[tt] Itzhak Bars: Standard model of particles and forces in the framework of two-time physics
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+++++++++++++++++++++++
Itzhak Bars: Standard model of particles and forces in the framework of
two-time physics
PHYSICAL REVIEW D 74, 085019 (2006)
Department of Physics and Astronomy, University of Southern California,
Los Angeles, California 90089-2535 USA
(Received 16 June 2006; published 20 October 2006)
In this paper it will be shown that the standard model in 3 + 1 dimensions
is a gauge fixed version of a 2T physics field theory in 4 + 2 dimensions,
thus establishing that 2T physics provides a correct description of nature
from the point of view of 4 + 2 dimensions. The 2T formulation leads to
phenomenological consequences of considerable significance. In particular,
the higher structure in 4 + 2 dimensions prevents the problematic F * F
term in QCD. This resolves the strong CP problem without a need for the
Peccei-Quinn symmetry or the corresponding elusive axion. Mass generation
with the Higgs mechanism is less straightforward in the new formulation of
the standard model, but its resolution leads to an appealing deeper
physical basis for mass, coupled with phenomena that could be measurable.
In addition, there are some brand new mechanisms of mass generation
related to the higher dimensions that deserve further study. The technical
progress is based on the construction of a new field theoretic version of
2T physics including interactions in an action formalism in d + 2
dimensions. The action is invariant under a new type of gauge symmetry
which we call 2T-gauge symmetry in field theory. This opens the way for
investigations of the standard model directly in 4 + 2 dimensions, or from
the point of view of various embeddings of 3 + 1 dimensions, by using the
duality, holography, symmetry, and unifying features of 2T physics.
I. THE SP(2, R) GAUGE SYMMETRY
The essential ingredient in two-time physics (2T physics) is the basic
gauge symmetry Sp(2, R) acting on phase space XM, PM [1], or its
extensions with spin [2,3] and/or supersymmetry [4-6]. Under this gauge
symmetry, momentum and position are locally indistinguishable at any
instant. This principle inevitably leads to deep consequences, one of
which is the two-time structure of spacetime in which ordinary one-time
(1T) spacetime is embedded. Some of the 1T physics phenomena that emerge
from 2T physics include certain types of dualities, holography, emergent
spacetimes, and a unification of certain 1T physics systems into a single
parent theory in 2T physics.
In the present paper a field theoretic formulation of 2T physics is given
in d + 2 dimensions. To construct the 2T field theory, first the free
field equations are determined from the covariant quantization of the 2T
particle on the worldline, subject to the Sp(2, R) gauge symmetry and its
extensions with spin. Next, an action is constructed from which the 2T
free field equations are derived, and then interactions are included
consistently with certain new symmetries of the action. The resulting
action principle for 2T physics in d + 2 dimensions is then applied to
construct the 2T standard model in 4 + 2 dimensions. It is shown that the
usual standard model in 3 + 1 dimensions is a holographic image of this 4
+ 2-dimensional theory. The underlying 4 + 2 structure provides some
additional restrictions on the standard model, with significant
phenomenological consequences, as outlined in the abstract. The 4 +
2-dimensional theory suggests new nonperturba-tive approaches for
investigating 3 + 1-dimensional field theories, including QCD.
Prior to this development, 2T physics had been best understood for
particles in the worldline formalism interacting with all background
fields [3], including gauge fields, gravitational field and all high spin
fields, and subject to the Sp(2, R) gauge symmetry, or its extensions with
spin. For the spinless particle, the three Sp(2, R) gauge symmetry
generators Qtj(X,P), i, j = 1,2, are functions of phase space and depend
on background fields (f>Mi-M2"'M'(X) of any integer spin s. The simplest
case of 2T physics corresponds to a spinless particle moving in the
trivial constant background field rjMN that corresponds to the metric in a
flat spacetime. In this case the Sp(2, R) gauge symmetry is generated by
the operators
(1.1)
<2n = ^x " x, Q22 = \p ' p>
<2i2 = G21 = tyx' p + p ' x),
where the dot product involves the flat metric rjMN. Similarly, for
spinning particles of spin s, phase space (jf, XM, PM), includes the
fermions /f, i = 1,2, " " " ,2s, so the gauge symmetry is enlarged to
the worldline supersymmetry OSp(2s|2) which includes Sp(2, R). In flat
spacetime, the generators of the gauge symmetry correspond to all the
spacetime dot products among the /f, XM, PM. These generators are first
class constraints that vanish, thus restricting the phase space (/f1, XM,
PM) to a OSp(2s|2) gauge invariant subspace.
To have nontrivial solutions for the constraints Qtj = 0, etc., the flat
metric rjMN, which is used to form the dot products in the constraints,
must have a two-time signature. So, in the absence of backgrounds, the 2T
particle action is automatically invariant under a global SO(d, 2)
symmetry in d + 2 dimensions, where the 2T signature emerges from the
requirement of the local gauge invariance of the physical sector. In the
presence of backgrounds, the 2T signature in d + 2 dimensions is still
required by the gauge symmetry. However, the nature of the spacetime
global symmetry, if any, is determined by the Killing vectors of the
background in d + 2 dimensions, and it may be smaller or larger than SO(d,
2).
It is well understood [1-7] that the gauge symmetry compensates for extra
dimensions in phase space (XM, PM, /f) and effectively reduces the d +
2-dimensional space by one-time and one-space dimensions, thus
establishing causality and guaranteeing a ghost free 2T physics theory.
The subtlety is that there are many ways of embedding the remaining time
and Hamiltonian in the higher space-time. Therefore, there are many 1T
systems that emerge in (d -- 1) + 1 dimensions as solutions of the
constraints with various gauge choices. Some examples are given in Fig. 1.
In these emergent spacetimes the Hamiltonian for each 1T system is
different, hence the dynamics appears different from the point of view of
1T physics. However, each 1T system holographically represents the
original 2T system in d + 2 dimensions. Of course, due to the original
gauge symmetry, the various 1T systems are in some sense equivalent. This
equivalence corresponds to dualities among the various 1T systems [7-11].
Hence 2T physics may be recognized as a unifying structure for many 1T
systems. The unification occurs through the presence of the higher
dimensions, but in a way that is very different than the Kaluza-Klein
mechanism since there are no Kaluza-Klein excitations but instead there
are hidden symmetries that reflect d + 2 dimensions, and also a web of
dualities among 1T systems that are holographic images of the 2T parent
theory in d + 2 dimensions.
As shown in Fig. 1, simple examples of such 1T systems in (d -- 1) + 1
dimensions, that are known to be unified by the free 2T particle in flat d
+ 2 dimensions, and have nonlinear realizations of SO(d, 2) symmetry with
the same Casimir eigenvalues, include the following systems for spinless
particles: free massless relativistic particle [1], free massive
relativistic particle [7], free massive nonrela-tivistic particle [7],
hydrogen atom (particle in 1/r potential in d -- 1 space dimensions) [7],
harmonic oscillator in d -- 2 space dimensions [7], particle on a sphere
Sd~l X R [11], particle on AdS^_j- X S* for k = 0, 1, " " " , (d -- 2)
[7], particle on maximally symmetric curved spaces in d dimensions [8],
particle on Banados-Teitelboim-Zanelli black hole (special for d = 3 only)
[9], and twistor equivalents [10,11] of all of these in d dimensions.
There are also generalizations of these for particles with spin [2], with
supersymmetry [4], with various background fields [3], and the twistor
superstring [5], although details and interpretation of the 1T physics for
gauges other than the massless particle gauge remain to be developed for
most of the generalizations.
FIG. 1 (color online). Some 1T physics systems that emerge from the
solutions of Qtj = 0.
The established existence of the hidden symmetries and the duality
relationships among such well-known simple systems at the classical and
quantum levels provide part of the evidence for the existence of the
higher dimensions. This already validates 2T physics as the theory that
predicted them and provided the description of the underlying deeper
structure that explain these phenomena.
Next comes the question of how to express these properties of 2T physics
in the language of field theory, and how to include interactions. This was
partially understood [12] in the form of field equations, including
interactions, as reviewed in Sec. II. But this treatment missed an action
principle from which all the equations of motion should be derived. The
field equations were classified as those that determine kinematics and
those that determine dynamics. The dynamical equations including
interactions could be obtained from an action as usual, but the
kine-matical equations, which determine how 3 + 1-dimensional spacetime is
embedded in 4 + 2-dimensional spacetime, needed to be imposed from outside
as additional constraints. This was considered incomplete in [12], because
a full action principle that yields all the equations is needed to be able
to study consistently the quantum theory and other properties of the
theory.
The new principles for constructing the 2T physics action in d + 2
dimensions are developed in Sec. II. These emerge from basic properties of
Sp(2, R) and its extensions as discussed in [12], supplemented with a hint
on the overall structure of the action that followed from a recent
construction of Becch- Rouet-Stora-Tyutin (BRST) field theory for 2T
physics [13]. The new action principle does not use the BRST formalism but
has a new type of gauge symmetry which we name as the 2T gauge symmetry
in field theory. From this action we derive both the kinematic and the
dynamical equations through the usual variational principle.
The standard model in 4 + 2 dimensions is constructed in Sec. III by
introducing the matter and gauge field content analogous to the usual
standard model but now derivatives and vector bosons are SO(4,2) vectors,
fermions are SO(4, 2) = SU(2, 2) quartet spinors, while all fields are
functions in 4 + 2 dimensions. The 2T gauge symmetry dictates the overall
structure and the form of the terms that can be included.
Next, in Sec. IV, the action for the standard model in 3 + 1 dimensions is
derived from the action in 4 + 2 dimensions by solving the subset of
equations of motion that determine the kinematics. This step is equivalent
to choosing a gauge for the underlying Sp(2, R) gauge symmetry in the
worldline formalism, as illustrated in Fig. 1. In particular, solving the
kinematics in a particular parametrization given in Eq. (4.1) corresponds
to the massless relativistic particle gauge denoted in Fig. 1. The
solution of the kinematic equations in this way provides a holographic
image of the 4 + 2-dimensional theory in the 3 + 1-dimensional spacetime.
The degrees of freedom in all the fields are thinned out from 4 + 2
dimensions to 3 + 1 dimensions. Both the 2T gauge symmetry and solving the
kinematical equations play a role in reducing the degrees of freedom from
4 + 2 dimensions to the proper ones in 3 + 1 dimensions. The remaining
dynamics in 3 + 1 dimensions is determined by the emergent 3 +
1-dimensional action. In the chosen gauge, the emergent theory is the
usual standard model action, however the emergent standard model comes
with some interesting restrictions on certain terms. The additional
restrictions are effects of the overall 4 + 2 structure and are not
dictated by working directly in 3 + 1 dimensions.
Having obtained the standard model in 3 + 1 dimensions, one may ask what
is new in 3 + 1 dimensions? Part of the answer includes the constraints
inherited from 4 + 2 dimensions that get reflected on the overall
structure of the emergent standard model in 3 + 1 dimensions. It is
fascinating that this has phenomenological consequences. First we
emphasize that all the basic interactions that are known to work in nature
among the quarks, leptons, gauge bosons, and Yukawa couplings are
permitted. The forbidden terms seem to coincide with unobserved
interactions in nature. In particular, a forbidden term in the 3 +
1-dimensional emergent action is the problematic FfiJ,FAo-sfJ'vA'T term in
QCD, or similar anomalous terms in the weak interactions that cause
unobserved small violations of B + L. This is because there are no
corresponding terms in the 4 + 2-dimensional action that would filter down
to 3 + 1 dimensions. As discussed in Sec. V, this provides a nice
resolution of the strong CP problem in QCD without the need for the
Peccei-Quinn symmetry or the corresponding elusive axion. This had
remained as one of the unresolved issues of the usual standard model (for
a recent review see [14]). So, the 4 + 2-dimensional theory seems to
explain more as compared to the usual 3 + 1-dimensional theory.
Mass generation is less straightforward in the emergent model than the
usual standard model, because a quadratic mass term for the Higgs boson is
not permitted by the underlying 4 + 2 structure. This is discussed in Sec.
VI. To obtain a nontrivial vacuum, one may either introduce interactions
of the Higgs with a dilaton or invoke dynamical breakdown of the SU(2) X
U(l) gauge symmetry through mechanisms such as extended technicolor. The
dilaton scenario offers an appealing deeper physical basis for mass and
generates new phenomena that could be measurable. There are also new
possibilities for mass in the emergent theory that involve the higher
dimensions. For this we only need to recall that a massive relativistic
particle also can come out from the 4 + 2 theory as illustrated in Fig. 1.
The mass in this case is analogous to a modulus that comes from the higher
dimensions in a non-trivial embedding of 3 + 1 in 4 + 2 dimensions. Since
mass generation is the obscure part of the standard model, and there are
new mechanisms in the 2T action, the mass generation deserves further
study of what the 2T approach has to offer.
Since our proposal is that the fundamental theory is formulated in 4 + 2
dimensions, one wonders if one can test the effect of the extra
dimensions. This can be explored by studying other gauges of Sp(2, R)
(equivalent to different forms of solutions of the kinematic equations)
that lead to 3 + 1-dimensional dual versions that are also holographic
images of the same 4 + 2-dimensional standard model in the sense of Fig.
1. The exploration of these dual theories corresponds to exploring the 4 +
2-dimensional space and is left to future work. Other remaining issues and
future directions will be discussed in Sec. VII.
II. PRINCIPLES FOR INTERACTING 2T FIELD THEORY ACTION
The construction of the proper action principle for 2T physics has
remained an open problem for some time. Equations of motion for each spin,
including interactions were available, and even the standard model in 4 +
2 dimensions in equation of motion form was outlined [12]. The main
stumbling block has been the fact that there are more equations to be
satisfied by each field than the number of equations which can be derived
from a standard action. The solution given in this paper will involve some
subtle properties of the delta function S(X2) that imposes the <2n ~ X2 =
0 constraint and already could have been attained in [12], but it was
missed. A crucial hint came from a recent BRST field theory formulation of
the problem, akin to string field theory, as discussed in [13]. In the
present paper we bypass the BRST construction and use only the tip as a
springboard to construct a simpler action for only the relevant fields of
any spin, although the full BRST field theory also may be useful to
consider for more general purposes. The 2T action principle given in this
section provides the proper minimal framework to consistently discuss new
symmetries, include interactions, and perform quantization in 2T field
theory. This principle is applied to construct the standard model in 4 + 2
dimensions in Sec. III.
