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NON-ABELIAN TENSOR GAUGE THEORY
ON LOWER RANK FIELDS
By
Spyros Konitopoulos
SUBMITTED FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
AT
NATIONAL TECHNICAL UNIVERSITY OF ATHENS
ATHENS, GREECE
MAY 2009
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NATIONAL TECHNICAL UNIVERSITY OF ATHENS
DEPARTMENT OFSCHOOL OF APPLIED MATHEMATICAL AND PHYSICAL
SCIENCES
The undersigned hereby certify that they have read and recommend
to the Faculty of Graduate Studies for acceptance a thesis entitled
Non-Abelian Tensor Gauge Theory on lower rank fields
by Spyros Konitopoulos for the degree of Doctor of Philosophy.
Dated: May 2009
Research Supervisor:Georgios Savvidis
Examing Committee:George Zoupanos
George Koutsoumbas
ii
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NATIONAL TECHNICAL UNIVERSITY OF
ATHENS IN COLLABORATION WITH THE NATIONALCENTER OF SCIENTIFIC RESEARCH DEMOKRITOS
Date: May 2009
Author: Spyros Konitopoulos
Title: Non-Abelian Tensor Gauge Theory on lower rank
fieldsDepartment: School of Applied Mathematical and Physical
Sciences
Degree: Ph.D. Convocation: May Year: 2009
Signature of Author
iii
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To Giannis Vasilogiorgakis (R.I.P.)
iv
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Table of Contents
Table of Contents v
Acknowledgements vii
Introduction 1
1 Basic formulation 101.1 Yang-Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 rank-2 tensor (part I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3 Geometrical Interpretation - Gauge bosons . . . . . . . . . . . . . . . 22
1.4 rank-2 tensor (part II) . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Unitarity of the theory at the non-interactive level 312.1 A method for counting the propagating modes of a gauge theory . . . 312.1.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.2 Vector Gauge Field . . . . . . . . . . . . . . . . . . . . . . . . 332.1.3 Symmetric Tensor Gauge Field . . . . . . . . . . . . . . . . . 34
2.1.4 Antisymmetric Tensor Gauge Field . . . . . . . . . . . . . . 36
2.1.5 Rank-2 tensor gauge theory . . . . . . . . . . . . . . . . . . . 37
2.2 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3 The Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . 44
2.4 Geometrical representation - Fermions . . . . . . . . . . . . . . . . . 48
3 Unitarity of the theory at the interactive level-Processes 523.1 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 ff TT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2.1 The Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . 583.2.2 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.3 Squared Matrix Element . . . . . . . . . . . . . . . . . . . . . 63
v
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3.2.4 Evaluation of Traces . . . . . . . . . . . . . . . . . . . . . . 67
3.2.5 Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.6 Helicity Structure . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 VV-TT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.1 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . 773.3.2 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3.3 Helicity Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 84
Conclusions 89
Appendices 91
A Free rank-2 Tensor Gauge Boson Equation Matrix 92
B Elements of Group Theory 95
C Evaluation of Traces 97
D Processes 101D.1 f f V V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
D.1.1 Feynman diagrams-the scattering amplitude . . . . . . . . . . 101
D.1.2 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . 102D.1.3 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
D.2 V V V V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107D.2.1 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . 107
D.2.2 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . 110
D.2.3 Scattering amplitudes . . . . . . . . . . . . . . . . . . . . . . 112D.2.4 Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
D.3 e+e W+W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114D.3.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
D.3.2 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . 115D.3.3 eRe
+L W+W . . . . . . . . . . . . . . . . . . . . . . . . . . 117
D.3.4 e+ReL W+W . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Bibliography 124
vi
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Acknowledgements
First of all I would like to thank my supervisor Prof. G.Savvidy for the pedagogical
guidance and support. Without his patience and tolerance I would not be able to
get in touch with the Russian-school perspective. In days of cultural crisis, where allaspects of human relations are overwhelmed by a commodity exchange value fetishism,
this man realizes a rebirth of the notion of the teacher. The least I can do is to express
him my deepest appreciation.
Many thanks to Prof. S.Bonanos for letting me use his Mathematica programm
for doing tensor algebra calculations. His program was an essential prerequisite for
my further programming.
The University of Crete has poisoned my brain irreparably. This was the place
where the seeds of doubt and method were sown upon me. S.Trahanas played an
important role on my decision to focus on theoretical physics, Prof. T.Tomaras hewho initiated me in its secrets and P. Ditsas the one who trusted and encouraged me
to carry on. I thank them all.
People outside the physics community have their share on my present devel-
opment. I would like to thank E.Karidi (pushed me off the beaten track ), E.Gavra
(opened my eyes to self-realization), D.Papaevagelou, D.Zaglis, F.Dilari, Th.Papagergiou
for uncovering and illuminating my inner contradictions.
Finally, I cannot resist the temptation to attribute my respect to some writ-
ers and artists whose influence was enormous to my cultural and mental evolution.
F.Nietzche, K.Marx, M.Bakunin, S.Freud, C.Castoriades, N.Kazantzakis, F.Dostoyevsky,
S.Beckett, J.S.Bach, L.W.Beethoven, P.Gabriel, I.Anderson, R.Fripp, Judas Priest,
Psychotic Waltz...
Demokritos, Athens, Greece Spyros Konitopoulos
May 24, 2009
vii
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Introduction
For more than 45 years the research in high energy physics has been conducted
under the guidance of the gauge principle. Demanding the localization of the global,
continuous symmetries of a theory one is straightforwardly led to the introduction
of some new vector gauge fields with uniquely determined dynamics. These fields
mediate the interactions between the initial fermionic fields in accordance with the
normative principles of locality, causality and Lorentz invariance [1, 2].
In the early 70s, a self consistent theory of the elementary particles and their
interactions was constructed, named the standard model (SM) [3, 4, 5, 6, 7, 11, 12].
The basic elements of the theory are the principle of local gauge invariance and the
Higgs mechanism [8, 9, 10].
Leptons and quarks are classified in 3 generations in each of which they are eigen-
states of 5 irreducible representations of the symmetry group SU(3)SU(2)LU(1)Y.In addition, a complex doublet of scalar fields should accompany the initially, free,
massless fermions, giving the vacuum a non-vanishing expectation value. Demanding
the theory be locally invariant under the symmetry group SU(3) SU(2)L U(1)Y,gives rise to 12 massless, bosonic fields and uniquely determine their dynamics. They
are the agents of the strong, weak and electromagnetic forces. It can be seen, that the
initial symmetry is spontaneously broken to SU(3) U(1)em and that the 3 degrees
1
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2
of freedom which would correspond to the massless Goldstone bosons are absorbed
by linear combinations of the SU(2)L U(1)Y gauge fields casting them massive. TheHiggs mechanism completes its mission when the initial Lagrangian is supplemented
by Yukawa interaction terms which will provide masses to the fermions. One finally
ends up with a theory which needs 25 external input parameters to work.
Up to energy scales of 100 GeV, the theory is more than successful. Most of
the predictions of the model have already been tested (neutral weak interactions,
W, Z0 parameters, top quark etc.)[53]. There is however one piece that escapes the
experimental confirmation: The Higgs boson. For perturbation theory to work, the
spontaneous symmetry breakdown is expected to occur at the regime of 250 GeV.
Hence, it is expected that the mass of the Higgs boson should be less than 1 T eV
[16, 17]. Fortunately this lies within the potentials of the LHC and hopefully it will
be detected.
Despite the experimental confirmations that the model has succeeded, and the
fact that there is no experimental evidence against it, there are theoretical arguments
that the SM does not provide us with a complete theory of particle physics and their
interactions.
First of all, gravity escapes the unification scheme. A renormalizable quantum
theory of gravity has become a torturing accomplishment due to the divergences the
theory exhibits. The theory breaks down at the singularities at energies of the order
of Mp 1019
GeV[15].
But even if one stays in lower energy scales, one will confront the so called natu-
ralness problem [16]. This problem rises from the very existence of the fundamental
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3
scalars of the theory. It can be seen that the the Higgs mass receives quadratic di-
vergent contributions from higher order diagrams. This means that the Higgs mass
is expected to be of the order of the cut off, above which yet undiscovered physics
prevail. But, as it was mentioned above, in order for the perturbation theory to work
-small coupling constants- the Higgs mass should be less than 1 GeV. The only way
to overcome this problem, within the framework on the SM, is to make some high
accuracy assignments (fine tuning) that will drop the value of the Higgss mass down
to the preferred energy scale.
Along with the naturalness problem comes the hierarchy problem [18, 19]. It
seems that something is missing. Something that would give an adequate explanation
why there is such a big gap between the energy scale at which the spontaneously
symmetry breakdown occurs and the scale at which new physics dominate. One
might legitimately guess that that there is a hidden symmetry which, broken in our
energy region, gives rise to non-zero, small parameters.
Finally, a theory with such a huge parameter space (25 input parameters) cannot
fulfill ones aesthetic demands. Many questions concerning the choice of the specific
pattern, the electric charge quantization etc, remain unanswered.
An interesting suggestion beyond the SM is provided by the grand unification
theory (GUT)[20]. The theory proposes that above the energy scale of 1015 GeV the
symmetry group of the SM is unified to a semi simple group (usually SU(5)). The
three interactions with their three coupling constants are combined in one, while the
particles are now classified in 2 representations of the symmetry group. It is proposed
that the initial gauge symmetry breaks spontaneously at 1015 GeV to the standard
model group SU(3) SU(2)L U(1)Y. Despite its magnificent tested prediction of
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4
the weak mixing angle, the theory predicts a proton lifetime of the order of 1031 years,
which contradicts with experiment. Another problematic issue of the model is that
the running coupling constants do not meet at an exact point, though they come very
close at 1014 GeV.
