular models in modern physics and differentiate them in order to choose few ones which
are more suitable for future experimental research. Obviously we need the theoretical calculations of
considered processes to be of high precision. Methods which give precise Lorentz-invariant expressions
will be of great importance. Also it should be noted that the analysis of particles polarization gives
additional opportunity for new physics searching.
In this paper the process e− + e+ −→ f + f¯ is considered in frame of a set of gauge models including
additional Z ′- exchange. Cross sections and asymmetries in frame of several non-minimal gauge models
are calculated and analyzed. Kinematic and energy regions appropriate for physics beyond Standard
Model research are suggested.
2. Matrix element and cross sections
There are two Feynman diagrams of process e−+ e+ −→ f + f¯ in the lowest order of perturbative
theory. One corresponds to electromagnetic interaction, another — to weak interaction. Additional
∗E-mail: tshishkina@tut.by
†E-mail: selfdenied@tut.by
Proceedings of the F&ANS-2008 Conference-School, (2008) 89 - 93 89
T. Shishkina, V. Andreev: Effects of new physics and gauge . . . 90
diagram including Z ′ appears in different expansions of SM. We include it in consideration in frame of
nonminimal gauge theories of electroweak interaction. We consider p1 and p2, k1 and k2 to be four-
FIG. 1. Diagrams of the process with γ and Z,Z ′-exchange .
momenta of initial and final particles respectively, λp1 and λp2 , λk1 and λk2 — helicity of initial and
final particles respectively.
In accordance with the Feynman rules one can obtain matrix element of the process:
M = Mγ +MZ +MZ′ , (1)
where
Mγ = i Nk1Nk2Np1Np2e
2(2pi)4U¯(k1, λk1) γν V (k2, λk2)
1
q2
V¯ (p2, λp2)×
γνU(p1, λp1) δ(k1 + k2 − p1 − p2), (2)
MZ = i Nk1Nk2Np1Np2
g2
4 cos2 θW
(2pi)4U¯(k1, λk1) γ
µ (gV + gAγ5) V (k2, λk2)×
gµν − qµqνm2Z
q2 −m2Z
V¯ (p2, λp2) γ
ν (gV + gAγ5)U(p1, λp1) δ(k1 + k2 − p1 − p2), (3)
MZ′ = i Nk1Nk2Np1Np2
g˜2
4 cos2 θ′W
(2pi)4U¯(k1, λk1) γ
µ (g˜V + g˜Aγ5) V (k2, λk2)×
gµν − qµqνm2
Z′
q2 −m2Z′
V¯ (p2, λp2) γ
ν (g˜V + g˜Aγ5)U(p1, λp1) δ(k1 + k2 − p1 − p2), (4)
Here Np = 1/
(
(2pi)
3
2
√
2p0
)
, p0 =
√
p2 +m2, q = p1 + p2 = k1 + k2, gV = −12 + 2 sin2 θW , gA = −12 ,
θW — Weinberg angle, U and V — Dirac bispinors, g˜, g˜A, g˜V — additional coupling constants, which
depend on the model considered.
As one can see matrix element (1) includes additional term connected with Z ′-exchange. That is
why the reduction of matrix element to scalar functions in frame of the method of basis spinors (MBS)
was used.
This method allows to reduce matrix element itself using no traces. Due to its block structure, this
method can be easily programmed using a computer.
Proceedings of the F&ANS-2008 Conference-School, 2008
T. Shishkina, V. Andreev: Effects of new physics and gauge . . . 91
When evaluating a Feynman amplitude involving fermions, the amplitude is expressed as a sum
of terms which have the form
Mλp,λk (p, sp, k, sk ;Q) == u¯λp (p, sp)Q uλk (k, sk) , (5)
where λp and λk are helicities of particles with four-momentum p, k and arbitrary polarization vectors
sp, sk. The operator Q can be expressed as a sum of products of Dirac γ-matrices. The operator Q is
equal to γν in case of matrix element Mγ , γµ(gV + gAγ5) in case of MZ and γµ(g˜V + g˜Aγ5) in case of
MZ′ .
The matrix element (5) with Dirac spinors should be expressed in terms of scalar functions formed
from spin and momentum four-vectors of Dirac spinors, including p, sp, k, sk and operator Q. We use
MBS based on spinors which are connected with vectors of isotropic tetrad.
We investigate the expressions for differential and total cross sections in the center-of-mass system.
This system was chosen to obtain simple expressions for cross sections in accordance with the kinematics
of future experiments on linear colliders (ILC, for example).
The differential cross section in the center-of-mass system can be written as:
dσ
dθ
=
βf
βe
|R|2
16(2pi)2s
, (6)
where θ is the scattering angle between the direction of initial and final particle.
Total cross section can be written as:
σ = 2pi
∫ pi
0
dσ
dθ
sin θ dθ. (7)
The next section is devoted to numerical analysis of differential and total cross sections, asymmetries
for a set of gauge models.
3. Asymmetries and discussion
In this paper forward-backward, polarization asymmetries and other additional types of observ-
ables are considered.
Polarization asymmetry can be presented as:
ALR =
dσ+− − dσ−+
dσ+− + dσ−+
, (8)
where dσ+− = dσ/dθ (λp1 = 1; λp2 = −1), dσ−+ = dσ/dθ (λp1 = −1; λp2 = 1).
Forward-backward asymmetry can be introduced as:
AFB =
∫ pi/2
0 dσ −
∫ pi
pi/2 dσ∫ pi/2
0 dσ +
∫ pi
pi/2 dσ
. (9)
But the main purpose of this paper to investigate additional observables:
A1 =
∫ pi/2
0 dσ+− −
∫ pi
pi/2 dσ−+∫ pi/2
0 dσ+− +
∫ pi
pi/2 dσ−+
, A2 =
∫ pi/2
0 dσ−+ −
∫ pi
pi/2 dσ+−∫ pi/2
0 dσ−+ +
∫ pi
pi/2 dσ+−
,
A3 =
∫ pi/2
0 dσ−+ −
∫ pi
pi/2 dσ+−∫ pi/2
0 dσ+− +
∫ pi
pi/2 dσ−+
, A4 =
∫ pi/2
0 dσ+− −
∫ pi
pi/2 dσ−+∫ pi/2
0 dσ−+ +
∫ pi
pi/2 dσ+−
(10)
Proceedings of the F&ANS-2008 Conference-School, 2008