In this section we first review the derivation [12] of the 2T free field
equations in d + 2 dimensions for scalars, fermions, vectors, and the
graviton from the worldline properties of the Sp(2, R), OSp(l|2),
OSp(2|2), and OSp(4|2) gauge symmetry, respectively. Then we introduce an
action principle in d + 2-dimensional field theory from which these field
equations are derived through the variational principle. The action has a
new kind of gauge symmetry that we call the 2T gauge symmetry.
The free field equations are obtained as follows. The OSp(2s|2) are the
gauge symmetry groups in the worldline formulation of the 2T particles of
spin s in flat d + 2
dimensions. In SO(d, 2) covariant first quantization, physical states are
identified as those that are gauge invariant by satisfying the first class
constraints which form the OSp(2s|2) Lie superalgebra. In position space
the constraint equations turn into field equations in d + 2 dimensions.
So, in principle the number of field equations for each spinning particle
of spin s is equal to the number of generators of the gauge groups
OSp(2s|2), since all generators must vanish on the physical gauge
invariant field. By some manipulation the number of equations in d + 2
dimensions can be brought down to a smaller set but still there are more
field equations as compared to the familiar field equation for
corresponding spinning fields in d dimensions. This must be so, because
only with the additional equations it is possible to have an equivalence
between the d + 2-dimensional field equations and their corresponding ones
in d dimensions. The familiar looking d + 2 field equation, that is
similar to the cf-dimensional equation, is interpreted as the dynamical
equation, while the additional equations can be interpreted as subsidiary
kinematical conditions on the field in d + 2 dimensions.
The same subsidiary kinematic equations for fields of any integer spin s
also were obtained by considering only the spinless particle propagating
in background fields (pM^M^"M'(X) of any integer spin s. In this case the
Sp(2, R) generators Qtj(X, P), i, j = 1,2 are functions of the background
fields (f>M^M^'"M*(X) [3]. Requiring closure of the Qtj(X, P) under
Poisson brackets into Sp(2, R) demands conditions on the background
fields. These conditions are identical to the kinematic equations. In this
case the background fields are off shell and are not required to satisfy
the dynamical equation. The kinematic equations by themselves are
sufficient to reduce the degrees of freedom in the fields <pM^Mi'"M*(X)
from d + 2 dimensions to d dimensions both in the spacetime XM dependence
of the field and in the components of spinning fields labeled by MhM2, %
% % ,MS.
The 2T physics equations that emerge from OSp(2s|2) constraints in d + 2
dimensions, or from the backgrounds with Sp(2, R) gauge symmetry, coincide
with Dirac s equations [15] in the case of s = 0, 1/2, 1 in d = 4 and
their generalizations to spin 2 and higher [3,12,16] in any dimension.
While there were attempts in the past to write down a field theory action,
the subsidiary kinematic conditions have been treated as external
conditions not derived from the same action. The proper action principle
will be given in this paper.
In any case, it has been known [12,15,17-23] at the level of equations of
motion, that the ensemble of the d + 2-dimensional equations correctly
reproduce the massless Klein-Gordon, Dirac, Maxwell, Einstein, and higher
spin field equations of motion in d dimensions. In 2T physics this is
interpreted as an example of a more general holography from d + 2
dimensions to (d -- 1) + 1 dimensions that emerges from gauge fixing
Sp(2, R).
A particular parametrization given in Eq. (4.1) that corresponds to the
Sp(2, R) gauge indicated as the mass-less particle gauge in Fig. 1
will be used to derive the standard model in 3 + 1 dimensions from the 4 +
2-dimensional theory. The massless Klein-Gordon, Dirac, Maxwell, Einstein
field equations in d dimensions emerge in the parametrization of Eq.
(4.1). The novelty in 2T physics is its more general property that all the
other 1T interpretations (massless, massive, curved spaces, etc.),
outlined in Fig. 1 also emerge from the same 2T field equations as
different holographic images in different Sp(2, R) gauges, as explained
before both in particle theory [7] and field theory [12]. In this paper we
will use only the massless particle interpretation related to the
parametrization of Eq. (4.1) in order to connect to the usual form of the
standard model. However, it should be evident that dual versions of the
standard model, including interactions, will emerge by taking advantage of
the more general properties of these equations.
A. Scalar field
1. Free field equations for scalars
For the spinless 2T particle, the vanishing of the Sp(2, R) generators
implies that the physical phase space must be gauge invariant. In
covariant first quantization, physical states |<P) are identified as those
on which the generators Qij vanish X2|<P) = 0, P2|<P) = 0, (X " P + P "
X)\<&) = 0. This means that the physical states form the subset of states
that are gauge invariant under Sp(2, R). The field O(X) is defined as the
probability amplitude of a physical
footnote: Dirac and followers regarded the equations of motion in 4 + 2
dimensions as a formulation of the hidden conformal symmetry SO(4,2) of
massless field equations in d = 4. The 4 + 2-dimensional spacetime was not
emphasized as being anything other than a trick. 2T physics developed
independently from the opposite end without awareness of Dirac s approach
to conformal symmetry, and deliberately focusing on signature (d, 2) with
2 times. The (d, 2) signature specifically started with a hunch [24]
developed from M theory that there are higher dimensions with signature
(10,2) (see also [25] for an independent idea). This path developed
through various papers [26,27], gathering hints on how to correctly
formulate a theory with signature (d, 2) consistent with M theory,
causality, and dynamics without ghosts in d + 2 dimensions. The
multiparticle symmetries in [27] provided the dynamical setup that
eventually led to the introduction of the Sp(2, R) gauge symmetry [1] for
a single particle. The new features that emerged in 2T physics include the
underlying fundamental role of the Sp(2, R) gauge symmetry, the existence
of the other 1T interpretations as holographic images in (d -- 1) + 1
dimensions of the same 2T system, with the same original SO(<i, 2)
symmetry that is not interpreted as conformal symmetry in the various
holographic images, the duality among the multiple 1T solutions, and the
corresponding interpretation of the hidden symmetries and dualities as
evidence for the higher dimensions. Only after these properties were
discovered, it was recognized in [12] that Dirac s approach to conformal
symmetry, that had been forgotten, could be seen as part of 2T physics in
a particular gauge.
state in position space O(X) = (X|<P). Since momentum is represented as a
derivative in position space (X\PM = -- idM(X\, the gauge invariance
conditions applied on physical states |<P) give the free field equations
for the field O(X) as
(2.1)
dMdM&(X) = 0,
X2®(X) = 0, vM
XMdM&(X) + dM(XM&(X))
0.
The general solution of the first equation is
(2.2) 0(X) = S(X2)(£>(X),
where O(X) (without the hat)2 is any function of XM which is not singular
at X2 = 0. We have used the property X28(X2) = 0 of the delta function. We
also note the following additional properties of the delta function that
we use repeatedly below:
(2.3)
X
d dXM d dX
S(X2) = 2XM8'(X2), S(X2) = 2X2S'(X2) =
(2.4)
28(X2),
d2S(X2) = 2(d + 2)8'{X2) + AX2S"{X2) = 2(d -- 2)8'(X2).
Here S'(u), 8"(u) are the derivatives of the delta function with respect
to its argument u = X2. So we have used uS'(u) = -- 8(u) and u8"(u) = --
28'(u) as the properties of the delta function of a single variable u to
arrive at the above expressions. These are to be understood in the sense
of distributions under integration with smooth functions.
Inserting the solution <P(X) = 8(X2)^(X) into the other two equations in
(2.1), and using Eqs. (2.3) and (2.4), gives
(2.5)
8{X2)(
d -- 2 X " d +-------KP = 0,
(2.6)
d2[<5(X2)<P] = <5(X2)d2<P + 48'(X2)l
d -- 2 X % d +--------KP = 0.
Here the derivatives must first be taken in the full space XM before the
condition X2 = 0 is imposed. It is easy to see that these equations are
invariant under the following gauge transformations
(2.7)
<5A<P = X2A(X)
for any function A(X). The 4S'(X2) part of the second equation in Eq.
(2.6) with the given coefficient 4 is crucial
2We distinguish between the symbols <f>, tya, AM and <t>, *f?a, AM to
emphasize that <t, ^a, AM include the delta function factor. In comparing
notes with Ref. [ 12] one should compare the 4>, ^a, AM in that paper to
the <t>, tya, AM in this paper, not to the <fr, $ ,, AM.
for this invariance. If we define
(2.8)
O(X) = O0(X) + iP~iX),
where O0 = [*J)(X)]X2=0, and X2 ~ is the remainder, the gauge symmetry
implies that O0 is gauge invariant while ~ is pure gauge freedom and
completely drops out. Hence, the nonsingular gauge invariant function
O0(X) satisfies the following equations in d + 2 dimensions
d -- 2 X % d +--------KP0
x2=0
0, [d2O0]X2=0 = 0, (2.9)
where we have substituted the gauge invariant part O0 instead of the full
O. This is verified directly by substituting Eq. (2.8) in Eq. (2.5) and
(2.6) and noting that O drops out.
The first equation in Eq. (2.9), together with the X2 = 0 condition, are
the kinematic equations, and the second one is the dynamical equation. The
kinematic equation is solved by any homogeneous function of degree -- (d
-- 2)/2. Namely, its general solution must satisfy the scaling property
O0(£X) = t~(d~2^2<&0(X). Note that this homogeneity condition is much more
than assigning a scaling dimension to a field in usual field theory
because it is a restriction on the spacetime dependence of the field. With
a particular parametrization of XM that satisfies the other kinematic
constraint X2 = 0, as given in Eq. (4.1), plus the homogeneity condition,
one can show [12,15] that the dynamical equation for O0(X) in d + 2
dimensions re-
duces to the massless Klein-Gordon equation
0
in d dimensions, with a definite relationship between O0(X) and 4>{x). We
will return to this detail of holography in the following section.
2. Action for scalars with interactions
We now propose the following interacting field theory action that
reproduces both the kinematical and dynamical equations of motion Eqs.
(2.9). The construction of a proper action principle in 2T physics field
theory has eluded all efforts before, even though one could write
equations of motion as shown above and [15] [12], including interactions.
We will argue that we obtain the interactions uniquely through a gauge
principle directly connected to the underlying Sp(2, R) symmetry.
The inspiration for the following form came from a BRST formulation for 2T
physics field theory including
The scaling dimension alone does not require homogeneity. For example, a
Klein-Gordon field in four-dimensional usual field theory has scaling
dimension -- 1, but it is not homogeneous. This is because the dimension
operator is not only the part X % d that acts on coordinates but also
includes a part that acts on canonical field degrees of freedom. In
particular, note that the usual plane wave solutions with definite
momentum exp(ik % X) are not homogeneous.
interactions [13]. Here we do not use the BRST version but only extract
from it a partially gauged fixed version which has just sufficient
leftover gauge symmetry for our purposes here. Thus, the key ingredients
that go into the proposed action below are first that it should possess
the gauge symmetry in Eq. (2.7), and second that it should have additional
gauge symmetry to reduce the theory to only the O0 degree of freedom,
including interaction. The action is
(2.10)
S(O) = / dd+2X{B(X)d2[<$>S(X2)]
-- §(x2)[5(x)y'(0) + ay(o)]}.
Notice the delta function that imposes the vanishing Sp(2, R) generator X2
= 0 condition. The function V(O) will be the potential energy for the
field s self interactions; its derivative is V'(O) = dV/d&. The role of
the constant coefficient a will become evident in the discussion below. We
will see that V(O) will be uniquely determined by the gauge symmetries of
the field B(X). The field B(X) emerged from the BRST point of view as a
combination of auxiliary fields associated with the kinematical and
dynamical Sp(2, R) generators X % P and P2.
Let us first discuss the gauge symmetries of this action. The 8(X2)
structure makes it evident that we have the gauge symmetry of Eq. (2.7)
SA& = X2A(X), hence if O is written in the form of Eq. (2.8) O = O0 + X2 ~
, the remainder O automatically drops out. Therefore, this action really
depends only on O0 automatically. We will continue to write O everywhere,
but it should be understood that any mode of O proportional to X2 is
decoupled, and this fact will be used below.
Next we show that there is nontrivial gauge symmetry associated with the
field B(X) under the following transformation with gauge parameter b(X)4
(2.11)
( d ~ 2\, 1 2/ 2, H/^W
obB = X % d +-------b -- -- X(db -- oV ($)),
24
any &(X).
The transformation of the action under this b symmetry gives
(2.12)
,,,. d -- 2\ dd+2 X X- d +-------\b
c
§&5(0) = / dd+2X
l
1 / 2, , »/^\\ r 2r^ o/ \n
-A2(d' b -- by"(<P)){d [(PolX2)] 4
<5(X2)y'(0)}
4This transformation was extracted from the BRST formalism in [13].
(2.13)
d -- 2\
" d +
c
/ dd+2X S(X2) X % d
+ (d2b -- W'(0))( X % d + + 48'(X2)l
b[d2& -- V(O)]
d -- 2
O
X % d +
d -- 1>
, d -- 2 bX % d +-------- KP
<P
f ^+9 9n^ a -- 2 / a X
9MXM 45'(a'z)<P X " db +-------b
(2.14)
8(X2)V'(<&)b
, n,d -- 2--8{X2)-------b
d + 2
<PV (O) -- -------V (<P) . (2.14)
a -- 2
In going from Eq. (2.12) and (2.13) we have used Eq. (2.6) to evaluate
d2[&S(X2)] and then set X2S(X2) = 0 and X28'(X2) = -- 8(X2). To reach Eq.
(2.14) we do integrations by parts taking into account the delta functions
and noting the identity
(2.15)
9 9 d -- 2 d <5(a ) A " d +-------- o
c 55(0) =--/ dd+2x\ 8{X2)8B{d2<S> + V'(O)}
9 / cf -- 2\ + 45 (X jofi X " a +-------KP
+ 8(X2)8<£>[d2B + y"(<P)5 + ay'(O)] .
The 45''(X2) comes from evaluating d2[&S(X2)] as in Eq. (2.6). Since the
distributions S(X2), S'(X2) are linearly independent, the coefficients of
S(X2)SB and S'(X2)SB should vanish independently
(2.18)
d -- 2
a " a$ +-------<p
2 x2=o
0,
[a2o -- y'(o)]X2=0 = o.