The next revolutionary step comes with the advent of Supersymmetry[21, 22].
The theory is motivated by the naturalness and hierarchy problems of the SM. It is
suggested that for every SM particle, a Supersymmetric partner should be introduced
and that above an energy scale, the world is symmetric under the interchange of
bosons with fermions. The gauge bosons (S=1) are accompanied with gauginos (S= 12
)
and the chiral fermions(S= 12) with complex scalars (S=0). Self consistence of the
theory demands the introduction of an additional complex scalar doublet, together
with their fermionic superpartners (S=12
). The role played by the partners is amazing!
They contribute in the higher order corrections of the Higgs mass, in such a way that
they cancel the quadratic divergences. Now, the value of the Higgs mass drops down
to the scale where the spontaneous symmetry breakdown
SU(3) SU(2)L U(1)Y SU(2)L U(1)Y
occurs, rendering the model natural[23].
Not only does Supersymmetry provide stability and naturalness to the Standard
Model, but also corrects the problems encountered in the framework of GUT. The
contributions of the Supersymmetric partners to vacuum polarization lead to an accu-
rate unification of the running coupling constants at 1016 GeV. Also, the prediction
for the life time of the proton extends to more that 1032 years, thus overcoming the
conflict with experiment.
Many ambitions have been invested to Supersymmetic theories, the greater of
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5
which lies on its local version: Supergravity. It is believed that this is the direction
towards the unification of gravity with all other forces.
A different proposal for a theory of everything comes from string theory. To
solve the divergence problem of quantum gravity, it is suggested that the interactions
are spread out by means of increasing the dimensions of the elementary particles.
Assuming that the structural elements of nature are 1 dimensional objects (strings),
a short distance cut-off is provided in harmony with Lorentz invariance and causality
[15].
Many models in string theory predict an infinite tower of particles of arbitrary
high spin in their spectrum [14, 15]. Especially, in the low energy limit of the open
string theory with Chan-Paton charges [27], the massless states can be identified
with Yang-Mills quanta. It would be interesting if these states would appear in the
spectrum of a Quantum Field Theory Lagrangian. This motivates a generalization of
Yang-Mills symmetry to include non-Abelian tensor gauge fields[31].
Under such a generalization, Quantum Field Theory is extended to a theory which
includes in its description fields of arbitrary high spins. Following the Yang-Mills
paradigm, the extended gauge transformations for higher rank fields are suitably
chosen, in order to form a closed algebraic structure [31]. Interpreting geometrically
the tensor gauge fields as coefficients of the expansion of an extended gauge field
over its tangent space, it has been seen that there are two series of gauge invariant
Lagrangians, each of which sums up all corresponding invariant Lagrangians of every
rank [31]. A similar procedure has been followed in the description of fermionic and
bosonic matter tensor fields, making possible an extension of the symmetry breaking
mechanism and thus of the electroweak theory to include particles of arbitrary high
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spins [31, 33].
It was further seen that the linear combination of the two Lagrangians that de-
scribe the free rank-2 tensor gauge bosons, exhibits an enhanced gauge symmetry. In
addition to the original gauge transformations of tensor fields, the theory is shown
to be symmetric under a complementary transformation for a special choice of the
ratio of the weights with which the two gauge invariant Lagrangians participate in
the linear combination. It is because of the enhancement of the gauge symmetry that
the cancellation of the pure gauge modes is possible [31]. The same program has been
followed in the series of Lagrangians that describe fields up to the fourth rank. It has
been shown that for a suitable choice of the coefficients, the total Lagrangian exhibits
an enhancement of the gauge symmetry, a condition which may possibly be proven
enough to guarantee the unitarity of the theory [34],[35].
It remains an open subject if the theory so far described is self consistent. Inspired
from the complementary gauge symmetry the theory of free gauge fields exhibits,
two dual rank-2 Lagrangians have been constructed which are invariant under the
full complementary gauge transformations [37]. It has been seen that these two
Lagrangians are mapped to the original ones by a dual transformation. Perhaps
in a complete theory, one has to include both the original series of gauge invariant
Lagrangians and their dual counterparts.
On this thesis, we will focus our attention on tensor gauge fields of rank-2. The
theory is extensively described and its self-consistence is examined.
In the first section, we present the fundamentals of Tensor Gauge Theory. Gauge
principle is generalized to tensor fields of rank-2 and 3 and two series of gauge invariant
Lagrangians describing the gauge fields of rank-2 are constructed.
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7
Further, we check the unitarity of the theory on the free level. In previous articles
it has been shown that for the special choice of the ratiog
2g2 = 1, the free theory
(g = 0) exhibits an enhanced gauge symmetry [31]. It is because of this symmetry
enhancement that the cancellation of longitudinal modes occurs, allowing the free
propagation of three physical, transversal degrees of freedom. Two of them are the
polarizations of a symmetric tensor gauge field of rank-2, which behaves like a particle
of spin 2, while the third describes an antisymmetric gauge field of rank-2, behaving
like a scalar. What was not known however, was the fact that the specific choice of
the ratiog2
g2= 1, far from being arbitrary and imposed externally, is directed by the
Bianchi identities of the free theory [36].
After having specified the value of this ratio, one can determine the number of the
physical propagating modes, my the standard methods of gauge fixing [31]. Neverthe-
less, since the number of the physical modes of a gauge theory is an experimentally
tested result, and thus independent of the specific choice of the gauge, we consid-
ered instructive to develop a general method for counting the propagating modes
of a gauge theory which, based on the rank of the free equation matrix, leaves the
gauge freedom of the theory unfixed. Implementing this method in the case of rank-2
tensor gauge theory, we finally arrive at the same, previously mentioned result [36].
As a next step, the energy momentum tensor for the free 2nd rank tensor gauge field
is constructed, confirming from another point of view the fact that there are three
physical propagating modes contributing to the energy of the free field. Finally, the
generalization of the gauge principle to fermions of spin 3/2 is exposed and the full
Lagrangian is constructed.
In the next section we pass from the free theory to the interactive. Our aim is to
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check the unitarity on the interacting level. For this, we focus on the processes of pair
production of rank-2, symmetric, tensor gauge bosons, first through the annihilation
of two fermions [44, 45] and next through the annihilation of two vector gauge bosons
[46]. In both the processes, to check unitarity, one of the outgoing tensor particles
is put in longitudinal polarization leaving all other particles transversal. In the first
case the scattering amplitude indeed vanishes for the choice of the L3/2 coefficientf = g24 . However, in the second process the scattering amplitude does not vanish.
This indicates the fact that contributions of higher rank particles should be added in
the rank-2 propagator. In both cases, the cross section for the production of physical,
transversal outgoing particles is calculated and the symmetry properties of every
contributing helicity combination term are studied.
The cross section for the annihilation process where the colliding particles are SM
fermions, exhibits a very simple scattering angle dependence:
d =2
s
C2(r)C2(G)
64d(r)sin2 d, ,
Its sin2 behavior is dramatically different from from the QED case, where as out-
going particles one has two photons. In that case the cross section maximizes at
small angles and has a minimum in perpendicular directions. On the other hand,
things bare much resemblance with the case of the standard electroweak theory pro-
cess, where the outgoing particles are longitudinal Ws. As it is known, the scattering
angle dependence of the cross section is just sin2 .
The cross section for the annihilation process with two gauge bosons incoming,
should be compared to the annihilation process with the same incoming particles, but
with two gauge bosons as final products. The brand new result for the process with
2 tensor gauge particles outgoing is:
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9
d = g222
sC22(G)d(G)
419 76cos2 + 9cos 4512
d
As se know, in the Yang-Mills case the scattering favors forward and backward
directions, where in the case of rank-2 tensor bosons, preference is shown to perpen-
dicular directions.
These results in the context of Tensor Gauge Theory are the main results to be
defended on this thesis.
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Chapter 1
Basic formulation
On this chapter we present the fundamentals of tensor gauge theory. In section 1.1
we give a brief review of Yang-Mills theory. In section 1.2 the free gauge theory
for tensors bosons of the second rank is constructed and a first guess for the free
Lagrangian is made. To determine uniquely the free rank-2 tensor Lagrangian, a
geometrical picture of the theory is needed to be presented. This is done in section
1.3 where we find that the most general Lagrangian which describes free rank-2 tensor
gauge bosons is linear combination of two terms. In section 1.4, the coefficient of the
linear combination is evaluated, with the aid of the Bianchi identities of the theory.
1.1 Yang-Mills
Yang and Mills proposed a generalization of the principle of local gauge invariance,
from the Abelian group U(1) to the group SU(2), in an attempt to describe strong
interactions [1]. This extension can be made more general to include special unitary
groups of arbitrary dimension (SU(N))[2]. The idea is to represent the fermionic wave
function as an N-dimensional multiplet each element of which is a usual 4-component
Dirac field.
10
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11
=
1
2
.
.
.