T. Shishkina, V. Andreev: Effects of new physics and gauge . . . 92
We consider a set of gauge models:
a) Sequential Standard Model (SSM) based on the group SU(2)L × U(1)Y × U ′(1)Y ′ ;
b) Two Left-Right Symmetric Models (LR) based on the group SU(2)L × SU(2)R × U(1)B−L, both
with certain value of free parameter α;
c) Three grand-unified E6-motivated models (E6) (λ, ψ, η models);
d) SU(3)C × U(1)Y -motivated model.
Differential and total cross sections were calculated for the considered set of gauge models (see
Figs. 2,3). As an example of final fermions a pair of muon - antimuon was considered. As a possible
value of Z ′ mass 1000 GeV was taken. As one can see the behavior of differential cross section is
FIG. 2. Differential cross sections for SSM, LR and SU(3)C × U(1)Y -motivated gauge models
similar for all models considered. In the case of λ model and LR model with free parameter α =
√
2/3
the value of differential cross sections is 2-3 times greater than that of Standard Model (SM). So one
can draw a conclusion that in the case when above mentioned models are realized, the deviations from
SM could be easily found in the forward kinematic region (0 ≤ θ ≤ pi/2).
In the case of other gauge models such deviations are much more difficult to found because the
difference between values of differential cross sections is less than 5 fb.
The situation for total cross sections is quite similar. It’s evident that in contrast to SM, all models
considered possess a Z ′ peak near 1000 GeV (Z ′ mass). It should be easily found during experiments.
The only problem is that the mass of Z ′-boson can be greater than 2, 3 or even more TeV .
The other feature that can be noticed is connected with the behavior of total cross sections before
Z ′ peak. One can see that total cross section behavior in λ, η models, LR model with α =
√
2/3 and
SU(3)C × U(1)Y -motivated model differs from that of the others. For example, in the case of λ model
deviations from SM can be found in the energy region from 600 GeV to 1000 GeV . In the behavior of
the function one can see a downward excursion and the minimal value is near 840 GeV . For LR model
the situation is almost similar. The only difference is that minimum of the function is near 800 GeV .
For SU(3)C × U(1)Y -motivated model this minimum is close to Z ′ peak and can be detected in the
region from 900 GeV to 1 TeV . For η model the location of minimum is situated in the similar region
as that of the latter model, but it’s not so expressed.
Proceedings of the F&ANS-2008 Conference-School, 2008
T. Shishkina, V. Andreev: Effects of new physics and gauge . . . 93
FIG. 3. Total cross sections for E6-motivated gauge models
So it’s very difficult to differentiate all the models considered judging only by differential and total
cross section. In order to succeed one has to look for some other magnitudes.
Additional extremely important information may be provided by additional types of observables
(10). On Figs. 4 one can see values of A1 and A2 observables for E6-motivated models. It’s obvious
that the results obtained during the experiment may indicate that deviations from SM occur according
to one or another model. Some model predictions are very close and hard to differentiate. That’s why
we need many observables to discriminate such models.
(a) (b)
FIG. 4. Additional observable A1 and A2 for E6-motivated gauge models
In the case of λ model A1 observable deviates at most from SM in the energy region between 750
and 850 GeV . In addition, it has the opposite sign than that of SM. But very close results are provided
Proceedings of the F&ANS-2008 Conference-School, 2008
T. Shishkina, V. Andreev: Effects of new physics and gauge . . . 94
by ψ and SSM models as well. So one have to look at A2 observable. For λ model A2 observable has
the same sign as that of SM on the most part of energy region while the behavior of function for ψ
and SSM models almost hasn’t changed. That is, calculation of different types of observables and their
further analysis provides us quite good method of gauge models discrimination.
In this paper calculations of differential and total cross sections, polarization and other types of
observables were made. Corresponding graphs were constructed and analyzed. As a result one can
see several problems in searching physics beyond Standard Model. It is clear that many non-minimal
gauge models provide very small deviations from Standard Model, so it’s difficult to register them.
To overcome this problem somehow one should investigate special observables where effects beyond
Standard Model are more essential. Additional types of observables presented and investigated in this
paper may be a useful tool for solving this problem.
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Proceedings of the F&ANS-2008 Conference-School, 2008


Effects of new physics and gauge theory construction in polarized lepton annihilation on ILC

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Effects of new physics and gauge theory construction in polarized lepton annihilation on ILC

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