In these equations we really have O = O0 with no remainder O due to the
gauge symmetry 5^*5 = X2A as discussed above. But we may allow any
remainder as long as it is homogeneous since this does not change the
equations of motion. Thus, our action did provide the two desired
equations of motion for O0, while the remainder O is gauge freedom and can
still be taken as nonzero as long as it is homogeneous. In addition, the
equation of motion for B is
(2.16)
[d2B -- y"(<P)5 -- ay'(O)]X2=0 = 0. (2.19)
S(X2)X % d + 48'(X2)l
d + 2 9 " d + -- z-- d b
d ~ 22, X % d +-------- b.
The total derivative in Eq. (2.14) can be dropped, and then we see there
is a gauge symmetry 5&5(<P) = 0 provided the potential energy y(O)
satisfies
(2.17)
oy"(o)
d + 2 d -- 2
y'(O) -- y'(O) = \<&(d+2/d-i)_
Thus, except for the overall constant A, the y'(O) is uniquely determined
as the given monomial, as a consequence of imposing the &-gauge symmetry.
This gauge symmetry is required to reduce the O, B degrees of freedom to
only O0. But interestingly, it also fixes the interaction uniquely.
Now let us verify that this action gives the equations of motion that we
require. The general variation of the action is obtained from Eq. (2.10)
This can be understood in a bit more detail by exhibiting the remainder of
B in the form B = B0 + X2B. Then the B equation really is
(2.20)
d2B0 -- y"(o0)50 -- ay'(o0)
/ d + 2\ ~ 1 + 4iX- d +-------B
0.
We see that this is an equation that determines the remainder B in terms
of B0, O0, without fixing the dynamics of B0 at all. So there remains one
fully undetermined function among the Bo, B. This is of course related to
the b symmetry given in Eq. (2.11). Using the b symmetry we can choose the
function B0 at will. As in [13], we make the convenient gauge choice B0 =
y<P0 where y is an overall constant to be determined consistently. In that
case the
In this paper we are very careful when we make such statements. The
equation S(X2)F(X) + 8'(X2)G(X) = 0 has the more general solution (G)^=0 =
0 and (F -- G)X2=0 = 0, rather than merely (F)X2=0 = 0 for the second
equation. Here G is the remainder when one writes G = G0 + X2G. So
generally F need not vanish on its own. However, in the present case we
have already argued that <t> = <t>0 since the remainder drops out.
Therefore, the two terms do vanish separately.
remainder B must be determined from Eq. (2.20) after inputting B0 = yO0
and recalling the dynamical equation for O0 up to the proportionality
constant y[d2O0 -- y'(O0)]X2=0 = 0. After using OV'(O) = jz22V'(O) as
given in Eq. (2.17) we obtain B
(2.21)
( d + 2\ ~ 4y
4X % d +-------B =------
2 a -- 2
V'(O0) + aV(O0).
And of course B0 = yO0. Finally, we can insert the fully fixed B0, B as
well as O = O0 into the action and obtain an action purely in terms of O0
(2.22)
c r " ^(^0) = / dd+2X\ (B0 + X2B ~\8(X2)(d2^0
-- V'(O0))
+ 4<5'(A2) A " d + --------KP0 -- 6(A2)flV($0)
(2.23)
r r = / dd+2X\ 8(X2)\B0(d2^0 -- y'(O0)) -- ay(O0)
+ 4<P0 a " d +------- ~
2 / cf -- 2\ "1 + 48 (X )B0 X % d +-------KP0
(2.25)
S(O) = 2y / c^+2X<5(X2) -Od20
2
i_______(U(2d/d-- 2)
2d \
The potential energy V(O) has now been fixed uniquely as the monomial
(2.26)
y(O) = a--------<$,(2d/d-2)_
v 7 2d
The constant 2y is an overall normalization factor that will be absorbed
later into the normalization of the volume in (d -- 1) + 1 dimensions.
In our derivation the O in the action of Eq. (2.25) was strictly O0. But
we can add a remainder to O0 + X2 ~ without changing the physics, as long
as the remainder is homogeneous and satisfies (X % d + ^^)~ = 0. Then the
full O satisfies (X % d + ^2)0 = 0 when it is on shell. We have the
freedom to add a homogeneous remainder because the action in Eq. (2.25),
including the homogeneous remainder but off shell O0, has a leftover gauge
symmetry <50 = X2A as long as A is homogeneous (X % d + ^-^)A = 0. To
demonstrate this symmetry, observe the general variation of the action in
Eq. (2.25):
(2.27)
f ,,2 2 f (1 2 ,/^\\
<5S(<P) = 2y / d X 8 (X ) <5<P--d <P -- V (<P)
2
+ -Od2(<50H
(2.24)
2
/ dd+2X8(X2) yO0d2O0 + a -- y-------O0y'(O0) = 2y / dd+2X
8<& S(X2)-d2<£> -- y'(O)
d -- 2
2
-- ay(o0) .
(2.28)
+ -- d2(8(X2)Q)
2
Equation (2.22) is the original action Eq. (2.10) rewritten in terms of
the components O0, B0, B. The form in Eq. (2.23) follows after using
X2S(X2) = 0 and X2S'(X2) = -- S(X2) and performing an integration by
parts in the middle term to get the structure (X % d + ^-^)B. Inserting
the gauge fixed B 0 , B we get Eq. (2.24), where the S'(X2) term of Eq.
(2.23) becomes the total derivative 2ydM(A'M<5'(A'2)020) and drops out in
this gauge.
We must require that the equation of motion for O0 that follows from this
gauge fixed action be the same as 32O0 -- V'(<&0) = 0 as given by the
original action. For this to be the case, the coefficient y that had
appeared in the gauge fixing B0 = yO0 must be determined self-consistently
in terms of the constant a as y = ^pa, so that the potential energy terms
in Eq. (2.24) sum up to being -- 2yy(0) and match the normalization of
the kinetic terms as follows:
(2.29)
2y / dd+2X8<$> 8(X2)[d2^ -- y'(O)]
r d -- 2 1] + 28'(X 2)X % d<P +-------<P L
If we substitute the gauge transformation <50 = X2A in Eq. (2.29) we get
the gauge variation <5AS(0) which becomes
(2.30)
c
<5AS(0) = 2y / dd+2XA8(X2) X % aO
d -- 2 +-------O
(2.31)
= 2y / dd+2XdM(XMA<£>8(X2)) = 0.
In the first line we have already dropped a term due to
X28(X2) = 0 and used X2S'(X2) = -- S(X2). The resulting form is a total
divergence as given in the second line as long as A is homogeneous (X % d
+ ^^)A = 0.
Furthermore, the action in Eq. (2.25), including the homogeneous
remainder, has a leftover ^-symmetry Sb& given in Eq. (2.11) as long as
the b parameter is homogeneous (X % d + ^Y^b = 0. In that case, from Eq.
(2.29) we derive
(2.32)
c 5^,5(0) = y(d -- 2) / dd+2X8{X2)b\ Oy"(0)
d + 2 d -- 2
y'(o) .
So, requiring the symmetry SbS(&) = 0 fixes the potential uniquely.
In conclusion, in the gauge fixed action in Eq. (2.25) we can allow any O
whose remainder O has the homogeneity property stated. Of course this
permits gauge fixing off shell all the way to O = O0 if so desired, but we
will rather keep the homogeneous gauge freedom as the remainder of the A
and b symmetries.
Now we show that the action in Eq. (2.25) is adequate to generate both the
kinematic and dynamical equations of motion. Using Eq. (2.29) we impose
the variational principle <5S(0) = 0 which gives
(2.33)
^ d ~ 2 X % d<P +-------<P = 0.
<5(X2)[d20 -- V'(O)] + 2<5'(X2) X % dO
This results in two different equations on O, not just one because the
coefficients of both S(X2) and S'(X2) must vanish separately. By contrast,
in an ordinary field theory the variation of a single field would result
in a single equation. This is one of the crucial observations that was not
appreciated in our previous attempts to construct an action principle that
gave both equations of motion.
Being careful as explained in footnote 5, the coefficients of S(X2) and
S'(X2) that vanish are
32O0 -- V(O0) + X % d
d + 2 ~ " d +------- O
x2
0,
d -- 2 (2.34)
X % d + -- --- O0 = 0.
The resulting equations are precisely the desired ones, provided O is
homogeneous, (X % d + ^4^)0 = 0 as is
6Note that the last coefficient in Eq. (2.29) is 2S'(X2) and not 48'(X2).
If it had been 48'(X2) there would have been a greater symmetry with
arbitrary A and arbitrary b rather than homogeneous A and homogeneous b.
Of course, the original action in Eq. (2.10) has the greater symmetry
before gauge fixing.
the case in our gauge fixed action as explained above. In that case we can
write the equations of motion without splitting O into components in the
form
(2.35)
d -- 2 X % d +--------
O = 0 and d20--V'(O) = 0.
We have made the point that it is crucially important that it is
understood that the action 5(0) in Eq. (2.25) is a gauge fixed version of
the original action Eq. (2.10) that contains only the gauge fixed O up to
an arbitrary homogeneous remainder, rather than the most general remainder
O. For the most general remainder O the action in Eq. (2.25) would give
the wrong dynamical equation.
Therefore, the correct action is either the simplified form Eq. (2.25)
with the gauge fixed O up to a homogeneous remainder that corresponds to
remaining gauge freedom, or it is the more general gauge invariant form in
Eq. (2.10) that includes all the degrees of freedom in O as well as those
of B. Recall that the &-gauge symmetry uniquely determined the interaction
V(O).
The advantage of the gauge fixed version in Eq. (2.25) is its simplicity
in terms of a single field O, but we must point out a subtle feature. To
arrive at the two equations of motion from this gauge fixed action, we
note that we have applied a slightly unconventional variational approach.
Specifically, note that two equations follow from the fact that the
general variation <50 = <5O0 + X2<50 contains two general variational
parameters <5O0, <50, in which neither <5O0 nor <50 are homogeneous,
although the remainder O is required to be homogeneous after the
variation. The unconventional part is the requirement of a homogeneous O
(as determined in our gauge fixing discussion), but a general <50 to yield
the second equation for O0, namely, (X % d + ^^)O0 = 0. By taking a
homogeneous O but a general <50, we have devised a tool to keep track of
the effects of the 2T gauge symmetry off-shell, whose utility is
demonstrated in Eq. (2.32), and which will come in handy in later
investigations. The alternative to the above is to take from the beginning
a gauge fixed action with a homogeneous O0 as well as homogeneous <5O0,
and have no O, <50 at all, however, in so doing we completely lose track
of the 2T gauge symmetry.
There is another physically equivalent gauge fixed form of the action that
makes the underlying Sp(2, R) symmetry more evident. This is given in the
appendix.
We have now shown that the field O(X) described by our action satisfies
precisely the same free field equations of motion in Eqs. (2.9) that
follow from the Sp(2, R) constraints, plus consistent interactions. So the
physical degrees of freedom and gauge symmetries of the 2T spinless free
particle are correctly described by our action principle.
In addition, we have introduced a gauge principle that leads to unique
self interactions.
In this process we also have discovered a new gauge symmetry that we will
call the 2T gauge symmetry. This includes both the A and the b gauge
symmetries. These gauge symmetries are responsible for removing gauge
degrees of freedom and identify the physical field as O0(X) =
[*J)(X)]X2=0. We will see that the 2T gauge symmetry persists in the
presence of all interactions of the field. Furthermore, for each field in
the theory there is an extension of this symmetry, so it is a rather
general symmetry that dictates the structure of the action.
3. Interactions among several scalars
Let us now describe interactions among several scalar fields. For
convenience we will do this in the gauge fixed version7 by using directly
O' = O0 for all the fields labeled by i = 1,2, % % %. However, for
brevity we will omit the zero subscript in O0. We identify as S0(O!) = --
12Zf dd+2XS(X2)(£>'d20! the quadratic part of the action in Eq. (2.25) at
zero coupling constant. The interaction term is then identified as
S1CO') =--/ dd+2X8(X2)V{&). (2.36)
The b symmetry requires y(O') to be overall homogeneous of degree jz2. For
example, if there are two scalar fields, say O(X) and H(X), the total
action must be taken in the form
S(0, H) = 50(O) + S0(H) -- I dd+2X8(X2)V((f), H),
(2.37) {I'm not moving the equation number to the top any more. Too lazy,
esp. since it's not too likely to make sense anyhow.]
where the allowed potential energy can only be of the form
V(0,#) = A0O(M/^2) + AHH(2d/d-2)
+ y Aj^i^i? ti dfc+l -- (2d/d-- 2)' k,l
(2.38)
The coupling constants are all dimensionless in any dimension. Note that
in four dimensions d = 4 only quartic interactions are allowed V(0, H) =
A^O4 + A.HH4 + Y.k,i^ki&kHlSk+i-4. No quadratic mass terms with
dimen-sionful constants can be included. This will impact our discussion
of mass generation as will be seen below.
7The fully gauge invariant version must have a B'(X) field corresponding
to each <t>'(X). Furthermore, there is a separate A' and V gauge parameter
for each i. After the gauge fixing procedure described in the previous
section for each i, we end up in the physical sector without the B'(X) and
only with the (p'(X) whose remainder ~'(X) is gauge fixed to be
homogeneous.
It is possible to modify the rigid result on the form of the potential
discussed above by modifying the action with another type of term that
includes S'(X2) instead of only S(X2). To illustrate this, consider again
the single field case and include an additional term in the action of the
form
52(0) =--/ dd+2X8'(X2)W{<&). (2.39)
The equations of motion as well as the gauge symmetries are altered with
the total action
StotC^) = S0(O) + 51(0) + s2(o).
(2.40)
Therefore, the final potential V(O) could be different. After a few trials
with a few such functions, it becomes clear that the resulting equations
of motion may have only trivial solutions except for special combinations
of V and W that must be chosen consistently to avoid a trivial system. So
far we have found only one very simple non-trivial case, given by a
quadratic W = 12aO2. This changes the kinematic equation for O and
requires O to be homogeneous O(fX) = tk^<&(X) with a degree k{a) that
depends on the constant a. Consistent with the new homogeneity degree of
O, the potential energy is again a monomial V(O) ~ Opl^ with a new power
p(a) so that V(O) has total homogeneity degree -- d just as before. Thus,
the power p(a) of the monomial in the potential V(O) can be changed
arbitrarily as a function of the coefficient a in W = 12aO2. Currently we
do not know of other examples for which the coupled kinematic and
dynamical equations have nontrivial solutions for other func-tionals W(O).
It has now become clear that V(O) could be altered by tinkering with the
additional term W(O). However, it is not a priori clear what forms of V(O)
exist consistently with the coupled differential equations.