N
(1.1.1)
The free fermionic Lagrangian
LF
1/2 = i
remains invariant under the global gauge transformation
= U()
where U() = eig is an element of the group SU(N) ,a are arbitrary constant
parameters and ta are the group Lie generators which satisfy the algebra:
[ta, tb] = ifabctc (1.1.2)
and the Jacobi identity:
[ta, [tb, tc]] + [tb, [tc, ta] + [tc, [ta, tb] = 0 (1.1.3)
Locality demands the constant parameters to be space-time dependent and the sym-
metry still to hold. This can be achieved via the introduction of a number of d(G) 1
gauge fields Aa. These fields A = Aata can be used as a connection to parallel trans-
port vectors inside this isospin, N-dimensional, internal space. The transformation
properties of the gauge fields are so chosen that the covariant derivative of
D = ( igA), (1.1.4)1The dimension of the adjoint representation of the group. For SU(N), d(G) = N2 1
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12
transforms under a gauge transformation the same way that does.
(D) (D) = U()(D) (1.1.5)
It can be easily seen that A should transform as:
A = U AU1 i
g(U)U
1 (1.1.6)
Then the Lagrangian
L1/2 = LF1/2 + LF B1/2 = i D = i + g A (1.1.7)
is locally gauge invariant.
Until now the gauge fields appear non-dynamical. In order to become such, a
third term must be added to the Lagrangian which carries derivatives of the fields
A. The standard way to find this is to compute the commutator of the covariant
derivative on a vector. This must give the analogous Riemann tensor. It can be seen
that
[D, D] = igG, (1.1.8)
where G = A A ig[A, A], is the field strength tensor.Since
([D, D]) = U()[D, D],
it is straightforward to show that the field strength tensor transforms as the connec-
tion does but homogenously.
G = U()GU()1 (1.1.9)
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13
Nevertheless, the trace of the product of two strength tensors is gauge invariant.
Indeed
tr(GG) = tr(GG
) = tr(GG). (1.1.10)
Having normalized the basis elements of the Lie space to unity:
tr(tatb) = ab,
one gets
tr(GG) = GaG
a
The full Yang-Mills Lagrangian is then given as follows:
LY M = L1/2 + L1 = i + g A 14
GaGa. (1.1.11)
Let us now be more specific in order to reveal some basic properties of the Yang-
Mills Lagrangian 1.1.11. First, we focus our attention on the free bosonic part:
L1 = 14
GaGa (1.1.12)
where
Ga = Fa
+ gfabcAbA
c,
or in terms of the Lie generators ta,
A = taAa , G = t
aGa
G = F ig[A, A]
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14
The infinitesimal form of (1.1.6), the gauge transformation under which the La-
grangian is invariant, is:
Aa =
a + gfabcAbc , A = ig[A, ],
which induces the homogenous transformation on the field strength tensor:
Ga = gf
abcGbc , G = ig[G, ].
It can be easily checked that the local gauge transformations form a closed algebraic
structure.
[, ]A = igA, (1.1.13)
where = [, ].
To get the equations of motion one must vary the action over Aa.
L
(Aa) L
Aa= 0 (1.1.14)
We see that
LAa
= 12
GcGcAa
GcAa
= gfacd(g Ad g Ad)
Hence,
L
Aa
= gfacdAdGc (1.1.15)
Analogously
L(Aa)
= 12
GcGc
(Aa)
Gc(Aa)
= (gg g g )ac
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15
Hence,
L(Aa)
= Ga (1.1.16)
Thus, (1.1.14),(1.1.15),(1.1.16)
Ga + gfabcAbG
c = 0 (1.1.17)
Or in terms of the Lie generators ta
DG = G ig[A, G] = 0. (1.1.18)
Equivalently, (1.1.17) can be rewritten as
Fa
= ja (1.1.19)
where
ja
=
gfabcAb
Gc
gfabc
(Ab
Ac
). (1.1.20)
The currents in the above equations stem from the fact that Yang-Mills equations
are not linear. They manifest that the gauge bosons curry color and are sources of
themselves. In an Abelian theory (fabc = 0) this is not the case.
Expressed in terms of the As the free bosonic Lagrangian (1.1.12) appears as:
L1 = 14
FaFa
gfabc(Aa)AbAc 1
4g2fabefcdeAaA
bA
cA
d (1.1.21)
The last two terms give us the Feynman rules for the bosonic self-interactions.
The 3-vertex (VVV) is (all momenta inwards):
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k p
q
a, b,
c,
= Vabc (k,p,q ) = gfabcFY M (k,p,q ) =
= gfabc
(k p) + (p q) + (q k)
(1.1.22)
while the 4-vertex (VVVV) is:
c, d,
a, b,
= Vabcd = ig2
fabefcde( ) +
+ facefbde( ) +
+ fadefbce( ) (1.1.23)
The propagator of the gauge boson field can be found by the standard Fadeev-
Popov quantization method [13]. In the Feynman gauge it is:
ka, b, = D;abF (k) = ik2 ab. (1.1.24)
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Let us take into account the terms of the Lagrangian that include fermions. Now,
to get the equations of motion for and Aa, one varies the full Yang-Mills Lagrangian
(1.1.11). We need the terms:
LAa
= gfacdAdGc + gt
a
L(Aa)
= Ga
L
= i + g A
L()
= 0
Hence for the fermionic field one has:
i + g A = 0 (1.1.25)
While for the gauge field:
Ga + gf
abcAbGc + g
ta = 0 (1.1.26)
or
Fa
= ja
where
ja(A, ) = gfabcAbGc gfabc(AbAc) gta.
In addition to the pure bosonic vertices (1.1.22),(1.1.23) the fermion-boson term
g A gives us the vertex rule:
a,
= igta (1.1.27)
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The propagator of the pure fermionic field is the well known:
k = DF (k) =
ik . (1.1.28)
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1.2 rank-2 tensor (part I)
Tensor Gauge theory is constructed under the restrictions posed by the fundamental
principle of local gauge invariance. What we need for such a construction is to extend
the gauge principle to higher spin gauge fields. On this thesis we will concentrate on
the lowest non-trivial tensor gauge fields; those of the 2nd rank. We will see how this
constructive procedure works [31, 32].
Actually, to describe the field A we will also need to introduce an auxiliary 3rd
rank tensor gauge field A. Although the 2nd rank tensor field have no symme-
try properties between its indices, the 3rd rank tensor field is symmetric under the
interchange of its last two indices.
The higher field strength tensors are defined as follows:
2nd-rank field strength:
Ga, = Aa Aa + gfabc( Ab Ac + Ab Ac) (1.2.1)
or in terms of the Lie generators ta
G, = A A ig
[A, A] + [A, A]
= DA DA, (1.2.2)
3rd-rank field strength:
Ga, = Aa Aa + gfabc( Ab Ac + Ab Ac + Ab Ac + Ab Ac )
(1.2.3)
or
G, = A A ig
[A, A] + [A, A] + [A, A ] + [A, A]
=
= DA DA ig
[A, A] + [A, A]
. (1.2.4)
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It should be noted that in the definition of the higher rank field strength tensors
apart from the tensor gauge fields of the same order, lower rank fields are also needed.
We now come to the crucial point to define the gauge transformations for the
higher rank tensor gauge fields. They should be defined so as to form a closed algebraic
structure. This means that the commutator of two extended gauge transformations
should lead to another gauge transformation. To do this, we need to introduce higher
rank gauge parameters: The vector = at
a and the symmetric second rank tensor
= at
a. We define the extended gauge transformations as follows:
A = ig[A, ]
A = ig
[A, ] + [A, ]
A = = ig
[A, ] + [A, ] + [A, ] + [A , ]
(1.2.5)
It is not hard to see that the above defined extended gauge transformations form
a closed algebraic structure. Indeed, the commutator of two gauge transformations
acting on a 2nd rank tensor gauge field is:
[, ]A = ig
[A, ] + [A, ]
[A, ] ig[A, ]
= ig
[, ] + [, ]
ig[A, ([, ] + [, ])] ig[A, [, ]]
= ig { ig[A, ] ig[A, ] } = ig A
This is again a gauge transformation with gauge parameters , which are given bythe following expressions:
= [, ], = [, ] + [, ].
The commutator of two gauge transformations acting on a rank-3 tensor gauge
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21
field is:
[, ]A = ig[A, ] + [A, ] + [A, ] + [A, ]
[A, ] + [A, ] + [A, ] + [A, ]
=
= ig
[, ] + [, ] + [, ] + [ , ]
ig[A, ([, ] + [, ] + [, ] + [ , ])]
ig[A, ([, ] + [, ])] ig[A, ([, ] + [, ])] ig[A, [, ]]
=
= ig { ig[A, ] ig[A, ] ig[A, ] ig[A, ] } == igA,
where
= [, ], = [, ] + [, ], = [, ] + [, ] + [, ] + [ , ].
Note that the gauge parameters are not separately closed on each order. This
happens only for the lowest, Yang-Mils, case. As we depart from this, we see that the
parameters of a given order mix not only between themselves but also with those of
lower orders.
The gauge transformations (1.2.5) induce the homogenous gauge transformations
on the field strength tensors.
Ga, = gfabc( Gb,
c + Gbc )
G, = ig[G, ] + [G,, ] (1.2.6)Ga, = gf
abc( Gb,c + Gb,
c + G
b,
c + G
b
c )
G, = ig
[G, ] + [G,, ] + [G,, ] + [G,, ]
(1.2.7)
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What remains now is to find the Lagrangian which describes the dynamics of
the 2nd rank tensor gauge field. One might guess that the most general one is the
following:
L2 = 14
Ga,Ga,
1
4GaG
a, (1.2.8)
Indeed it can be easily checked that the above Lagrangian remains invariant under
the gauge transformations (1.2.6) and (1.2.7). However, we will prove that (1.2.8) is
not the most general Lagrangian that can be constructed out of the field strengths
G, and G,. To do this, a geometrical picture of the above construction is needed
[31, 33, 36].
