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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

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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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    6

    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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    8

    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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    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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    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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    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)


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