If there is more than one scalar, such as O, H, then the system of
equations derived from W(0, H) and V(0, H) gets more complicated. A
general study of which forms of V(O) or V(0, H) can consistently be found
through this procedure is not currently available. We will return to this
topic when we discuss the mass generation mechanism in the standard model.
B. Spinor field
1. Free field equations for fermions
For particles of spin 1/2 on the worldline the phase space (XM, PM, M)
includes the anticommuting fermions M. The worldline gauge symmetry acting
on phase space is enlarged from Sp(2, R) to the supergroup OSp(1|2) [2].
The generators of this symmetry in the flat background are proportional to
X2, P2 (X % P + P % X), X " , P " . The quantized fermions M form a
Clifford algebra and therefore are represented by gamma matrices8 TM, TM
acting on the two chiral spinors of SO(d, 2) (assuming even d). The gamma
matrices satisfy TMTN + TNTM = 2rjMN, which is equivalent to the
quantization rules for if/*1.
The physical states |^P) correspond to the gauge invariant subset of
states on which all of the OSp(l |2) generators vanish. It is sufficient
to impose X " /\^) = P " /\^) = 0 because the remaining constraints
follow from these. The chiral field ^a(X) is defined as the probability
amplitude of a physical state in position and spinor space ^a(X) = {X,
a|^P). The second chiral spinor field ^ 00 = (X, a\ty) is associated with
the second spinor labeled with a instead of a. The number of components of
each chiral spinor of SO(d, 2) is 2dl2. In the case of d + 2 = 6 these
4-component spinors form the two fundamental representations of SU(2, 2) =
SO(4, 2).
Both chiral spinors must satisfy the physical state conditions. Thus,
defining "ira = "ira, it satisfies
"&a(X) = chiral spinor of SO(d, 2).
(2.41)
8An explicit form of SO(d, 2) gamma matrices in even dimensions labeled by
M = 0', 1', /x and /x = 0, i is given by T° =
-- iTy XI, T1 = t2 X 1, T° = 1 X 1, P = t3 X y, where y' are the SO(d --
1) gamma matrices. It is convenient to use a lightcone type basis
by defining r± = 4= (r° ± T1) =
-- ;'V2t± X 1. The TM are the same as the TM for M = ±', i, but for M = 0
we have F° = -- T° = -- 1 X 1. From these we construct the traceless TMN
as
\*ILV --
yp
0
i'V2
0 -p 00
r _!_/_/ --
r1- 'f1
10
1
ijl
00
yV 0
where y^ = (1, y'), y^ = (-- 1, y'), noting the lower /x indices. Then
\TMNJMN = -T+'-'j+'-' + \J^Y^--T+'^J-'f -r_'/ttJ+'i takes an explicit
matrix form. We can further write y1 = a1 X 1, y2 = a2 X 1, and yr = a3 X
pr, where the pr are the gamma matrices for SO(<i -- 3). It is possible
to choose Hermitian pr for SO(<i -- 3). In d = 4 the pr are replaced just
by the number 1 and then the y^, y^ are just 2X2 Pauli matrices. These
gamma matrices are consistent with the metric in spinor space 7] = -- ir\
X 1 = T0' = f° that is used to construct the contravariant spinor WL>R =
^lrV. The metric rj has the properties r/(r )r/ = equivalently
(r]TA
/T-'M'vt and y.^MN y.-- l
(MNi
o(v*t
and (r/r ) = (r/P
MN\f t =
Therefore, we have the Hermiticity properties (/\iTM'Pi
2R^M/\L and (/uXMN 2L)^ = 2L^MN/iL. We can also define the charge
conjugation matrix C by C = Tj X o-2 = -- Cr\, with C = Cr\ = -- 1 X
ia2. The property of C is such that
-- (TMN)T, CYMNC~l = -- (YMN)T. Then crM are antisymmetric matrices and
group theoretically corresponding to (4 X 4)antisymmetric = 6 for SU(2,2)
representations.
PHYSICAL REVIEW D 74, 085019 (2006) We use the notation 0 = TMdM, X --
TMXM, and similarly
j--rMdM, x--rMxM.
The general solution of X"& = 0 is
VQQ = 8(X2)Xy(X),
(2.42)
where we have used XX = X2 and X28(X2) = 0. Here ^a(X) (without the haPand
with a instead of a) is labeled like the other chiral spinor of SO(cf, 2).
Next we examine the second equation $"& = 0 which can be put into several
forms as follows:
0 = ffi = ^[§(X2)X^] = <5(X2)[#f(X^) -- 2^P] (2.43)
9 r .-. ( d\ 1
o\XL) -- Xp^V + 2X % d + -- W
2
(2.44)
= six2)] --
2z'
1 ~MN d d
1 LMN + -- yV + X " d + -- yV
2
(2.45)
In the first line we have used dM8(X2) = 2XM8'(X2) and 2<5'(X2)XX =
28'(X2)X2 = -28(X2) to obtain an overall delta function. In the second
line we changed the order $X = ~X0 + 2X % d + d + 2. In the third line we
have used the definition TMN = ^(TMTN -- TNTM), while i(XMdN -- XNdM) is
the SO(d, 2) orbital angulat mn
1 AM0N
momentum. The structure ±TMNLMN ~X0 + X % d can be regarded as the analog
of spin-orbit coupling, where the SO(d, 2) spin angular momentum is given
by SMN = ArM7V. As we will see in the case of the scalar in the appendix,
the appearance of LMN is naturally expected from the point of view of
Sp(2, R) symmetry.
By applying X on Eq. (2.44) we can derive a homogeneity condition for ^ as
follows:
0
C-/Tr9\ TATA^ T .7., ^ T
= 8(X ) -- XXp^V + 2X X % d + -- T
2
9 d + 2 -28(XZ) X % d +------- (X^),
(2.46)
(2.47)
where we have used XXS(X2) = X28(X2) = 0. According to the last line, (X^)
is homogeneous of degree -- ^-^. Then from Eq. (2.43) we learn that ^ =
j^(X^) is homogeneous of degree -- | since the right- hand side has this
homogeneity degree. This requires the second terms in Eqs. (2.44) and
(2.45) to vanish, hence the two terms of Eqs. (2.44) and (245) vanish
independently. Therefore we learn that the OSp(l|2) gauge invariance
conditions of Eq. (2.41) require ^ to satisfy the following equations:
X % d + -- ^a
^- x2=o
M
0,
[xm#=,
x2=o
1 ,,, d
0 or -- lM"LMN + -- t 2i 2
0.
x2=0
(2.48)
The derivatives must be taken before the condition X2 = 0 is imposed.
This analysis is performed independently for the two spinors tya =
8(X2)(XiPR)a and^^ = 8(X2)(X^L)a. So, the free field equations for the two
chiral fermions of SO(d, 2) are of the form of Eq. (2.48), except for
interchanging TM <-» fM to describe the L, R sectors.
The first equation in (2.48) is the kinematical equation and the second is
the dynamical equation. Both will be derived from an action in the next
subsection, and consistent interactions will be introduced after that.
These 2T physics equations for chiral fermions in d + 2 dimensions [12]
coincide with the equations proposed by Dirac [15] for d + 2 = 6. So it
has been known for a long time that they reproduce the massless Dirac
equation in d = 4. More precisely, from ^L(X) we reproduce the massless
Weyl equation for a left handed SL(2, C) spinor and from ^R(X) we
reproduce the massless Weyl equation for a right handed SL(2, C) spinor.
We will return to this detail explicitly when we derive the standard model
in 3 + 1 dimensions from the one in 4 + 2 dimensions.
For the application to the standard model, it is essential to classify the
left/right handed chiral fermions differently under SU(3) X SU(2) X U(l).
This is why we were careful in our analysis above to distinguish between
left/right spinors directly in d + 2 dimensions.
One may ask how do we do away with the extra spinor components in going
from the 4-component SU(2,2) spinors ^ , ^R in six dimensions, to the
2-component SL(2, C) chiral spinors in four dimensions. The explanation is
rooted in the 2T gauge symmetry extended to fermions. The following
fermionic-gauge symmetry was first noted in [12]:
b2>
8^L = X2l,\ + X$, 8^R = X2lR + ttk
SO(d, 2) fermionic spinors. (2.49)
L,R
/ --
6i,:
The easiest way to notice this symmetry is through Eq. (2.42), where it is
evident that the transformations above leave the physical states ^ =
<5(X2)X^R or ^ = 8(X2)X^L invariant. One may follow the symmetry down to
the Eqs. (2.48) written in terms of ^ (rather than "&) where it is
sufficient to have homogeneous £L,R. This symmetry will be an automatic
fundamental symmetry in the fermion action proposed below.
Note that in the discussion above the gauge parameters l['2 are
independent. When we introduce all the fermions in the standard model,
each chiral fermion will naturally have its own independent local
fermionic symmetry pa-
PHYSICAL REVIEW D 74, 085019 (2006)
rameters l[f with their SU(3) X SU(2) X U(l) charges identical to those of
the corresponding fermion. So the fermionic 2T gauge symmetry will be just
large enough to remove all ferminoic gauge degrees of freedom. This will
be a symmetry that is elegantly built in the structure of the action for
fermions.
The fermionic 2T gauge symmetry can eliminate only half of the components
in each ^L-R because, despite appearances, X£f, XCk contain only half as
many independent degrees of freedom as ^L-R. This fact, that will become
more evident below by constructing X,X as explicit matrices, is due to the
condition X2 = 0. So, half of the components in each ^L-R(X) can be gauge
fixed, while their dependence on the d + 2 dimensions XM can be reduced to
(d -- 1) + 1 dimensions x^ by solving the kinematic conditions in (2.48).
This leaves the correct physical degrees of freedom for chiral fermions in
d dimensions.
Thus, in the application to the standard model in the following sections,
where we use a specific embedding of d dimensions as given in Eq. (4.1),
four component SO(4,2) spinors ^L-R(X) in six dimensions will be
equivalent to two component chiral fermions L'R(x) in four dimensions
[SL(2, C) doublets] after eliminating the gauge components of each quark
and lepton via fermionic-gauge fixing.
We emphasize that in 2T physics this reduction to d dimensions is
understood as one of the possible gauge choices that provides a
holographic image of the d + 2-dimensional theory in the sense of Fig. 1.
2. Free field action for fermions
We now propose the following action whose minimization gives the fermion
Eqs. (2.48)
1
5o(t) =--i
2
(dd+2X)8(X2)(^t^ + * i^X*)-
(2.50)
This action is manifestly Hermitian. It is possible to add to Sq(^) the
term j f(dd+2X)S(X2)^fr(-- d % X + X % d)^P with an arbitrary real
coefficient a. However, as remarked after Eq. (2.48) above, and after Eq.
(2.57) below, this term does not change any part of the discussion, hence
we will suppress a in this paper for simplicity, but it should be kept in
mind in future investigations. The action in Eq. (2.50) can be rewritten
only in terms of LMN as in Eq. (2.45), so we can easily argue that this
action is invariant under Sp(2, R). Here the contravariant ^" is defined
as ^ = ty^rj by using the SU(2,2) metric r\ given in footnote 8,
and we have used the notation ^ 0 = dM"^TM. Upon general variation this
action gives
<5So(^) = i l {dd+2X)8"^r\-- <5(X2)Xi^
2
3M[8(X2)TMXH
+ He. (2.51)
1
2
= i / (dd+2X)8ty\8(X2)\ -(X0 ~ 0X)ty + ty \\ + H.c.
2
(2.52)
= i (d X)8(X )Sty Jt$ty -- X % d + -- ty + H.c.
.j d
X$ty -- X % d + -- ty
2
(2.53)
In the first line, an integration by parts has been performed to collect
the coefficient of Sty, while H.c. stands for the Hermitian conjugate term
that contains Sty. The last term in the second line comes from taking the
derivative of the delta function -- j(dMS(X2))TM]tty = -- ^ S1
(X2)]t]tty = -X2S'(X2)ty = S(X2)ty, thus obtaining an overall factor of
S(X2). To derive the third line we interchange orders 0X = ~X0 + 2X % d +
d + 2.
Next we point out the fermionic 2T gauge symmetry when we substitute
S^ty = X2Ci + l2t instead of the general variation Sty
8^S0(ty) = -i f(dd+2X)8(X2)(X2i1 + <f2X)
X X0ty -- X % d + -- ty + H.c.
2
(2.54)
(2.55)
i /
i (dd+2X)dM[XMg2Xty8(X2)] + He. = 0.
(2.56)
In the first line the 8(X2)X2li terms and the <5(X2)^2XX^P = S(X2)X2£2$ty
vanish trivially, while the remaining term S(X2)l2]t(X ' d + f)ty can be
written as the total divergence in the second line provided £2 satisfies
the homogeneity condition
/ -- d + 2 -(X- d£2 +-------£2
0.
x2=o
(2.57)
If the term proportional to a noted following Eq. (2.50) had been
included, the same arguments would still hold for any a. Note that the
symmetry holds off shell without requiring ty or li to be restricted in
any way. By applying the operator (X % d + |) on both sides of Eq. (2.54)
and using the homogeneity of l2 we note that (X % d + f)Sgty = X2(X % d
+ ^y)I,\ is proportional to X2. This implies that, while the off shell ty
in general has only pure gauge degrees of freedom in its parts
proportional to X2 (symmetry X2^), only homogeneous parts of the off shell
ty with homogeneity degree of -- i contain gauge degrees of freedom of
the type X&.
We now return to the general Sty and require the varia-tional principle
SS0(ty) = 0. The resulting equation of motion S(X2)[]t$ty -- (X % d +
f)ty] = 0 is identical to Eq. (2.44), which is also equivalent to Eqs.
(2.43) and (2.45) by the same arguments supplied when studying those
equations. As we have argued before, the two terms must vanish separately,
and therefore the action principle gives the correct equations of motion
derived in Eq. (2.48). Since the on shell free field is already
homogeneous, we can eliminate half of its degrees of freedom by fixing the
fermionic 2T gauge symmetry £2. Hence our action reproduces the correct
degrees of freedom, and the field equations required by the 2T physics
constraints that follow from OSp(l|2) gauge symmetry, namely, $ty = 0,
]tty = 0 with ty(X) = S(X2)Xty(X).
3. Interactions for fermions
The interaction terms for fermions must respect the fermionic 2T gauge
symmetry as well as the SO(d, 2) invariance. Interaction with gauge fields
constructed by replacing ordinary derivatives dM with covariant
derivatives DM = dM + AM are automatically consistent with these
symmetries. The AM are associated with the adjoint representation of an
internal gauge symmetry, and the fermions are in some representation of
the gauge group, as usual in non-Abelian gauge theories.