1.3 Geometrical Interpretation - Gauge bosons
We can think of the tensor gauge fields A, A, A ,... as x-dependent coefficients
appearing in the expansion of the extended gauge field A(x, e) over the tangentvector e [28, 29, 30].
A(x, e) =
s=0
1
s!Aa1...s(x)L
ae1 . . . es (1.3.1)
The same holds for the field strengths Gaa...s.
G(x, e) =
s=01
s!Ga,1...s(x)L
ae1 . . . es (1.3.2)
The extended gauge field A(x, e) is the connection, while the field strength G(x, e)the curvature tensor on an extended vector bundle X, with a structure group G the
elements of which U() = eig(x,e) can be parametrized through the gauge parameters
a1...s . These can similarly be considered as appearing in the expansion of the big
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(x, e) over the tangent vector e.
(x, e) = s=0
1s!
a1...s(x)Lae1 . . . es (1.3.3)
The extended algebra space is spanned by the infinite many generators La1...s =
Lae1 . . . es, which obey the commutation relations:
[La1...s, Lb1...r
] = ifabcLc1...s1...r (1.3.4)
We will now prove that
G(x, e) = A A ig[A, A]
A A ig[A, A] =
=
s=0
1
s!(A
a1...s
Aa1...s)Lae1 . . . es
ig
s,r
1
s!r!Aa1...sA
b1...r
[La, Lb]e1 . . . ese1 . . . er
=
=
s=0
1s!
(Aa1...s
Aa1...s)Lae1 . . . es
igs
1
s!
sk=0
n
k
Aa1...kA
bk+1...s
[La, Lb]e1 . . . ekek+1 . . . es
=
=
s=0
1
s!
(A
a1...s
Aa1...s) +
+ gfabcs
k=0n
k Ab1...k
Ack+1...sLae1 . . . es =
=
s=0
1
s!Ga,1...sL
ae1 . . . es = G(x, e) (1.3.5)
The extended covariant derivative is defined as:
D = igA
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Now we see that
[D, D] = [ igA, igA] = igA + igA g2[A, A] =
= ig
A A ig[A, A]
= igG (1.3.6)
The operators D, D, D obey the Jacobi identity:
[D, [D, D]] + [D, [D, D]] + [D, [D, D]] = 0,
which with the aid of (1.3.6) is transformed into the generalized Bianchi identity
[D, G ] + [D, G] + [D, G] = 0. (1.3.7)
Let us now expand equation (1.3.7) over e up to linear terms. We have,
[ igA igAe, G + G,e] + cyc.perm. + O(e2) = 0
In zero order the above equation gives the standard Bianchi identity in YM theory:
[D, G] + [D, G] + [D, G] = 0,
where D = igA. The linear term in e gives:
[D, G,] ig[A, G] + [D, G,] ig[A , G] + [D, G,] ig[A, G] = 0(1.3.8)
Using the explicit form of the operators D, G and G, one can independently
check the last identity and get convinced that it holds. Now, if we expand the above
equation over g, the zeroth order gives the Bianchi identity for the field strength
tensor F,:
F, + F, + F, = 0. (1.3.9)
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We define the extended gauge transformation of the extended gauge field
A(x, e) = U()A(x, e)U1() ig
U()
U1()
The gauge transformation over the A(x, e) fields induces the corresponding homoge-nous gauge transformation on the field strength tensors G(x, e). Indeed,
G (x, e) = A
A
ig[A
, A
] =
= (U)AU1 + U(A)U1 + UA(U1)
(U)AU1 U(A)U1 UA(U1)
ig
(U)U
1 + (U)(U1) (U)U1 (U)(U1)
ig
UAU1 ig
(U)U1, UAU1 i
g(U)U
1
=
= U(A A)U1 ig(U AAU1 UAAU1) +
+i
g
(U)U
1 (U)U1 + (U)U1U1 (U)U1U1
=
= U(A A ig[A, A])U1
= UG
U1
(1.3.10)
For infinitesimal transformations:
G = (1 + ig)G(1 ig) = G + igG igG
G = ig[G, ] (1.3.11)
La
Ga = ig
b
Gc[L
b, Lc] =
gfbcab
GcL
a
Ga = gfacbbGc (1.3.12)
Similarly,
A = ig[A, ] (1.3.13)
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Expanding (1.3.13) one can compute term by term the gauge transformations of the
higher rank tensor fields and reproduce the ones defined in (1.2.5). Ineed,
A =
Aa + (A
a)e +
1
2(A
a)ee + . . .
La (1.3.14)
=
a + (a)e +
1
2(
a )ee + . . .
La (1.3.15)
[
A, ] = gf
abc Abc + (Abc + Abc)e ++
1
2(Ab
c + Abc + A
b
c + A
b
c)ee + . . .
La
(1.3.16)
For each order we have the equation for the gauge transformation of the tensor field
of the corresponding rank.
Aa
= a + gfabcAb
c,
Aa = a + gf
abc(Abc + A
b
c),
Aa = a + gf
abc(Abc + A
b
c + A
b
c + A
b
c). (1.3.17)
In a similar way we can obtain the homogenous gauge transformation induced on the
field strengths (1.2.6), (1.2.7).
We can now form invariant Lagrangians:
L(x, e) = GaGa (1.3.18)
Indeed,
L = gfacbb(GaGc + GaGc) = 0
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Expanding L(x, e) to orders of e we can find invariant Lagrangians for every rank.
We are interested in Lagrangians describing 2nd rank tensors. So we see:
L(x, e) = GaGa
s=0
1
s!(L)1...s(x)e1 . . . es =
s=0
1
s!
sk=0
s
k
Ga,1...kG
a,k+1...s
e1 . . . es
L(0) + L(1),1e1 +1
2L(2),12e1e2 + . . . =
= GaGa + (G
aG
a,1
+ Ga,1Ga)e1 +
+12GaGa,12 + Ga,1Ga,2 + Ga,2Ga,1 + Ga,12Gae1e2 (1.3.19)
Comparing the terms, we see that:
L(2),12 = GaGa,12 + Ga,1Ga,2 + Ga,2Ga,1 + Ga,12Ga (1.3.20)
Since we need scalar Lagrangians, we have to contract the indices. There are 15 ways
of contracting them. Nevertheless, there are only two independent scalar Lagrangians
which we call:
LI2 = 1
4Ga,G
a,
1
4GaG
a, (1.3.21)
and
LII2 =1
4Ga,G
a, +
1
4Ga,G
a, +
1
2GaG
a, (1.3.22)
Hence, the Lagrangian which describes the dynamics of the free rank-2 tensor gauge
field must be a linear combination of the above Lagrangians:
L2 = g2LI2 + g
2LII2 (1.3.23)
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1.4 rank-2 tensor (part II)
We are now ready to examine the properties of the rank-2 tensor Lagrangian (1.3.23).
It will be seen that the primary constraints of the gauge system -Bianchi identities-
(1.3.9) will determine the ratio of the coefficientsg2
g2[36]. Further, this particular
choice of the ratio, as dictated by the zeroth order expansion over g of the Bianchi
identity (1.3.8), enhances the gauge symmetry at the free level and makes possible
the cancellations of all the negative norm states.
In the previous section it was seen that the Lagrangian which describes the dy-
namics of the rank-2 tensor gauge boson is:
L2 = 14
Ga,Ga,
1
4GaG
a, +
g
2
g2
1
4Ga,G
a, +
1
4Ga,G
a, +
1
2GaG
a,
(1.4.1)
Let us find the Euler-Lagrange equations for the rank-2 tensor gauge field Aa. that
stem from this Lagrangian.
L2
(Aa)
L2
Aa= 0 (1.4.2)
We see thatGb
(Aa)= 0 ,
Gb,(Aa)
= 0
Gb,
(Aa)=
Fb,
(Aa)= (
)
ab (1.4.3)
L2(Aa)
= 12
Gb,Gb,
(Aa)+
1
2
g
2
g2
Gb,
Gb,(Aa)
+ Gb,Gb,
(Aa)
=
= Ga, 1
2
g
2
g2(Ga, + G
a, + G
a, + G
a, ) (1.4.4)
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Just like in Yang-Mills case, the equations of motion that describe the rank-2 tensor
gauge boson are not linear. We isolate all the g-terms which contribute to the source-
self current on the right hand side of the equation.
Fa
, 1
2
g
2
g2
(F
a, + F
a, + F
a, + F
a,) = J
a (g). (1.4.5)
Now, let us ignore the self-interactions, taking g = 0. There are two free indices in
equation (1.4.5). It consists of two parts: Fa
, and (Fa
, + Fa
, + Fa
, +
Fa
,), which arise from LI2 and LII2 respectively. The derivatives, over the firstfree index , of both terms in the equation are equal to zero separately. Indeed, due
to the antisymmetric properties of the field strength tensor Fa, under the exchange
of and , we have
Fa
, = 0,
as well as
{Fa, + Fa, + Fa, + Fa,} = 0.
What about the second index ? Here the cancellation is not that trivial. Let us
take the derivative over of the left-hand side of the equation (1.4.5),
We see that
Fa
, = 0,
as well as
{Fa,+Fa, +Fa, +Fa,} = {Fa, +Fa, + Fa,} = 0.
Thus, it is not obvious to verify the cancellation alone from the antisymmetric prop-
erties of the field strength tensor. However, let us take advantage of the Bianchi
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identities (1.3.9) derived in the previous section. Taking the derivative of the Bianchi
identity over and setting = we get
2F, + F, + F, 0 (1.4.6)
Hence, with the aid of (1.4.6) we get
0 = Fa
, 1
2
g
2
g2
(F
a, +
2Fa, + Fa
,) =
= 1 g
2
g2Fa
, (1.4.7)
For arbitrary tensors Fa, the above equation implies: g2 = g2.
Hence, the Bianchi identity forces LI2 and LII2 to appear with the same weight inthe linear combination ofL2.
L2 = 14
Ga,Ga,
1
4GaG
a, +
1
4Ga,G
a, +
1
4Ga,G
a, +
1
2GaG
a,.
(1.4.8)
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Chapter 2
Unitarity of the theory at thenon-interactive level
Now that we have the final form of the Lagrangian which describes the dynamics of
the rank-2 tensor gauge boson, we can study its properties. Our first task will be to
check the unitarity of the theory at the free level (g = 0). We will see that at the
non-interactive level, the theory is endowed with an enhanced gauge symmetry [31].
It is because of this enhancement that the negative norm states do not appear in the
experimentally testable quantities [36].
2.1 A method for counting the propagating modesof a gauge theory
Before we begin the study of the properties of the free rank-2 tensor gauge theory, it
will be instructive to present a general method for counting the propagating modes of
a gauge theory. In the bibliography this is usually done in a straightforward gauge-
fixing approach. Nevertheless, when the number of degrees of freedom becomes large
such a procedure becomes rather cumbersome. A general, gauge fixing independent
method was considered an essential missing piece of the bibliography and we present
31
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it here.
2.1.1 The method
The vector space of independent solutions of a system of equations crucially depends
on the rank of the equation matrix. If the matrix operator H, the equation matrix of a
free gauge theory in momentum space, has dimension dd and its rank is rankH = r,then the vector space has the dimension
N= d r.
Because the matrix operator H(k) explicitly depends on the momentum k, its
rankH = r also depends on momenta and therefore the number of independent
solutions Ndepends on momenta
N(k) = d r(k) . (2.1.1)
Analyzing the rankH of the matrix operator H one can observe that it depends on
the value of momentum square k2. When k2 = 0 - off mass-shell momenta - the
vector space consists of pure gauge fields. When k2 = 0 - on mass-shell momenta -
the vector space consists of pure gauge fields and propagating modes. Therefore the
number of propagating modes can be calculated from the following relation
of propagating modes =N(k)|k2=0 N(k)|k2=0 = rankH|k2=0 rankH|k2=0.
(2.1.2)
Let us consider for illustration some important examples.
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2.1.2 Vector Gauge Field
The kinetic term of Lagrangian which describes the propagation of free vector gauge
field is
K = 14
FF (2.1.3)
and the corresponding equation of motion in momentum space is
H e = (k2 + kk)e = 0, (2.1.4)
where A = e exp(ikx). We can always choose the momentum vector in the third
direction k = (, 0, 0, k) and the matrix operator H takes the form
H =
k2 0 0 k
0 2 k2 0 00 0 2 k2 0
k 0 0 2
.
If 2
k2
= 0, the rank of the 4-dimensional matrix H
is rankH
|2
k2
=0= 3
and the number of independent solutions is 4-3=1. As one can see from the relation
H (k)k = 0 this solution is proportional to the momentum e = k = (, 0, 0, k)and is a pure-gauge field. This is a consequence of the gauge invariance of the theory
e e + ak. If2k2 = 0, then the rank of the matrix drops, rankH|2k2=0 = 1,and the number of independent solutions increases: 4-1=3. These three solutions of
equations (2.1.4) are
e(gauge) =1
2
10
0
1
, e(1) =
0
1
0
0
, e(2) =
0
0
1
0
,
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from which the first one is a pure gauge field ( k), while the remaining two are the
physical modes, perpendicular to the direction of the wave propagation. The general
solution at 2 k2 = 0 will be a linear combination of these three eigenvectors:
e = ak + c1e(1) + c2e
(2) ,
where a, c1, c2 are arbitrary constants. We see that the number of propagating modes
is
rankH
|2
k2
=0
rankH
|2
k2=0 = 3
1 = 2,
as it should be.
2.1.3 Symmetric Tensor Gauge Field
The free gravitational field is described in terms of a symmetric second-rank tensor
field h and is governed by the Einstein and Pauli-Fierz equation:
2
h
h
h + h
+ (
h 2
h
) = 0, (2.1.5)
which is invariant with respect to the gauge transformations
h = + , (2.1.6)
that respects the symmetry properties of A. The corresponding matrix operator is:
H(k) =
{
1
2
( + )
}k2
kk
kk +
+1
2(kk + kk + kk + kk) (2.1.7)
and is a 10 10 matrix in four-dimensional space-time with the property H =H = H = H.
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If 2 k2 = 0, the rank of the 10-dimensional matrix H (k) is equal to
rankH|2k2=0 = 6 and the number of independent solutions is 10 6 = 4. Thesefour symmetric solutions are pure-gauge tensor fields. Indeed, if again we choose the
coordinate system such that k = (, 0, 0, k), then one can find the following four
linearly independent solutions:
e =
2 0 0 00 0 0 0
0 0 0 0
0 0 0 k2
,
0 0 0 0 0 k
0 0 0 0
0 k 0 0
,
0 0 00 0 0 0
0 0 k
0 0 k 0
,
2 0 0 k0 0 0 0
0 0 0 0
k 0 0 0
pure-gauge field solutions of the form (2.1.6) e = k + k as one can see from
the relation
H (k)(k + k) = 0. (2.1.9)
When 2 k2 = 0, then the rank of the matrix H(k) drops and is equal torankH|2k2=0 = 4. This leaves us with 10 4 = 6 solutions. These are thefour pure-gauge solutions (2.1.6) and two additional symmetric solutions representing
propagating modes: the helicity states of the graviton
e(1) =
0 0 0 0
0 1 0 0
0 0
1 0
0 0 0 0
, e
(2) =
0 0 0 0
0 0 1 0
0 1 0 0
0 0 0 0
. (2.1.10)
Thus the general solution of the equation on mass-shell is:
e = k + k + c1e(1) + c2e
(2),
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where c1, c2 are arbitrary constants. We see that the number of propagating modes
is
rankH|2k2=0 rankH|2k2=0 = 6 4 = 2,
as it should be.
2.1.4 Antisymmetric Tensor Gauge Field
The antisymmetric second-rank tensor field B is governed by the equation [39, 40,
41, 42]:
2B B + B = 0 (2.1.11)
which is invariant with respect to the gauge transformations
B = , (2.1.12)
that respects the symmetry properties of B. The corresponding matrix operator is:
H(k) = 12
( )k2
12
(kk kk + kk kk) (2.1.13)
and is 6 6 matrix in four-dimensional space-time with the property H =H = H = H.
If 2 k2 = 0, the rank of the 6-dimensional matrix H (k) is equal torankH|2k2=0 = 3 and the number of independent solutions is 6 3 = 3. Thesethree antisymmetric solutions are pure-gauge fields. Indeed, in the coordinate system
k = (, 0, 0, k), one can find the following three solutions:
e =
0 0 0
0 0 0 0
0 0 k0 0 k 0
,
0 0 0
0 0 k0 0 0 0
0 k 0 0
,
0 0 0 1
0 0 0 0
0 0 0 0
1 0 0 0
, (2.1.14)
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pure-gauge fields of the form (2.1.12) e = k k as one can see from the
relation
H (k)(k k) = 0. (2.1.15)
When 2 k2 = 0, then the rank of the matrix H(k) drops and is equal torankH|2k2=0 = 2. This leaves us with 62 = 4 solutions. These are the three pure-gauge solutions (2.1.12) and the antisymmetric solution representing the propagating
mode: the helicity zero state
e(A) =
0 0 0 0
0 0 1 0
0 1 0 00 0 0 0
. (2.1.16)
Thus on mass-shell the general solution of the equation is:
e = k k + c3 e(A) ,
where c3 is arbitrary constant. We see that the number of propagating modes is
rankH|2k2=0 rankH|2k2=0 = 3 2 = 1.
After this parenthetic discussion we shall turn to the tensor gauge theory.
2.1.5 Rank-2 tensor gauge theory
We begin with the pure kinetic term of the 2-rank bosonic Lagrangian. The coupling
constant is set to zero (g = 0) and all all fields of rank other than 2 are ignored. We
have:
K = 14
Fa,Fa
, +1
4Fa,F
a, +
1
4Fa,F
a,. (2.1.17)
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We notice here that apart from the the usual gauge symmetry
Aa =
a
the theory is also invariant under the complementary gauge transformation
Aa =
a.
Indeed, the field strength tensor Fa, transforms with respect to these transforma-
tions as follows:
Fa
, = 0,
Fa
, = (a a). (2.1.18)
Therefore the kinetic term K is obviously invariant with respect to the first group ofgauge transformations (K = 0), but it is less trivial to see that it is also invariantwith respect to the complementary gauge transformation . The transformation of
K is
K = 12
Fa,(a a) +
1
2Fa,(
a a) +
1
2Fa,(
a a) =
=1
2Fa,
a +
1
2Fa,(
a a) (2.1.19)
where we combined the first, the second and the forth terms and used the fact that
the third term is identically equal to zero. Just from the symmetry properties of the
field strength tensor it is not obvious to see why the rest of the terms are equal to
zero. Nevertheless, we can use the g-zeroth order Bianchi identities (1.3.9) for the
field strength tensor Fa,. We use the identity in the form
Fa
, + Fa
, + Fa
, = 0. (2.1.20)
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Ignoring total divergence terms, equation (2.1.19) is equivalent to
K = 12
(F
a,)
a + (F
a,)(
a a)
(2.1.21)
Which with the aid of (2.1.20) can be shown to nullify.