In the application to the standard model, the AM are the SU(3) X SU(2) X
U(l) gauge fields, and the SU(2, 2) = SO(4, 2) fermions tyL'R are
classified under this internal group exactly like left/right handed quarks
and leptons are classified in the usual standard model.
The analysis in Eqs. (2.50), (2.51), (2.52), (2.53), (2.54), (2.55),
(2.56), and (2.57) goes through unchanged in every step except for
replacing the covariant derivative DM in all steps instead of dM. Note
that this modifies the homogeneity (or kinematic) conditions on the
fermionic-gauge parameters in Eq. (2.55) to include the gauge field (X %
D + ^-^)l2 = 0, since the ^12 are classified just like ty under the gauge
group. However, as seen below, it is possible to choose a gauge X % A = 0
in which X % D = X % d so that in this gauge the kinematic equations
revert to the ordinary homogeneity condition.
We now consider interactions with scalars. Because of the fermionic, as
well as the SO(d, 2) symmetry, the interactions must have the fermionic
structures tyL^tyR, tyR]CtyL coupled to the scalars and multiplied with
the delta function
Sint(A> scalars) = f(dd+2X)S(X2)[tyLXtyR X (scalars)
+ He.]. (2.58)
The group theoretical explanation is as follows. First, these structures
are SO(d, 2) invariant because the product of a left spinor and a right
spinor makes an invariant when dotted with the vector XM. For example, for
SO(4, 2) = SU(2, 2) which is our main interest, the left spinor ^L is a 4
of SU(2,2), and the right spinor ^R is a 4*, while the left spinor "fyL is
again a 4*. The SU(2,2) product 4* X 4* antisymmetrized is exactly the
SO(4,2) vector in six dimensions. So the structure ^LTM^R must be dotted
with the vector XM to make the SO(d, 2) invariant ^LX^R. Second, under the
fermionic-gauge transformations of Eq. (2.49) the variation of ^>LX^R
produces a factor of X2, as in
+ yLX(X2£R + tCh) (2.59) = x\{l\pvR + ^LtC\) + ilR^R + ^L^)l (2.60)
When multiplied with the delta function this vanishes X28(X2) = 0.
Therefore, the interaction S-mt(/, scalars) is invariant under the
fermionic 2T gauge symmetry.
Now we turn to the internal gauge group to see how to couple the scalars
in Eq. (2.58). Let us assume that ^X^ is not invariant under the internal
gauge group. For example, if we classify tyL'R under SU(3) X SU(2) X U(l)
as in the standard model, then ^|>LX^R is invariant under SU(3) but
transforms as a doublet under SU(2) and has a nontrivial phase under U(1).
Therefore, to make an invariant we must introduce a complex scalar H that
has just the opposite classification under the gauge group to be able to
construct a gauge invariant in the form ^LX^RH, exactly like the Higgs in
the standard model. We will see that, in the reduction from d + 2 to d
dimensions by solving the kinematic equations, this term will become the
familiar Yukawa coupling ipL Rh in d dimensions and will lead to the
correct Yukawa couplings of the Higgs field in the standard model in d = 4
dimensions.
There is one more point to consider to obtain nontrivial interactions with
scalars, if d i= 4. This is not immediately evident from the action but
becomes clear when we examine the equations of motion. The dynamical
fermion equation of motion including interaction with scalars has the
formX(i^L + H^R + " " " ) = 0. In addition, there are the kinematic
equations of motion which demand definite homogeneity for each field. We
learned above that the on shell tyL'R have homogeneity degree -- d/2 and
the scalar H has homogeneity degree -- {d -- 2)/2 [assuming the W term
analogous to Eq. (2.39) is zero for the H field]. We see that, unless d =
4, the homogeneity of ^P which is -- (d + 2)/2 does not match the
homogeneity of the term H^ which is -- d + 1. Then each term must vanish
separately, and the interaction becomes trivial. To avoid this we need to
multiply H^ with a factor that establishes the same homogeneity for both
terms. Thus, in addition to H we consider another scalar O that is neutral
under the SU(3) X SU(2) X U(l) gauge group. Assuming that O has
homogeneity degree -{d- 2)/2, we see that #^o-(rf-4)/(rf-2) has degree --
(d + 2)/2, which is the same degree as that of $^L. Hence the action that
produces nontrivial interactions between fermions and scalars has the form
S-mt(/, H, O) = gH Udd+2X)8(X2)(^Lt^RH
+ ~<frR]ttyLH)(£>-(d-4:'>/(d-2\ (2.61)
with a dimensionless coupling gH in any dimension. The exponent in
0~(rf~4V(rf~2) would be shifted to another value if W(O) ^ 0 in Eq.
(2.39), since in that case the homogeneity of O would be different, but
still some power of O will be needed as long as d i= 4. With this form of
interaction we also find that the interactions in the dynamical equations
of motion for H and O are also consistent with their homogeneity. We see
here that the role of O is similar to the role of a dilaton. In d = 4 the
O disappears and the interaction reduces to a renormalizable Yukawa
interaction.
C. Gauge field
The gauge field equations of motion can be obtained from two different
approaches in 2T physics. We will not provide the derivation here because
it would require too much space and instead refer to past work. The first
approach uses OSp(2|2) as the gauge group on the worldline that acts
linearly on the phase space superquartet {pf, sf, XM, PM) which contains
two worldline fermions ipfir) [2]. In d + 2 = 6-dimensional flat
spacetime, the physical states in the covariant quantization of this
theory are precisely the gauge bosons that we need in d + 2 = 6 dimensions
[12]. The second approach is to consider a spinless particle moving in
background fields in any dimension d + 2, and subject to the Sp(2, R)
gauge symmetry. In this case, phase space contains only (XM, PM) but the
action of Sp(2, R) is nonlinear in a way that depends on the background
fields (f>M^M^'"M*(X) of all integer spins s = 0,1,2, 3, " " " . The
background fields include gauge fields AM(X). The requirement of the
consistent closure of the Sp(2, R) Lie algebra generates kinematical
equations fo
There are also other invariant couplings that occur in the symmetric
product of (4* X 4*)s = 10 of SU(2,2). This product takes the form ^T^^^^
and it can be coupled invariantly to the gauge field and Higgs field in
the form ^l^mnkFMNXk^rH consistently with the SU(3) X SU(2) X U(l) gauge
symmetry of the standard model. Furthermore, this term is invariant also
under the fermionic-gauge symmetry after taking into account the property
of the gauge field XMF = 0 given in the next section. So this interaction
is permitted by all the symmetries of the theory. However, this term
cannot be included in the action of the standard model because it leads to
a nonrenormalizable interaction. Instead, it is expected to arise with a
calculable coupling from the quantum effects in the theory and contribute
to the anomalous magnetic moment [28].
all background fields, including the gauge fields [3]. The two approaches
give the same kinematic equations for AM, namely,
XMFMN = 0, where FMN = d^^/v ~ ^n^m ~ ^SaIAm^ Awl
(2.62)
while also demanding X2 = 0. The dynamical equation follows from the
OSp(2|2) approach for 4 + 2 dimensions as given in [12] and can be
extended to any d + 2 dimensions by including a dilatonic factor
02(rf~4)/(rf~2)
(D M(02(rf~4)/(rf~2)'FMN))X2=0 = sources. (2.63)
These equations have been constructed to be covariant under Yang-Mills
transformations for some non-Abelian gauge group G, so DM is the usual
covariant derivative. The dilaton factor 02(rf~4)/(rf~2) is a singlet
under the gauge group G. It disappears in four dimensions but is
nontrivial in dimensions other than d = 4. Its role is to provide
consistency with the kinematical conditions for all the fields in the
theory. Recall that the kinematical equations follow from gauge invariance
under the Sp(2, R) generator (X % P + P % X). The exponent 2^5f is
determined by homogeneity which comes from this condition, with a
reasoning similar to the one that led to the dilaton factor in Eq. (2.61).
This exponent would be shifted to another value if W(O) ^ 0 in Eq. (2.39),
since in that case the homogeneity degree of the dilaton O would be
different, but the exponent still vanishes if d = 4.
In the rest of this section we discuss the new 2T gauge symmetry beyond
the usual Yang-Mills gauge symmetry G. To do so we will first use G to
choose some gauges for the Yang-Mills field and then discuss the 2T gauge
symmetry in the fixed Yang-Mills gauge. We could discuss the 2T gauge
symmetry of the equations of motion without choosing any Yang-Mills gauge,
as will be done for the action in the next subsection. However, here we
want to take the opportunity to point out some useful gauge choices.
Using the Yang-Mills type gauge invariance we may impose various gauge
conditions. In the gauge XMAM = 0, the kinematic constraint (2.62) reduces
to a homogeneity constraint as follows:
For general d + 2, the quantum spectrum of the OSp(2|2) theory gives the
gauge fields AM ...M for a p-brane with p = (d -- 4)/2 [12]. Thus, for a
gauge field AM, we can use the OSp(2|2) setup only for d = 4. The
background field method does not have this limitation and applies to any
d. Thus, after obtaining the appropriate equations for gauge fields, we
have managed to extend the gauge field equations to any dimension d by
introducing the dilaton factor as in Eq. (2.63).
PHYSICAL REVIEW D 74, 085019 (2006) X % A = 0: XmFmn = (X % d + l)An =
0. (2.64)
Note that the homogeneity degree of Am is -- 1, which is the same
homogeneity as the derivative dM, consistent with the covariant derivative
Dm = dM + Am, in all dimensions. In this gauge the equations above, taken
for d = 4, agree with Dirac s equations [15] as generalized to non-Abelian
gauge fields [18], and so it has been known for a long time that after
solving the kinematic equation, the dynamical equation reproduces the
correct equations of motion for non-Abelian gauge fields in 3 + 1
dimensions. We will return to this detail when we discuss the derivation
of the standard model in 3 + 1 dimensions from the one in 4 + 2
dimensions.
Now we turn to the new 2T gauge symmetry with the transformation law
X A = V2/7 <f) [2(^~ 4)/(J-- 2)]
with [(X % D + d -- l)aN -- XnD % a]X2=0 = 0, (2.65) and X
% a = 0.
The local parameters aM are Lie algebra valued and must be in the adjoint
representation of the Yang-Mills gauge group G to be consistent with the
Yang-Mills classification of Am. We examine the dynamical equation to
verify the 2T gauge symmetry for on shell fields Am as follows:
8 a{Dm^ Fmn)}x2=o (2.66)
__ f__-(fj2(J-- 4)/{d-- 2)[V s \ \ pmn~l
+ Dm(<&2<-d~4:^<-d~2",D[m8aAn^)}X2=0 (2.67)
= {-iX2[aM,FMN]
+ DM(^d-4'>^d-2'>D[M(X2aN]^-2^-4^d-2'>))}X2=^
(2.68)
= 2{DM[(XMaN--XNaM)] + XM(DMaN--DNaM) + (d -- 4)a7V}X2=0
(2.69)
= 4{(d -- l)aN + X % DaN -- XND % a}xi=() = 0. (2.70)
In this computation all terms proportional to X2 have been dropped after
the derivatives are evaluated. In going from the third line to the fourth
we have used X % a = 0, and [X % dO + ^^O]X2=0 = 0 as in Eq. (2.35) to
obtain the term proportional to d -- 4. This shows that the dilaton
factor is required in the dynamical equation of the gauge field Eq. (2.63)
in order to have the 2T gauge symmetry in all dimensions d. Finally the
last line vanishes due to the property of the gauge parameter aM given in
Eq. (2.65).
The new gauge transformation says that the part of AM proportional to X2
contains gauge degrees of freedom. If we first identify different parts of
the gauge field as
AM(X) = A°M(X) + X2AM(X)
(2.71)
such that A°M(X) = [AM(X)]X2=0, then the remainder AM(X) can be completely
removed from the equations of motion by using this gauge symmetry,
provided the on shell AM satisfies the same conditions as aM as given in
Eq. (2.65).
Note that the Yang-Mills field strength FMN is not invariant under the new
gauge symmetry. But the equation of motion is invariant as seen above, and
also the action will be shown to be gauge invariant. The noninvar-iance of
FMN is welcome because this is how the d + 2-dimensional theory will
contain the same physical information that resides in the field strength F
in (d -- 1) + 1 dimensions.
1. Non-Abelian gauge field action
We now propose the action principle that gives both the kinematical and
dynamical equations of motion in Eqs. (2.62) and (2.63). In the following
we will assume that the physical gauge for the 2T gauge symmetry discussed
in the previous section is already chosen, and we will build the action
starting directly with the physical component A°M(X). By comparison to the
scalar action, this is analogous to building the action directly for O0,
skipping the fully gauge invariant treatment for AM(X) that we gave for
both the scalar and the fermions. Thus everywhere we write AM below should
be understood as being the gauge invariant part A^iX) = \_AM(X)'\x2=o.
The consistent 2T physics Yang-Mills type action in any dimension is
11
S{A) =---
I (Ad+7. Y'>liS('y'2\(f)2(J-- A)/{d-- 2) Tyf f? l?MN\
(2.72)
The dilaton factor <&^-A)/(d-2) is
necessary for the consistency of homogeneous terms in the equations of
motion. Evidently, this factor disappears for d = 4, which is the case of
interest for the application to the standard model. The gauge coupling gA
as defined in Eq. (2.62) is dimen-sionless in any dimension.
The general variation with respect to the gauge field gives
11It is also possible to construct a fully Sp(2, R) invariant action for
gauge fields. This is given in the second part of the appendix. The point
is that every derivative can appear in the form LMN to display fully the
invariance under the underlying Sp(2, R). The physical sector of either
treatment is identical.
SS(A)=~K w*2"-4*-2'
X Tr(FMNDM(8AN)) (2.73)
= / (dd+2X)Tr{SANDM[<$>2(d-4V(d-yS(X2)FMN]}
(2.74)
= / (dd+2X) Tx{8An[_8{X2)Dm{^2(d-^/(d-2^Fmn)
+ 2<$>2(d-4V(d-VS'(X2)XmFmn]}. (2.75)
In d i= 4, there is also a contribution to the equations of motion of the
dilaton O, through the variation <50 which is not shown. The equations
of motion that follow from this action include both the kinematical and
dynamical equations since the coefficients of S(X2), S'(X2) must vanish
separately. There are however subtleties in the delta functions that need
to be taken into account as in footnote 5. For this reason the A(X) that
appears in this action is already gauge fixed Am(X) = A^iX), excluding the
remainder AM(X), as emphasized in the beginning of this section. After
taking this point into consideration, we see that this action yields
precisely the correct equations given in Eqs. (2.62) and (2.63), so we
have the correct action principle for the physical sector Am(X) = A^iX).