This demonstrates the invariance of the action with respect to the and trans-
formations defined by (1.3.17) and (2.1.18) when g2 = g
2.
Hence the free Lagrangian (2.1.17) is invariant under the general gauge transfor-
mation:
Aa Aa = A
a +
a +
a. (2.1.22)
Now the question is the following. Is this enhanced gauge symmetry enough to
guarantee the the self consistence of the theory? Is it enough to guarantee that the
propagating modes it predicts are physical? To answer that we have to solve the
equations of motion and study the solutions. As we have seen in the previous section,the equations of motion for the free rank-2 bosonic Lagrangian (g = 0) are:
Fa
, 1
2(F
a, + F
a, + F
a, + F
a,) = 0. (2.1.23)
Or in terms of the As:
2Aa 1
2
Aa Aa 1
2
Aa Aa 1
2
Aa++
Aa
1
2Aa
+
1
2 (A
a 2Aa) = 0 (2.1.24)
Transforming the above equation to momentum space we get:
H (k) Aa = 0, (2.1.25)
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where H (k) is a matrix operator quadratic in momentum k. It has the following
form:
H(k) = ( + 12
+1
2)k
2 + kk + kk
12
(kk + kk + kk + kk), (2.1.26)
with the property that H = H.
The matrix operator (2.1.26), in four-dimensional space-time is a 1616 matrix. Inthe reference frame, where k = (, 0, 0, k), it has the form presented on Appendix A.
If 2 k2 = 0, the rank of the 16-dimensional matrix H (k) is equal torankH|2k2=0 = 9 and the number of linearly independent solutions is 16 9 = 7[36]. These seven solutions are pure-gauge tensor potentials of the form
e = k + k, (2.1.27)
e =
2 0 0 00 0 0 0
0 0 0 0
0 0 0 k2
, 0 0 0
0 0 0 0
0 0 0 0
k 0 0 0
,0 0 0
0 0 0 0
0 0 0 0
0 k 0 0
,0 0 0
0 0 0 0
0 0 0 0
0 0 k 0
,
0 0 k
0 0 0 0
0 0 0 0
0 0 0 0
,
0 0 0 0
0 0 k
0 0 0 0
0 0 0 0
,
0 0 0 0
0 0 0 0
0 0 k
0 0 0 0
(2.1.28)
as one can get convinced from the relation
H (k)(k + k) = 0, (2.1.29)
which follows from the gauge invariance of the action and can also be checked explic-
itly.
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When 2 k2 = 0, then the rank of the matrix H(k) drops and is equal to
rankH|2k2=0 = 6. This leaves us with 16 6 = 10 solutions. These include theseven pure-gauge potentials (2.1.28), (2.1.27) plus three new solutions representing
the physical propagating modes:
e(1) =
0 0 0 0
0 1 0 00 0 1 0
0 0 0 0
, e(2) =
0 0 0 0
0 0 1 0
0 1 0 0
0 0 0 0
, eA =
0 0 0 0
0 0 1 0
0 1 0 00 0 0 0
(2.1.30)
Thus the general solution of the equation on mass-shell is:
e = k + k + c1e(1) + c2e
(2) + c3e
(A) , (2.1.31)
where c1, c2, c3 are arbitrary constants. We see that the number of propagating modes
is three
rankH|
2
k2
=0 rankH
|2
k2
=0= 9
6 = 3.
These are propagating modes ofhelicity-two ( = 2) and helicity-zero = 0 chargedgauge bosons [31]. Indeed, if we make a rotation around the z-axis
=
1 0 0 0
0 cos sin 00 sin cos 0
0 0 0 1
,
we shall get
e(1)
= e(1)T =
0 0 0 0
0 cos2 sin2 00 sin2 cos2 00 0 0 0
, e(2)
= e(2)T =
0 0 0
0 sin2 cos20 cos 2 sin2
0 0 0
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Therefore the first two solutions describe helicity = 2 states. On the other hand,
the third, antisymmetric solution remains invariant under a Lorentz transformation,
therefore it describes a helicity-zero state.
This result can also be derived from the consideration of the equations of motion
for the symmetric and antisymmetric parts of the tensor gauge field Aa, as it was
done in [31]. Indeed, one can observe that for the symmetric part of the tensor gauge
fields Aa the equation reduces to the free equation of gravity [39], which describes
the propagation of massless tensor gauge bosons with two physical polarizations: the
= 2 helicity states. For the antisymmetric part of the tensor gauge fields Aa theequation reduces to the equation which describes the propagation of massless boson
with one physical polarization: the = 0 helicity state [40].
2.2 Propagator
In the previous section we saw that the equation matrix for the pure kinetic part of
the rank-2 tensor bosonic Lagrangian is:
1
g2H, = ( +
1
2 +
1
2)k
2 +
+kk + kk
12
(kk + kk + kk + kk) (2.2.1)
This can be decomposed into its symmetric, under the interchange ( ), andantisymmetric parts.
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43
1g2
HS,
(k) = 14
( + 2)k2 +
+1
4(kk + kk + kk + kk)
12
(kk + kk) (2.2.2)
1
g2HA
,(k) = 3
1
4( )k2 +
+
1
4(kk kk kk + kk) (2.2.3)Since, the matrix H is singular, gauge fixing is essential in the computation of the
propagator. A convenient gauge is the following:
Aa
1
2A
a = 0, A
a
1
2A
a = 0 (2.2.4)
In this gauge one gets:
1g2HS,fix,(k) = 14( + )k2 (2.2.5)
1
g2HA,fix
,(k) = 3
1
4( )
k2 (2.2.6)
It is remarkable that the antisymmetric component of H comes with a factor of
3 [46]. In the next section the energy-momentum tensor of the rank-2 tensor theory
will be given [36]. There we will see this factor appears as the weight of the helicity
zero quantum in the total energy of the rank-2 field.
The equation that defines the propagator is:
Hf ix,
(k) DF,,(k) = i (2.2.7)
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from which we find:
D,;abF (k) = i3g2k2 (4 + 2 3)ab (2.2.8)This can be decomposed into a symmetric and an antisymmetric part:
ka, b, =
D,;abF (k) =
= ig2k2
( + ) + 1
3( )
(2.2.9)
We see that
HS,fix,
(k) DSF,,(k) = i2( + ) (2.2.10)HA,fix
,(k) DA
F,,(k) =
i
2( ) (2.2.11)
2.3 The Energy-Momentum Tensor
We would like to consider the contribution of the general solution (2.1.31) into the
energy-momentum of the tensor gauge field theory. This will test from another point
of view the unitarity of the theory [36]. One can expect that only transverse propa-
gating modes
e c1e(1) + c2e(2) + c3e(A) , (2.3.1)
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will contribute to the energy-momentum of the gauge fields and that the longitudinal,
pure gauge fields,
e k + k, (2.3.2)
will have no contribution. The free Lagrangian has the form (2.1.17):
L = 14
Fa,Fa
, +1
4Fa,F
a, +
1
4Fa,F
a, (2.3.3)
and the equation of motion for the A field are (2.1.23):
Fa
, 12(Fa
, + Fa
, + Fa
, + Fa
,) = 0 (2.3.4)
By definition, the energy momentum tensor for the A field is:
T = AL
(A) L (2.3.5)
In order to calculate the term L(A)
we need the expression for the derivative of
the field strength tensor:
F,(A)
= ( ),
hence it is easy to see that
L(A)
= F, + 12
(F, F,) + 12
(F, F, )
and finally get:
T = AF, + 14
F,F, + (2.3.6)
+1
2A(F, F,) 1
4F,F, +
+1
2(A F, AF,) 1
4F,F,
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With the aid of (2.3.4) one can compute the derivative of the energy-momentum
tensor T over its second index and demonstrate that it is zero:
T = 0. (2.3.7)
The energy-momentum tensor is not uniquely defined because one can add any term
of the form
T T +
where = without changing its basic property (2.3.7) and the total four-momentum of the system
P =
T0dV. (2.3.8)
We can use this freedom to express T solely in terms of the field strength tensor
F,. Choosing
= AF, 12
(AF, + AF, + AF, + AF,), (2.3.9)
which fulfills the property = , and using (2.3.4) we can get that
= F,A 12
(F,A + F,A + F,A + F,A).
(2.3.10)
The sum of (2.3.6) and (2.3.10) gives the final form of the energy momentum tensor
expressed in terms of field strength tensors
T = F,F, + 14
F,F, +
+1
2(F,F, + F,F,) 1
4F,F, +
+1
2(F,F, + F,F,) 1
4F,F,. (2.3.11)
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It is easy to see that the energy-momentum tensor is traceless
T = T = 0 (2.3.12)
as it should be in a massless and scale invariant theory. As it is also obvious from
the final expression it is not symmetric T = T . This only means that it can notbe used for the calculation of angular momentum of the fields (see paragraph 32 and
paragraph 96 of [43].
Now we can calculate the contribution of the general solution (2.1.31) into the
energy and momentum of the free gauge field. First of all we can find that
F, = i(ke + ke)
where e is a general solution (2.1.31)
e = k + k + c1e(1) + c2e
(2) + c3e
(A) .
Using the following orthogonality relations:
kk = 0, ke
= ke = 0
e(i)e
(j) = e
(i)e
(j) =
ij, f or i,j = 1, 2
e(A) e
(A) = 1, e
(A) e
(A) = 1
e(A) e
(i) = e
(A) e
(i) = 0, i = 1, 2,
it is straightforward to see that
T =1
2kk(c
21 + c
22 + 3c
23) (2.3.13)
Thus, we see that only the transverse propagating modes contribute to the energy-
momentum of the field. As expected no pure gauge fields appear in the expression
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(2.3.13). An interesting remark is the factor of three which accompanies the energy
contribution of the antisymmetric field. As we stated before, this coefficient owes its
existence in the form of the T-propagator derived in the previous section (2.2.9).
2.4 Geometrical representation - Fermions
The fermion spinor-tensor fields 1...s belong to the same representation of the
compact Lie group G and are considered to appear in the expansion of the extended
fermion field
i
(x, e) over the unit tangent vector e [33].
i(x, e) =
s=0
1
s!(x)1...se1 . . . es (2.4.1)
Under the extended gauge group with elements U() = eig(x,e) the fields are trans-
formed as:
(x, e) (x, e) = U()(x, e), (2.4.2)
Infinitesimally, the transformation which fermions undergo, is coupled to the
transformation of the gauge fields.
= ig , A = ig[A, ] (2.4.3)
Expanding the gauge equation (2.4.3), term by term, one can read out the transfor-
mations induced on every fermion field of every rank. The transformations are:
= igLaa,
= igLa(a +
a),
= igLa(a +
a +
a +
a). (2.4.4)
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The Dirac Lagrangian,
LF(x, e) = iD (2.4.5)
remains invariant under such transformations. Indeed,
(D) =
( igA)
=
= () ig(A) igA = ig( igA) =
= igD (2.4.6)
Hence,
(LF(x, e)) = ()iD + i(D) = 0 (2.4.7)
On expanding the extended Lagrangian density over e on can get the gauge
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invariant Lagrangian density of every rank.
LF(x, e) = s=0
1s!
1...se1 . . . es i ig
r=0
1r!
Aa1...rLae1 . . . es
q=0
1
q!1...qe1 . . . eq
=
=
+ e +
1
212e1e2 + . . .
i
ig
A + Ae +1
2A12e1e2 + . . .
+ e + 1212e1e2 + . . . == O(0) + O(1) +
+
i
1
2( igA) ig
2(A + A) ig
2A
+
+1
2
i( igA) + i( igA)
ig2
iA + iA
+
1
2i( igA)
ee
+O(3) + . . . LF,(2), =
1
2
(i + gA) + (i + gA) + (i + gA) +
+(i + gA) + gA + gA +
+gA + gA + gA
(2.4.8)
To get scalar Lagrangians the indices must be contracted in pairs. There are two
ways of doing this:
LI3/2 = LF,(2) = (i + gA) +1
2(i + gA) +
1
2(i + gA) +
+gA + gA +1
2gA (2.4.9)
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LII3/2 = LF,(2), =1
2(i + gA) + (i + gA) +
+ (i + gA) + +(i + gA) +
+ gA + gA + gA + gA +
+ gA
(2.4.10)
Hence, the total fermionic Lagrangian for spin- 3/2 fermions is a linear combina-
tion of the above two Lagrangians:
L3/2 = f1LI3/2 + f2LII3/2 (2.4.11)
Since in the case of the fermionic fields there are no Bianchi identities that would
determine the ratio f1f2
, for reasons of simplicity, in what follows we shall assume
f2 = 0, so that f1 = f.
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Chapter 3
Unitarity of the theory at theinteractive level-Processes
Up to now, we have checked the self-consistency of the lower rank tensor gauge theory
in the absence of interactions. As shown in the previous chapter, the theory which
describes the dynamics of the rank-2 tensor gauge bosons respects unitarity at the free
level (g = 0) and predicts 3 physical propagating modes. As a next step we confront
the gauge invariance of the theory at the interacting level (g
= 0). A straightforward
way to examine gauge invariance and hence unitarity, is to consider the full theory up
to rank-2 tensor bosons and spin 3/2 fermion fields and study scattering processes.
On this thesis two processes will be thoroughly studied. That of the production of
two tensor gauge bosons of helicity 2 via the annihilation of 2 spin 1/2 fermions
[44, 45], and that of the production of the same particles via the annihilation of
two Yang-Mills vector gauge bosons[46]. To check the consistency of the theory, the
scattering amplitudes of both processes will be calculated with one external particle
in longitudinal polarization. It is expected that the gauge invariance of the theory will
force both the amplitudes to vanish. After this purely theoretical test, the scattering
cross sections for both processes will be calculated and compared with those of the
52
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corresponding, more familiar QCD and Standard model results [47].
3.1 Feynman rules
The total Lagrangian which describes Yang-Mills vector bosons, rank-2 tensor gauge
bosons, spin 1/2, 3/2 fermions and their interactions is the following:
L = LY M + g2L2 + fL3/2, (3.1.1)
where
LY M = L1/2 + L1 = i + g A 14
GaGa
L2 = 14
Ga,Ga,
1
4GaG
a, +
1
4Ga,G
a, +
1
4Ga,G
a, +
1
2GaG
a,
L3/2 = (i + gA) + 12
(i + gA) +1
2(i + gA) +
+gA + gA +1
2gA (3.1.2)
From the above Lagrangian we can derive the following Feynman rules.
spin 1/2 fermion propagator:
k = DF (k) =
ik . (3.1.3)
spin 3/2 fermion propagator:
ka, b, = D;abF (k) =
1
f
ik
ab. (3.1.4)
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vector propagator:
ka, b, = D;abF (k) = ik2 ab (3.1.5)
2nd rank tensor propagator:
ka, b, = D,;abF (k) =
= ig2k2
ab
( + ) + 13
( )
(3.1.6)
Vertex: FFV
a,
= igta (3.1.7)
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Vertex: F F2T
a,
= ifgta (3.1.8)
Vertex: VVV
k p
q
a, b,
c,
= Vabc (k,p,q ) = gfabcFY M (k,p,q ) (3.1.9)
FY M (k,p,q ) = (k p) + (p q) + (q k) (3.1.10)
Vertex: VVVV
c, d,
a, b,
= Vabcd = ig2
fabefcde( ) +
+facefbde( ) ++fadefbce( )
(3.1.11)
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Vertex: TTV
k p
q
a, b,
c,
= Vabc,,(k,p,q ) = gg2fabcFT T V,,(k,p,q )
(3.1.12)
FT T V,,(k,p,q ) = FT T V(1)
,, (k,p,q ) 1
2F
T T V(2),, (k,p,q ) (3.1.13)
FT T V(1)
,, (k,p,q ) = FY M
(k,p,q ) (3.1.14)
FT T V(2)
,, (k,p,q ) = ( + )(k p) + (k p) + (k p) +
+( + )(p q) + (p q) + (p q) ++( + )(q k) + (q k) + (q k)
(3.1.15)
Vertex: VVTT
a, b,
c, d,
= Vabcd,,, =ig2g2
2FV V TT,abcd,,, (3.1.16)
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FV V TT,abcd,,, = (facefbde + fadefbce)(2 ) +
+ (fabefcde fadefbce)(2 )
(fabefcde + facefbde)(2 ) +
+ (fadefbce fabefcde)( + )
(facefbde + fadefbce)( + ) +
+ (f
abe
f
cde
+ f
ace
f
bde
)(
+
) (3.1.17)
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3.2 ff TTOn this section the leading-order differential cross section of spin-two tensor gauge
boson production in the fermion pair annihilation process ff T T will be calculated[44] and the angular dependence of the polarized cross sections for each set of helicity
orientations of initial and final particles will be analyzed [45]. The process receives
contribution from three Feynman diagrams which are similar to the QED and QCD
diagrams for the annihilation processes with two photons or two gluons in the final
state (see D.1). The difference between these processes is in the actual expressions of
the corresponding interaction vertices, the explicit form of which have been presented
in the previous subsection.
Below we shall present the Feynman diagrams for the given process and calculate
the transition amplitude. We will find that the transition amplitude is gauge invariant.
That is, if we the one of the outgoing tensor gauge bosons is taken to be in a physical -
transverse polarization - and the other one in an unphysical - longitudinal polarization- the transition amplitude vanishes [44]. This means that the unphysical - longitudinal
polarization states are not produced in the scattering process. Having made this
consistency check, we shall calculate the polarized cross sections for each set of helicity
orientations of the initial and final particles (3.2.35), (3.2.36) and compare them with
the corresponding cross sections for photons and gluons in QED and QCD (D.1), as
well as with the W-pair production in Electroweak theory (D.3).
3.2.1 The Feynman diagrams
Working in the center-of-mass frame, we make the following assignments:
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z
x
p
p+
q1
q2
E '
Figure 3.1: Incoming and outgoing momenta in the center-of-mass frame
p = E(1, 0, 0, 1), p+ = E(1, 0, 0, 1),
q1 = E(1, sin , 0, cos ), q2 = E(1, sin , 0, cos ), (3.2.1)
where p are the momenta of the fermions f, f and q1,2 the momenta of the tensor
gauge bosons T T. All particles are massless p2 = p2+ = q
21 = q
22 = 0. In the
center-of-mass frame the momenta satisfy the relations p+ = p, q2 = q1 andE = E+ = 1 = 2 = E. The invariant variables of the process are:
s = (p+ + p)2
= (q1 + q2)2
= 2(p+ p) = 2(q1 q2) = 4E2
t = (p q1)2 = (p+ q2)2 = s2
(1 cos )
u = (p q2)2 = (p+ q1)2 = s2
(1 + cos ), (3.2.2)
where s = (2E)2 and is the scattering angle.