Of course, S(A) has been built already to be gauge invariant under some
Yang-Mills type gauge symmetry group G. For example G = SU(3) X SU(2) X
U(l) in the application to the standard model.
Next we show that we can add to A°M(X) a remainder of the form Am(X) =
A^iX) + X2Am(X) without changing the physics, provided AM(X) satisfies the
same equations that aM satisfies as given in Eq. (2.65). For this we first
assume that the action (2.72) is already written for the more general A
that includes the special AM. We will then show that this action has the
2T gauge symmetry in which AM(X) can be changed by arbitrary amounts aM.
Therefore one can choose the gauge AM = 0 if so desired,
Thus, consider the transformation 8aAm of Eq. (2.65), and insert it in Eq.
(2.73) to show there is a gauge symmetry 8aS(A) = 0 as follows:
vw)=K2x,Tr{*^4,/Mx2
X aN[_8{X2)Dm{<S>2(d-AV(d-VFmn)
+ 2<$>2(d-4^d-VS'(X2)XmFmn]} (2.76)
= -2 \{dd+2X)8{X2)i:x{aN{X % d + l)An
-- aNDN(X % A)} (2.77)
= -2 (dd+2X)S(X2)Tr{[(X % d + 1)(A " a)
-- DN(X % AaN) -- ANX % daN + (X " A)(D % a)]}
(2.78)
= -2 (dd+2X)S(X2) Tr{(X % d + d)(A % a)
- dN(X % AaN) + AN[-(X % D + d- \)aN
+ XND % a]} (2.79)
= -- 2 / (dd+2X) Tr{dN[(XNA % a -- X % AaN)8(X2)] + ANDM[(XNaM --
XMaN)S(X2)]} = 0. (2.80)
The steps in this calculation are explained as follows. In Eq. (2.76) we
use X2S(X2) = 0 to drop the first term, and then use X2S'(X2) = -- SiX2)
and write out XMFMN in the form XMFMN = (X % d + l)AN -- DN(X % A) to
obtain Eq. (2.77). The form in Eq. (2.78) is equivalent to Eq. (2.77)
after evaluating the derivatives. To get to Eq. (2.79) we added and
subtracted the terms proportional to d and moved the non-Abelian term in
the covariant derivative from the first line to the second line. One can
show that this takes the form of Eq. (2.80). Indeed after evaluating the
derivatives in Eq. (2.80) we get back Eq. (2.79). Then we note that the
expression DM[(XNaM -- XMaN)8(X2)] vanishes by using the conditions on
the gauge parameters aM given in Eq. (2.65). Finally the total divergence
can be dropped.
We see that as long as AM satisfies the same equation as aM this gauge
symmetry can gauge fix it to zero. Thus the physics is the same for any
remainder X2A M of this type. We will see below that, in a special
Yang-Mills gauge, the class of allowed remainders AM(X) are those that are
homogeneous of degree -- 3.
The Yang-Mills gauge symmetry combined with the new 2T gauge symmetry
discussed in this section are just sufficient to reduce the degrees of
freedom in AM(X) to be <5aO = X2A, SAHl = X2Al, 8bB$, 8bBHi as in Eq.
(2.11)
physical and ghost free. The physical degrees of freedom will then agree
with the degrees of freedom of the gauge field A^(x) in (d -- 1) + 1
dimensions. This reduction will be discussed, in particular, for the
standard model in 3 + 1 dimensions in the following sections.
We emphasize that the Yang-Mills field strength FMN is not covariant under
the 2T gauge symmetry transformation 8aAM as seen in the computations
above. Therefore, using the new gauge symmetry it is possible to gauge fix
some of the components of FMN at will. We will use this freedom in the
reduction from 4 + 2 to 3 + 1 dimensions to show that only the 3 +
1-dimensional components F^v{x) survive as the physical field strengths.
III. THE STANDARD MODEL IN 4 + 2 DIMENSIONS
In the previous section we have constructed the action principle for field
theories in the framework of 2T physics in any d + 2 dimensions. The
theory contains scalars i/!,0, left/right handed chiral fermions ^La, ^f
and gauge bosons ArM classified according to any gauge group, and also can
be extended to include gravity in any d + 2 dimensions. Among the scalars
we distinguish one of them as the dilaton O. Although the dilaton factors
in Eqs. (2.61) and (2.72) disappear for d = 4, the dilaton may still
couple to the other scalars Hl as in Eq. (2.38) even if d = 4, so we keep
the dilaton as one of the fields in the theory.
The 2T gauge symmetry was derived in a tortuous way by starting from 2T
physics equations of motion based on Sp(2, R) local symmetry and its
extensions on the world-line. However, it is possible to reverse the
reasoning and suggest the 2T gauge symmetry directly in field theory as
one of the principles for building an action. This would then lead to the
action suggested above in a unique way. This is the point of view we take
in this section.
Thus, in addition to the field theoretic guiding principles for
constructing the standard model in four dimensions, we add a new one,
namely, the 2T gauge symmetry given by
/ spans all other scalar fields,
8^La
X2txa + t&,
x2^ + U\
~L a, p span all fermions,
(3.1)
(3.2)
_ p{ rt (T) \^\^ v/ ^ ^)
similar to Eq. (2.11)
spans all gauge bosons.
(3.3)
12The equations of motion for the gravitational field in 2T physics is
derived in [12]. The 2T action that generates these equations is
constructed using the methods of the present paper. This will be given in
a separate paper.
There is a separate 2T gauge symmetry parameter for each field. Thus,
degrees of freedom can be removed from every field in d + 2 dimensions,
such as to make it equivalent to a field in (d -- 1) + 1 dimensions.
Under the requirement of the 2T gauge symmetry the Lagrangian must include
an overall S(X2) or S'(X2) factor, and furthermore it acquires the form of
the actions given in the previous section. Then the theory must be
constructed in d + 2 dimensions. The action will not have translation
symmetry in d + 2 dimensions since XM appears explicitly through the delta
function SiX2) and in the fermion terms, but the action will have SO(d, 2)
symmetry. When the SO(d, 2) is interpreted as the conformal symmetry in (d
-- 1) + 1 dimensions, which is the case in one of the embed-dings of (d
-- 1) + 1 in d + 2, then Poincare invariance, that includes translation
invariance and Lorentz invariance in (d -- 1) + 1 dimensions, emerges as
part of conformal symmetry. Thus, we take the point of view that the added
2T gauge symmetry principle requires the two-time structure in field
theory, and this is fully consistent with everything we know in (d -- 1)
+ 1 dimensions.
In addition to the 2T gauge symmetry there is of course the principles of
Yang-Mills gauge symmetry and renor-malizability requirements for d = 4.
The Yang-Mills gauge symmetry is straightforward as discussed in the
previous section. By renormalizability, in the present paper, we mean that
the emergent 3 + 1-dimensional field theory should be renormalizable. This
amounts to requiring that the emergent theory in four spacetime
dimensions, at the classical level, should not contain any terms of
dimension larger than 4. In turn, this becomes a principle for restricting
the types of terms that can be included in the classical 2T physics field
theory in 4 + 2 dimensions. For example we cannot include high powers of
fields in the classical 2T physics action. Eventually, when we develop the
techniques of computation with the quantum theory directly in 4 + 2
dimensions, we need to replace the requirement of renormalizability to
mean the same directly in 4 + 2 dimensions at the quantum level.
Given the principles stated above, we now construct the standard model in
4 + 2 dimensions. The internal Yang-Mills group structure is identical to
the usual standard model, but the spacetime structure is different, thus
all the fields are six-dimensional fields instead of four-dimensional
fields. The six-dimensional structure will impose certain restrictions on
the emergent standard model in
13The last one, SbBAr , is similar to Eq. (2.11). This is the gauge
symmetry required in order to allow arbitrary remainders AM for the gauge
field AM = AQM + X2AM. In the presence of this gauge symmetry there would
be no conditions on the parameters aM(X). We bypassed this more general
setting that would include the additional field BAr and considered the
gauge fixed form of the action after the extra field BAr is eliminated by
a gauge choice. In this _gauge only to a specialized subset of aM and
corresponding AM play a role as discussed in the text. This was sufficient
for our purposes here.
3 + 1 dimensions as outlined in the abstract. To be completely explicit we
write out the details below.
The Yang-Mills gauge group is G = SU(3) X SU(2) X U(l), so we have the
corresponding Yang-Mills fields AM = (GM, WM, BM), namely, gluons GM and
electroweak gauge bosons (WM, BM). These are in the usual adjoint
representations denoted by the dimensions for SU(3) X SU(2) and by the
charge for U(1) written as a subscript, as follows:
vectors of SO (4, 2):
GM = (8, l)o, WM = (1, 3)o, BM
(1, l)o. (3.4)
The SO(4,2) scalar fields include the dilaton O which is neutral under
SU(3) X SU(2) X U(l) and the Higgs doublet H classified as usual
O
scalars of SO (4,2): (l,l)o, H' = =(l,2)1/2.
H+
H°
(3.5)
1/2
Of course more scalars can be included, but for now we will assume a
minimal number as above. The fermionic matter fields are the three
generations of quarks and leptons ^La(X), ^Rfi(X) taken as the left/right
quartet spinors of SU(2, 2) = SO(4, 2), and in the usual representations
of the Yang-Mills gauge group G, as follows:
4 of SU (2, 2):
uL dL
1/6
1/6
pL
-(1/2)
CL SL
/ULL _
4-L
bL
(1/6)
(3.6)
-(1/2)
TL
-(1/2)
4* of SU (2, 2):
(uR)2/3 (cR)2/3 (tR)2/3
(dR)-{m (^)o
(**)_(1/3)(**) i^R)-l' (^)-l
(1/3)
(3.7)
We have included the right handed neutrinos assuming these particles
develop Dirac- or Majorana-type masses. To describe the fermions in a more
compact notation we further introduce the following definitions. The three
left handed quark and lepton doublets are defined as (2L0i/6, (LLj)-(i/2),
respectively, with / = 1, 2, 3 denoting the three families. Similarly, we
define the family labeling j = 1,2, 3 for the right handed quarks and
leptons as urj =
\ U C t )l /Q Cl J ^= id
S U ) f1 /Q^ 0 J ^=
(eR, jAR, tr)_1, and vrj = (vR, pR, pr)q. All quarks are triplets and all
leptons are singlets under color SU(3). The left handed quarks and leptons
QLi, LLi are doublets and the right handed quarks and lepton urj, dRj,
eRi, vrj are singlets under SU(2) as listed above. Furthermore, each field
is charged under U(1) with the charges marked as subscripts above. The
electric charge of each field is then given by Q = /3 + Y, where Y is the
U(1) charge and /3 is the third generator of SU(2) represented as \a3 on
all doublets.
The covariant derivatives for each field is then straightforward, as usual
in gauge theories. Then the action for the standard model in 4 + 2
dimensions is given by
S(A, tyL-R, H,<$>) = Z I (d6X)8(X2)L(A, tyL-R, H, O),
(3.8)
L(A, tyL'R, H, O) = L(A) + L(A, ^L-R) + L(^L-R, H)
+ L(A, O, H), (3.9)
where Z is an overall normalization factor that will be chosen below, and
A = (G, W, B) is a shorthand notation for the gauge fields. Thus, after
peeling off the overall volume element Z f(d6X)S(X2), the various parts of
the Lagrangian are given as follows. The factor of Z will be fixed later
to normalize the emergent volume element in 3 + 1 dimensions.
To get to the physics as simply as possible, we assume the 2T gauge
symmetry has already been gauge fixed to simplify the action as discussed
in the previous sections. This means that the simplified actions given
below for the standard model contain fields whose remainders (i.e. parts
proportional to X2) are not the most general. For example, all remainders
can be fixed to be zero. More generally, they are gauge freedoms that have
appropriate homogeneity properties to be consistently removable by the
remaining 2T gauge symmetry. The latter version is not only more general,
but can also be simpler for understanding the reduction from 4 + 2 to 3 +
1 dimensions discussed in the next section.
The Lagrangian for the gauge bosons L(A) is then
L(A) = -lTr3(GMNGMN)
In dMN 4DMND
\Tv2(WMNWMN)
(3.10)
Note that there is no dilaton factor in the Yang-Mills action since d = 4.
Each field strength is of the form AMN = dMAN -- dNAM -- j&aIAm, Ajy],
for A = (G, W, B) with corresponding gauge groups SU(3) X SU(2) X U(l),
and with different dimensionless coupling constants g3, g2, g\ appearing
instead of the gA. Of course for the Abelian BMN there is no quadratic
term proportional to the U(1) coupling
§i.
The Lagrangian L(A, O, H) is of the form Eq. (2.37)
L(A, O, H) = -Od20 + -(H^D2H + (D2H)^H)
2 2
-- V(0, H)
where the covariant derivative DMH is given by
/ T gi \
DMH = dM -- ig2WM % -- -- i -- BM H. (3.12)
2 2
The potential energy can be written in the following gauge invariant form:
A , , , ,
y(<P, H) = -- (H'H -- a <P ) + Vi^) (3.13)
4
where the couplings A, a are dimensionless. Recall that quadratic mass
terms for H are not allowed in the potential by the consistency of the
kinematic equations of motion that require homogeneity as discussed in Eq.
(2.38). For this reason we needed to introduce a coupling to the dilaton
field O. Now y(0, H) has a nontrivial minimum for the Higgs field H where
the minimum occurs atH^H = a202 o
H
H+
aO
(3.14)
This breaks the SU(2) X U(l) gauge symmetry down to the electromagnetic
U(1) subgroup. Next we need to discuss V(O) that stabilizes the dilaton O
at some constant expectation value (O) j= 0 that sets the scale for the
weak interactions as (H°) = a(O) = v in the range of 100 GeV. For this we
refer to the later discussion on the topic of mass generation in Sec. VI.
The Lagrangian L(A, tyL'R) for fermions is of the form given in Eq. (2.50)
but otherwise has all the terms in parallel to the standard model in 3 + 1
dimensions as follows:
L(A, "¥L-R) = ^(QL'X0QLl + QL'0XQL')
+ ULL>t0LL> + LL>0tLL>) (3.15)
- {{dR>t0dR> + dR>0XdR>)--\{eRit0eR> + eRi0teR>)
(3.16)
-\(uRi%0uRi + uRJ0tuRj)--{{vRit0vRi + vRJ0tvRj).