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It is convenient to write the differential cross section in the center-of-mass frame
with
d =1
4(p+ p) |M|2d, (3.2.3)
where the final-state density for two massless tensor gauge bosons is
d =
d3k1
(2)321
d3k2(2)322
(2)4(p+ + p k1 k2) = 1322
d,
so that
d =
1
2s |M|2 1
322d. (3.2.4)
We shall calculate the polarized cross sections for this reaction, to lowest order in
= g2/4. The lowest-order Feynman diagrams contributing to fermion-antifermion
annihilation into a pair of tensor gauge bosons are given below. In order g2, there
are three diagrams. Dirac fermions are conventionally drawn as thin solid lines,
and Rarita-Schwinger spin-vector fermions by double solid lines. These diagrams
are similar to the QCD diagrams for fermion-antifermion annihilation into a pair ofvector gauge bosons (see D.1). The difference between these processes is in the actual
expressions for the corresponding interaction vertices (3.1.12).
iM,;abA =
p
q1
p+
q2
a, b,
= (ig)2v(p+)ta
if
p
q2tbu(p)e(q1)e
(q2
(3
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iM,;abB =
pp+
a,,q1 q2,b ,
= (ig)2v(p+)tb
if
p q1 tau(p)e(q1)e
(q2
(3
iM,abG =
p
q1 q2
q3
p+
a, b,
=
= (ig)2v(p+)g2fbactc 1q23
F,,T T V (q2, q1, q3)u(p)e(q1)e(q2)
(3.2.7)
The probability amplitude of the process can be written in the form
M,e(q1)e(q2) =
(ig)2v(p+)
ta
f
p q2 tb + tb
f
p q1 ta +
+ ig2fbactc
1
q23
F,,T T V (q2, q1, q3)u(p)e(q1)e
(q2),
(3.2.8)
where u(p) is the wave function of spin 1/2 fermion, v(p+) of antifermion, the final
tensor gauge bosons wave functions are e(q1) and e(q2) and q3 = p + p+. The
Dirac and symmetry group indices are not shown.
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3.2.2 Gauge Invariance
At this point, we can check unitarity at the interacting level imitating the standard
procedure followed in the Yang-Mills case (Appendix D.1). The total scattering am-
plitude should cancel when one of the outgoing gauge bosons is taken longitudinal
(eL) and the other in transversal polarization (eT). This is a consequence of the
generalized Ward identity and expresses the fact that the unphysical - longitudinal
polarization - states are not produced in the scattering process.
Indeed, let us consider the last term in (3.2.8). We take the rank-2 tensor e(q2) inlongitudinal polarization. In the center of-mass-frame the following relations should
hold. e(q2) = q2 + q2, and the tensor e(q1) in transversal polarization:
q1 e(q1) = q1 e(q1) = 0, q
2 e(q2) = q
1e(q1) = 0. (3.2.9)
Taking into account that the tensors representing rank-2 tensor gauge bosons must
be traceless (e(q) = 0), we get:
FT T V(1)
,, (q2, q1, q3)(q2 + q2)e(q1) = q23e(q1)
FT T V(2)
,, (q2, q1, q3)(q2 + q2)e(q1) = 3
2q23
e(q1)
FT T V,,(q2, q1, q3)(q2 + q2)e(q1) = q23e
1 12
32
= q
23
4e(q1).
Then the last term becomes:
ig2fabc v(p+)t
cu(p) 14
e(q1) . (3.2.10)
Now let us consider the first two terms in (3.2.8). Taking again the polarization
tensors e(q2) to be longitudinal, using relations (3.2.9) for the wave function e(q1)
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and the Dirac equation for the wave-functions u(p) and v(p+) we shall get:
fv(p+){tatb + tbta}u(p)e(q1) =
= iffabcv(p+)tcu(p)e(q1). (3.2.11)
This term precisely cancels the contribution coming from the last term of the ampli-
tude (3.2.10) if we demand:
f =g24
(3.2.12)
Thus the cross term matrix element between transverse and longitudinal polarizations
vanishes: MeTeL = 0. Our intention now is to calculate the physical matrix elementsMeTeT for each set of helicity orientations of initial and final particles.
3.2.3 Squared Matrix Element
Using the explicit form of the vertex operator F,, (3.1.12) and the orthogonality
properties of the tensor gauge boson wave functions:
q1 e(q1) = q1e(q1) = q
2 e(q1) = q
2e(q1) = 0, (3.2.13)
q2 e(q2) = q2 e(q2) = q
1 e(q2) = q
1e(q2) = 0,
where the last relations follow from the fact that q1 q2 in the process of Fig.D.2.2,we shall get
M;abe(q1)e(q2) = (ig)2g2 v(p+)ta 14p q2 tb + tb 14p q1 ta ++ifabctc
(q2 q1)q23
( 12
)
u(p)e(q1)e
(q2).
(3.2.14)
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The complex conjugate of the scattering amplitude (3.2.8) is
M = (ig)2g2u(p) tb 14gp q2 ta + ta 14gp q1 tb ++ifabctc
1
q23F,,T T V (q2, q1, q3)
v(p+)
and we can calculate now the squared matrix elements in the form
MM = (ig)2v(p+)
ta14
p q2 tb + tb
14
p k1 ta
ifabctc
1
q23
F,,T T V (
q2,
q1, q3)u(p)
(ig)2u(p)
tb14g
p q2 ta + ta
14g
p q1 tb +
+ifabctc1
q23F,,T T V (q2, q1, q3)
v(p+)
For unpolarized fermions-antifermion scattering the average over the fermion and
antifermion spins is defined as follows:
|M|2
=
1
2
1
2 spin 1/2 |M|2.Using the completeness relations
s
us(p)us(p) =p ,
s
vs(p+)vs(p+) =p+ .
and averaging over spins of the fermions we shall get:
MM = g4
4T r
p+
ta14
p
q2tb + tb
14
p
q1ta
ifabctc 1q23
F,,T T V (q2, q1, q3)
p
tb
14
p q2 ta
+
ta
14g
p q1 tb
+
+ ifa
b
c
tc
1
q23F
,,
T T V (q2, q1, q3)
.
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Contracting the last expression with the transversal on-shell polarization tensors of
the final tensor gauge bosons e(k1) and e(k2) we get the probability amplitude in
the form:
M;abM ;abe(q1)e(q2)e (q1)e (q2) =
=g4
4T r
p+
ta14
p q2 tb + tb
14
p q1 ta +
+ ifabctc1
q23
(q2
q1)(
1
2
g)p
tb
14
p q2 ta
+
ta
14
p q1 tb
ifabc tc1
q23(q2 q1)( 1
2
)
e(q1)e (q2)e (q1)e (q2).
As a next step we shall calculate the sum over transversal tensor gauge bosons po-
larizations. The sum over transversal polarizations of the helicity-two tensor gaugeboson is [48, 31, 44]
r
er(q1)er
(q1) =1
2
( +
q1q1 + q1q1
q1q1)( +
q1q1 + q1q1
q1q1) +
+( +q1q1 + q1q1
q1q1)( +
q1q1 + q1q1
q1q1)
( + q1q1 + q1q1q1q1
)( +q1 q1 + q1 q1
q1q1)
,
where q1 = (1, q1) and q1 = (1, q1). The explicit form of the transversal polar-ization tensors, when the momentum is aligned along the third axis, is given by the
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matrices [48, 31, 36]:
e1 =1
2
0, 0, 0, 0
0, 1, 0, 0
0, 0, 1, 00, 0, 0, 0
, e2 =1
2
0, 0, 0, 0
0, 0, 1, 0
0, 1, 0, 0
0, 0, 0, 0
.
From the kinematics of the process in Fig.D.2.2 it follows that 2 = 1 and k2 = k1therefore
q1 = q2, q2 = q1
and the average over polarizations can be rewritten ase(q1)e (q1) =
1
2(EE + EE EE ), (3.2.15)
where
E = +q1q2 + q2q1
q1 q2 .
Thus the average over tensor gauge boson polarizations gives
M;abM ;ab
e(q1)e (q2)
e (q1)e (q2) = (3.2.16)
=g4
4T r
p+
ta14
p q2 tb + tb
14
p q1 ta +
+ ifabctc1
q23(q2 q1)( 1
2 )
p
tb
14
p q2ta
+
ta
14
g
p q1tb
ifabc tc 1
q23(q2 q1)( 1
2
)
aa
d(r)
bb
d(r)
1
2(EE + EE EE )
1
2(EE + EE EE ).
In the next section we shall evaluate these traces and sum over the polarizations.
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3.2.4 Evaluation of Traces
In order to evaluate the squared matrix element in the last expression (3.2.16) we
have to calculate the traces and then perform the summation over polarizations. We
shall use convenient notations for the different terms in the amplitude. The whole
amplitude will be expressed as a symbolic sum of three terms:
M = A + B + G,
exactly corresponding to the three Feynman diagrams in (3.2.5),(3.2.6), (3.2.7) so
that the squared amplitude (3.2.16) shall have nine terms
MM = (A + B + G)(A + B + G).
The first contribution can be evaluated in the following way:
(GG) ;
=g4
4d2(r)T r{p+ifabctc 1
q23( 1
2 )(q2 q1)
p(i)fabc tc 1q23
(
12
)(q2 q1)}aabb =
=g4
4d2(r)tr(fabcfabc
tctc
)T r{p+(q2 q1)