(3.17)
The overall signs of the L/R kinetic terms in the 4 + 2 theory are chosen
to insure the correct signs of the time derivative terms in the kinetic
terms of the emerging 3 + 1 theory. Note that TM <-» TM are interchanged
in comparing the left/right sectors. Also, we have replaced the ordinary
derivative dM^L,R by the Yang-Mills covariant derivatives DM as follows,
again in parallel to the usual standard model in 3 + 1 dimensions:
A"
T gi
j
1 T
2 6
(3.11)
(3.18)
A 2gi
DMu i = dM -- ig$G%,-- -- i-----BM u ', (3.19)
2 3
A g\ lR
DMd i = dM -- ig$G%,-- + i -- BM di, (3.20) 2 3
2 3
r T §1 DML ' =
dM -- ig2^M % -- + i -- BM L ',
2 2
(3.21) DMvRi = dMvRi, (3.22)
DMeRJ = (dM + igiBM)eRJ,
(3.23)
where 4r are the 3X3 matrices that represent the generators of SU(3) and |
are the 2X2 Pauli matrices that represent the generators of SU(2). This
part of the Lagrangian is invariant under a global family symmetry group F
that transforms only the fermions indicated as subscripts
F = (7(3)gi X U(3)ur X (7(3)^« X U(3)ll X U(3)eR
X U(3)vr. (3.24)
The Lagrangian S(^L'R, H) for Yukawa couplings is of the form in Eq.
(2.61) but without the dilaton factor since d = 4. This couples the three
families as follows, again in parallel to the usual standard model in 3 +
1 dimensions:
L(^L,R, H) = {g^)ijQLiSuR>Hc + (gl)jiHcuRjXQL'
+ (gd)uQL^dRJH + {gl)nH~dR'iQL'
+ {g^^tv^H0 + (gt)nHcvRiUL<
+ (g^jL^Xe^H + (gbjiHetjXL1
(3.25)
Here
Hc = 1T2H*
-H
-(1/2)
is the SU(2) charge conjugate of H, which transforms as an SU(2) doublet
and has opposite U(1) charge. The dimen-sionless Yukawa couplings (gu)ij,
(gd)ij, (gv)ij, (ge)ij are complex 3X3 constant matrices since this is the
most general permitted by the gauge symmetry SU(3) X SU(2) X U(l). These
couplings break the global family symmetry F of the Lagrangian L(A, ^L-R)
mentioned above. As is well known, by using the freedom of the global
family symmetry F, it is possible to choose a basis for the quarks and
leptons such that gu and ge are real and diagonal, while gd and gv become
Hermitian but non-diagonal. This relates to the Kobayashi-Maskawa matrices
for the quarks and for the neutrinos. The off diagonal entries mix
families so that a separate family number is not conserved. This leads to
the explanation of how the
more massive families decay to the less massive ones, and how neutrino
mixing occurs. Using the symmetry F to its maximum to eliminate phases in
the remaining off diagonal entries, the mixing between quark families
(1,2) can be chosen real, but the mixing between families (1,3) and (2,3)
remain complex. The complex phases in gd and gv violate the discrete CP
symmetry.
Based on the principles of 2T gauge symmetry in field theory, that emerged
from the underlying Sp(2, R) gauge symmetry in the worldline formalism, we
have constructed the standard model in 4 + 2 dimensions. This action
generates consistently both the kinematic and dynamical equations of
motion for every on shell field in the theory. The kinematic equations,
together with the 2T gauge symmetry, are just the necessary ingredients to
make the 4 + 2-dimensional theory equivalent to the 3 + 1-dimensional
standard model. However the 4 + 2 structure imposes some restrictions on
the emergent 3 + 1-dimensional theory that relate to unresolved issues in
the standard model in 3 + 1 dimensions, including the issues of the strong
CP problem and the mass generation mechanism. These are discussed in the
following sections.
IV. THE EMERGENT STANDARD MODEL IN 3 + 1
In this section we demonstrate how the 3 + 1-dimensional standard model
emerges from 4 + 2 dimensions. The new 2T gauge symmetry in field theory
is essential to show that every field in the theory can be gauge fixed so
that it becomes independent of X2, as already assumed in the simplified
gauge fixed form of the Lagrangian given in the previous section. Then by
using the condition X2 = 0 imposed by the delta function we can eliminate
one of the components of XM from every field in the theory. A second
component of XM will also be eliminated from every field by putting every
field partially on shell by satisfying the kinematic equations that follow
from the action. These two conditions are precisely the two Sp(2, R)
generators in the worldline formalism X2 = (X % P + P % X) = 0 that are
solved explicitly in a fixed gauge to obtain a holographic image of the 4
+ 2-dimensional system in 3 + 1 dimensions, as in Fig. 1. We now discuss
how this happens in field theory for the standard model. We emphasize that
the third Sp(2, R) generator is modified by the interactions and will be
left off shell in the discussion below.
We start by choosing a lightcone type basis in 4 + 2 dimensions so that
the flat metric takes the form ds2 = dXMdXNT]MN = -- 2dX+ dX~ +
dX^dX1'rj^v, where rj^y, with pi, v = 0, 1, 2, 3 is the Minkowski metric
and X- = 4= (X0> ± X1') are the lightcone coordinates for the
extra one space and one-time dimensions. Furthermore we choose the
following parametrization which defines the emergent 3 + 1-dimensional
spacetime x^
X+'
K,
X = kA, X^ = kx^,
(4.1)
X'
(4.2)
X X^ \ =____ xm =____
x+" x+!'
This provides one of the many possible embeddings of 3 + 1 dimensions in 4
+ 2 dimensions. Each such embedding corresponds to a Sp(2, R) gauge choice
in the underlying 2T physics worldline theory. The present one corresponds
to the one labeled as the relativistic massless particle in Fig. 1. In
other embeddings, we will obtain a different 3 + 1-dimensional view of the
4 + 2-dimensional theory, as in the examples of Fig. 1.
The fields are parameterized as 0(X) = 0(k, A, xM), and similarly for the
others, where A, xM are homogeneous coordinates which do not change under
rescaling &(tX) = &(tK, A, x^). The kinematic equations will be solved in
this parametrization to reduce the theory from fields in the spacetime XM
to fields in the smaller spacetime x^. After this, there remains to
satisfy the dynamical equations, including interactions, only in terms of
the fields in the smaller spacetime x^. The dynamics in the reduced space
is described by the reduced action which holographically captures all of
the information in the 4 + 2-dimensional theory.
With the parametrization of Eq. (4.1) we get X2 = -- 2X+ X~ + X^X^ = k2(--
2A + x2). So the volume element that appears in the action Eq. (3.8) takes
the form
(d6X)S(X2) = K5di<d4xdAS(K2(2A -- x2)). (4.3)
When A = x2/2 is imposed, the 4 + 2-dimensional flat metric reduces to the
conformal metric in 3 + 1 dimensions ds2 = dXMdXM = K2(dx)2. This is how
the SO(4,2) in 4 + 2 dimensions becomes the conformal symmetry in 3 + 1
dimensions. The nonlinear realization of conformal transformations in 3 +
1 dimensions xM is nothing but the SO(4,2) Lorentz transformations in the
space XM.
Recall that derivatives must be taken before the X2 = 0 is imposed. Let us
now use the chain rule dM = (8mk)X %T" + (c*mA) 4t + (d jfX'O +7T to
compute derivatives as follows:
dK ' m ' dA v m ' OX1
r dM that appear in the fermion equations in 4 + 2 dimensions by using
explicitly the gamma matrix representation given in footnote 8
^MrM = -- r+'x~' -- r~'x+> + r^x^ (4.6)
x^cr^
is/2
i\l2A
(4.7)
and
rMdM = r+ d+i + r s_/ + r^s,
-i^2dA
-- z'v2(/cdK -- AdA -- x^d^)
(4.8)
(4.9)
For TMXM, TMdM we obtain the same structures as above, but replacing a^ by
a^ and vice versa.
A. Reduction of scalars
We now proceed to solve the kinematical equations. We start with the
kinematic equations of the scalars (X % d + ^^)0 = (k-^ + 1)0 = 0. The
Higgs scalar H also satisfies a similar equation but with the covariant
derivative replacing the ordinary derivative. We fix the Yang-Mills gauge
symmetry so that
X % A = 0 for all YM fields A = (G, W, B). (4.10)
In this gauge X % D = X % d; therefore, H satisfies the same kinematic
condition as the singlet O. These homogeneity conditions determine the
kappa dependence fully as an overall facto
O(X) = k :0(x, A), similarly for H.
(4.11)
d Id
___ -- _ ___
dX^ k dx^'
d 1 / d -----------T = K-------
dX+ k 3k
d dX~
1 d K dA'
d d \____ __ y/j,______
dA dx^
(4.4)
Using these we further compute XMdM = k^- and the Laplace operator in 4 +
2 dimensions
m 1d d 2 1 d
d
d dM = -- ~------+ x ---- -- -- ~2k----+ d -- 2 ----
K dx^ dA k 8k dA
(4.5)
1
+ -r(2A-x 2) --
We will also need the structures TMXM, TMXM, TMdM,
Now recall that according to Eq. (3.1) the part of the homogeneous scalar
field proportional to X2 is gauge freedom with respect to the 2T gauge
symmetry. We have already said that we can gauge fix the remainder to
zero, but let us look at the details of how this is done. Thus if we
define <p(x) = 0(x, 0) and write 0(x, A) = <f>{x) + (A -- y) <fi (x, A),
then the remainder </> (x, A) can be gauge fixed at will. It is convenient
to choose the gauge <p(x, A) = 0 which makes the field O(X) independent of
A
0(X) = /c"1 </»(*),
similarly H(X)
K-lh{x).
(4.12)
Then it is simple to compute the 4 + 2-dimensional Laplacian dMdM of Eq.
(4.5) since all derivatives +V vanish
K
1
K
and we obtain14
dMdM&(X) =
1 d2<p(x) k3 dx^dx
(4.14)
It will be argued below that the gauge fields will also get reduced to
four-dimensional fields, so that D2H{X) = \D^D^h will involve only the
four-dimensional component of the gauge field A^(x). In this way the
scalar Lagrangian L(A, O, H) is reduced to the form
1
L(AM(X), QyX), H(X)) = -- L(A (x), <p{x), h(x))
K4
(4.15)
where L(Alx(x),4>(x),h(x)) is purely a four-dimensional Lagrangian that
has the same form as L(AM(X), O(X), H(X)) except for the fact that only
four-dimensional fields and only four-dimensional covariant derivatives
appear. Replacing this in the action we obtain
S(A, O, H) = Z I \k\5di<d4xdA8(K2(2A -- x2))
X -jL(A (x), 4>(x), h(x))
K
(4.16)
If we do not choose the simplifying gauge (f>(x, A) = 0, we still obtain
the same result as follows. In computing dMdM<t>(X) we recall that A = y
is imposed by the delta function after all derivatives are computed. Then
the operator dMdM in Eq. (4.5) simplifies as follows. The last term drops
because of the factor (A -- y) = 0 even after differentiation. The second
term drops because of the form of the homogeneous solution (4.11). The
first term simplifies because derivatives with respect to x^ appear only
in the combination ^r + x^ -^. Then, setting A = x /2 after
differentiation with the derivative operator ^r + x^ -^ gives the same
result as setting A = x2/2 before differentiation and then differentiating
with respect to the total x dependence including the part coming from A =
x2/2
d d s, ,
-----+ x -- 4>(x, A)
ydx'* 3A
A=x2/2
___
3x^
4>
x,
(4.13)
Therefore, we can set <t>(x, A)|A=x2/2 = <f>(x, y) = <p(x) before
differentiation. This is equivalent to dropping the term ^ in the
derivative operato
+ X
Hence 3 3 in 4 + 2 dimensions reduces to the Laplace operator in 3 +
1 dimensions Eq. (4.14), in agreement with the simpler derivation based on
the fixed gauge.
1 As seen in Eq. (4.30) the gauge fixed form of AM includes the nonzero
component A~ =^x'M/tt(x)= -- A+i, while A+ = -- A-i = 0. Therefore, the
covariant derivatives in the extra dimensions D+i take the form D / = d_/
= -3. and D+i = 3+/--iA+, = dK + -Lx'lA!,. These »""«'
M
uDi -- DiDi + D^D. Since
D
a appear in DMDM ^ D , = ±3A vanishes on
the A independent h(x) and Att(x) we obtain the reduction DMDM -- » D^Dp
which then leads to the four-dimensional theory.
r c ~\ c
= \ Z I dKdu8{2\i<\u) I d4xL(A/J/(x), 4>{x), h{x)).
(4.17)
In the last step we have changed integration variable to u = A -- y. We
see that the action has an overall logarithmically divergent factor which
is cancelled by choosing the overall normalization Z in front of the whole
action in Eq. (3.8), so that
Z
I dKdu8(2\i<\u) = 1. (4.18)
The same factor will appear in all the terms of the action in the
reduction from 4 + 2 to 3 + 1 dimensions. The four-dimensional action S(A,
<p, h) = j d4xL(Alx(x), <p(x), h(x)) is translation and Lorentz invariant
and captures all of the information contained in the six-dimensional
action S(A, O, H) without losing any information. In this sense, the
four-dimensional action is a holographic image of the higher-dimensional
one.
B. Reduction of chiral fermions
We start by writing every fermion in the form ilrL-R(X) = ^Pq,r(X) +
X2"^rL'R(X), where ^Pq,r is defined as ^qR(X) = [<irL'R(X)]X2=0 = ^qR(k,x)
which is independent of A. We can gauge fix "^L,R{X) = 0 off shell by
using the X2^ part of the 2T gauge symmetry for fermions of Eq. (3.2). In
this gauge ilrL-R(X) = "^^(k, x) becomes fully independent of A. Next we
use the kinematical equations (X % d + |)^L'R = (kj^ + 2)^l-r = 0. This
determines the kappa dependence fully as an overall factor for all
fermions, and we get the homogeneous form
y$rL,R(X) = k-2xL-R(x).
(4.19)
Since we now have homogeneous fermions, half of the degrees of freedom can
be removed by using the fermionic 2T gauge symmetry of Eq. (3.2) with the
parameters X^2 ", X^ that have the same degree of homogeneity. It is
convenient to choose the lightcone type gauge r+'tyL-R = 0 in the extra
dimensions that requires the two lower components of ^L-R to vanish:
WL'R(X) = yL'R(X) =
1
91/4 i<-2
-- i
2l'4K
L-R(x) 0
(0 d/L'R(x))
(4.20)
Note that tyL'R is constructed by taking Hermitian conjugation and
applying the SU(2,2) metric rj = -- ir1 X I given in footnote 8. At this
point we remain with only four-dimensional fields L'R(x) written in the
form of SL(2, C) doublets.
For the gauge fixed form of tyL'R given above, the structures 0^L and ^LX
that appear in the action take the following forms after using Eqs. (4.7)
and (4.9):
^>LX
xc-- j-- (0,pL) "^
2 ' k w2
i\l2A -x^a
(4.21)
and
0ypL
= -^ {i^24>L, ~ ^ Lx^ <xj, (4.22)
1 / a^Dp -- i\f2(kDK -- AdA -- x^D^)
X
-- L(r)
K
0
91/41^-3
0
(4.23)
(4.24)
where we have used16 d^^ix) = 0. Note that only dA appears instead of Dk
due to a Yang-Mills gauge choice A_/ = -- A+ = 0 as explained below and
in footnote 15.
There remains to show that the dynamical equations of the original
fermions reduce to the usual four-dimensional massless fermion equations
in 3 + 1 dimensions. This can be done either for the equations of motion
or more concisely for the Lagrangian, with equivalent conclusions. Thus
consider the fermionic structures of the type ^LX0^L, ^RX0^R, g^LX^RH that
appear in the 4 + 2-dimensional action in Eqs. (3.15) and (3.25). From the
above gauge fixed expressions for tyL'R, tyL]t, and 0^L, we compute
i^X^1
-r9RX0<JrR
g^Lt^RH
g^Lt^LH*
1 -.
-- lt^Dp
K 1
K4 g
-- dfi
K4
L Lh*
(4.25)
Note that all explicit dependence on XM has disappeared. These emergent
forms are precisely the correct translation and Lorentz invariant kinetic
and Yukawa coupling terms that should appear in the four-dimensional
action. Therefore, we have shown that after using the kinematical
Even if we had not chosen the simplifying gauge ^'R(X) = 0 which made
^^(X) independent of A, we can still reach the same conclusion for Eq.
(4.24) for any ^'R(X), after first differentiating with respect to A and
then setting A = x2/2. The argument for this is similar to footnote 14.
equations, and imposing some gauge fixing, all the fermion terms in the 4
+ 2-dimensional action reduce to a four-dimensional theory:
S(fermions) = Z / \k\5di<d4xdA8(K2(2A -- x2))
1 X -- L (fermions) (4.26)
K4
d4xL
fermions
(AJx), h{x), L-R(x)\ (4.27)
where L
M,
Eq.
(4.18) is used. Here
fermionsv-^/iW^ Hx), L'R(x)) is the reduced form of the fermion terms L(A,
^L-R) + L(^L-R, H) that appear in the 4 + 2 theory in Eqs. (3.15) and
(3.25). This is precisely the usual chiral fermion terms in the standard
model Lagrangian in 3 + 1 dimensions interacting with the SU(3) X SU(2) X
U(l) gauge bosons and the Higgs field.
C. Reduction of gauge bosons
We start by gauge fixing the Yang-Mills gauge symmetry in the form of Eq.
(2.64) X % A = 0, so that the kinematic equations for the AM = (GM, WM,
BM) become the simple homogeneity condition XNFNM = (X % d + l)AM = (k8k
+ l)AM = 0. Thus in this gauge we can write
AM{X) = K-lAM{xix, A).
(4.28)
There is a leftover gauge symmetry given by Yang-Mills transformations
SAAM = DMA = dMA -- i[AM, A] that do not change the gauge condition X %
A = 0. This leftover symmetry corresponds to homogeneous A(X) of degree 0
X % S^A = 0-- > % X % 3A = 0.
(4.29)
This is just sufficient gauge symmetry to remove one full degree of
freedom in d + 2 dimensions from a homogeneous gauge field AM(X). Using
this freedom we choose a lightcone type gauge in the extra dimensions A+
(X) = 0 (note A-i = -- rj_/+iA+ = 0) and also use the fact that X % A =
0 to solve for the other lightcone component in terms of the four
Minkowski components A^{X). We find
AM(X): A+l = -- A-i = 0,
£-'
1
x^AJx^,A),
(4.30)
1
A^iX) =-Afi(xfi,A).
We now turn to the 2T gauge symmetry. In the present gauge its parameters
aM(X) of Eq. (2.65) must be restricted to maintain the gauge choice A+ (X)
= 0. Therefore, we must take a+ (X) = 0. Furthermore due to X % a = 0
they must have the same form as AM(X), namely, a+ = 0, a~' = X^a^. The
conditions that the remaining a^iX)
1
K
must satisfy in this gauge follow from Eq. (2.65) as (X " d + 3)alx{X) =
X^D " a, (X % d + 3)a± (X) = X± D % a.
(4.31)
Since a+> = 0 we must have D % a = 0, hence a^iX) must be homogeneous (X
% d + 3)afJ'(X) = 0, and therefore we have
1 / a»(X) = -- ra^(x, A).
K
(4.32)
We can now use this homogeneous degree of freedom in the 2T gauge symmetry
to eliminate the A dependence from the gauge field AM(x^, A).
To proceed we first identify different parts of the gauge field as
AJX)=AHX)+X2Ap
1
K
, , / xz . ,
A^yx) + A -- -- vM(x, A)
(4.33)
such that A^iX) = \_AIX(X)]X2=() = k~1A/j,(x). Then, we use the gauge
parameters a^{x, A) to gauge fix V^ix, A) = 0. In this way we have arrived
at a gauge fixed field AM(X) that is independent of A:
AM(X): A+l = -- A-i = 0,
£-'
A , --
x^A^x),
A>*(X)
AM(x).
(4.34)
We can now compute field strengths. Recall from the chain rule in Eq.
(4.4) that 3_/ = /c_1dA will vanish when applied on the A independent
fields. Therefore, we find
F pV(X) = K~2FfiJ/(x), with F^v(x) = d^Av -- dvA^ -- i[A^,Av],
F+'pOO = k~2xvF' pV{x), F-i^iX) = 0, F+i_i(X) = 0. The Lagrangian density
becomes
(4.35)
(4.36)
L(A(X))
1
4 1
4 k4
Tr (FMNFMN) (X) Tr(_FjttJ/_F'"F)(x).
(4.37)
We see that only the four-dimensional field strength F^v has survived as
the only independent field. It was possible to gauge fix the components
F-i^iX), F+i_i(X) to zero, because FMN(X) is not gauge invariant under the
2T gauge symmetry, although the action as well as the dynamical equations
constructed from FMN are gauge invariant.
Using these results we can now see that after using the kinematical
equations, and imposing some gauge fixing, all the gauge theory terms in
the 4 + 2-dimensional action reduce to a four-dimensional theory
S(A) = Z / \K\5dKd4xdAS(K2(2A--x2)) X ^ L{A^{x))
K
(4.38)
I cf4xL(AM(x)).
(4.39)
This is precisely the usual Yang-Mills terms in the standard model
Lagrangian in 3 + l dimensions for the SU(3) X SU(2) X U(l) gauge fields.
We have thus demonstrated that the 3 + l-dimensional standard model
emerges from 4 + 2 dimensions.
V. RESOLUTION OF THE STRONG CP PROBLEM
Recall that the strong CP problem in QCD is due to the fact that a term of
the form Sg = ^ X J dx4sfivAa Tr(GMFGAo") can be added to the QCD action
in 3 + l dimensions without violating any of the gauge or global
symmetries. Unfortunately this term violates CP conservation of the strong
interactions. So, phenomeno-logically speaking, if it is not absolutely
zero, it must be extremely small. However, there is no explanation of this
fact within the simple version of the standard model. This problem can be
circumvented by extending the standard model with an additional U(1)
symmetry, called the Peccei-Quinn symmetry [29], by doubling the Higgs
bosons. The spontaneous breakdown of this symmetry, along with SU(2) X
U(l) leads to the Goldstone boson called the axion. So far searches for
the axion have limited its parameters sufficiently to basically rule it
out. This leaves us with a fundamental problem to solve.
We will argue that there is a resolution of this problem in the 4 + 2
formulation of the standard model. The key point is that a term similar to
the form s^vA(T Tr(GMFGAo") that appears in the QCD Lagrangian in 3 + l
dimensions cannot be written down in 4 + 2 dimensions as an invariant
under the symmetries. This is because in 4 + 2 dimensions the Levi-Civita
symbol eMiM2M3M4MsM6 has six indices instead of four.
We may ask if there are any additional invariant terms that we could have
included in the 4 + 2-dimensional theory that could lead to Sg upon
reduction to 3 + l dimensions? In providing an answer to this question we
must take into account the 2T gauge symmetry as well as the principle of
renormalizability. The latter says that we should not include any terms in
4 + 2 dimensions that would lead to nonrenormalizable interactions in 3 +
l dimensions. Then one cannot find any terms that include the delta
function S(X2) in the volume element, except for the following one:
K
I (d6X)S(X2)XMidM2 T^(FM3M4FMsM6)sMlM2M3M4MsM6.
(5.1)
This term vanishes identically as follows. The ordinary derivative dMl can
be rewritten as the covariant derivative DM when applied on each of the
field strengths inside the trace, since the non-Abelian terms sum up to
become the commutator of matrices that vanish due to the trace. Then we
use the Biachi identities D[M FM M] = 0 to show that the term is null.
There remains the following type of term to consider without the delta
function:
I (d6X)Tr(FMiM2FM3M4FM5M6)sMlM2M3M4M5M6. (5.2)
The fact that it is cubic rather than quadratic seems to already violate
the renormalizability requirements. However, this is a topological term
that can be written as a total divergence, so it cannot violate
renormalizability. Furthermore, since it is a total divergence, it is
automatically invariant under all the infinitesimal gauge symmetries we
discussed before. Furthermore, it cannot contribute to the equations of
motion. For the SU(3) X SU(2) X U(l) gauge group there are several such
gauge invariants, namely,
I (d6X)[aTr(GMlM2GM3M4GMsM6)
+ bTr(WMiM2WM3M4WM5M6) + c(BMlM2BM3M4BMiM6)
+ dBMiM2 Tr(WM3M4WMsM6)
+ eBMiM2i:x{GM3M4GMiM)~\EM^M2M3M^MK (5.3)
The one that comes closest to producing Sg is the last one, since upon
reduction to 3 + 1 dimensions we might get the term
jdx4EfiJ/Ao-Tr(GfJ'1'GAo~) X f \i<\5dKdAB+i_i(x, A, k). However, we have
shown in the previous section that such a term is gauge dependent under
the 2T gauge symmetry. In particular in the reduction to 3 + 1 dimensions
we have shown that all F-i^iX) = 0, F+/_/(X) = 0 vanish not only for BMN
but for all others as well. Hence the topological term vanishes
identically.
This resolves the strong CP problem in QCD in the emergent standard model
in 3 + 1 dimensions.
VI. MASS GENERATION
We will mention briefly several mass generation scenarios, namely,
dynamical symmetry breaking, Coleman-Weiberg mechanism with only the SU(2)
X U(l) doublet field H and no dilaton, new mechanisms offered by the 2T
PHYSICAL REVIEW D 74, 085019 (2006)
physics formulation, and finally a discussion in more detail regarding the
dilaton assisted Higgs mechanism.
Dynamical symmetry breaking in the form of extended technicolor is
certainly one of the possibilities for mass generation. This would proceed
by adding all the ingredients of extra technimatter and
techni-interactions in parallel to what is done in the usual 3 +
1-dimensional theories of technicolor. The 4 + 2-dimensional theory seems
to proceed in the same way, and therefore we do not have new comments on
this possibility from the point of view of 4 + 2 dimensions.
The Coleman-Weiberg mechanism proceeds through radiative corrections in a
completely massless theory with quartic plus gauge interactions [30]. This
would apply in 3 + 1 dimensions to the Higgs field H in interaction with
the electroweak gauge bosons and produce an effective potential which does
lead to spontaneous breakdown. Of course, we would need to recompute these
effects directly in the 4 + 2-dimensional quantum theory, but for now let
us assume that the result is roughly similar. In the 3 + 1 theory this
mechanism predicts a definite mass ratio between the massive vector and
the massive Higgs. Using the values of the electroweak coupling constant
and an average of the W/Z masses, the mass of the Higgs comes out in the
range of about 10 GeV and seems to be ruled out already.
In 2T physics there are new ways of understanding mass as having a
relationship to some moduli in the embedding of 3 + 1 dimensions in the
higher space of 4 + 2 dimensions. This produces the massive relativistic
particle instead of the massless particle, as indicated in Fig. 1. This
effect, which has no relation to the Kaluza-Klein type of mass, has been
studied in the worldline formalism for particle dynamics [7,11]. It has
even been suggested as an alternative mechanism to the Higgs [20] but
remained far from being understood. The application of this approach to
understand the various corners of Fig. 1 in the context of field theory is
at its infancy [12] and needs to be studied in the presence of
interactions as formulated in this paper. This has not been developed so
far, but should be mentioned as a new possible source of mass that remains
to be investigated.
Next we turn to the Higgs mechanism, which is the most popular possibility
within the usual standard model. We find that there are new twists in the
4 + 2 formulation, and as a result there could be measurable
phenomenological consequences as described below.
The 4 + 2 action leads to the absence of quadratic mass terms in the Higgs
potential. Therefore, instead of the tachyonic mass, an interaction of the
Higgs H with a dilaton O was introduced in Eq. (3.13) in the form
A , , .,
V(0, H) = -- (H'H -- aZ(Pz)z + V(<P) (6.1)
to induce the spontaneous breakdown of the electroweak symmetry SU(2) X
U(l). The reason that only quartic interactions were allowed is intimately
related to the b symmetry [which ultimately comes from the underlying
Sp(2, /?)]. The b symmetry removes gauge degrees of freedom from the 4 +
2-dimensional fields and reduces them to the 3 + 1-dimensional degrees of
freedom.
The reason for the pure quartic interaction also can be understood
directly by studying the equations of motion. As shown in the previous
section, the solution of the kinematic equations of motion for the scalars
require that they must be homogeneous of degree -- 1. This means that
their dependence on the extra dimension k must be of the form
H(X)=-H(x), 0(X)=-0(x). (6.2)
K K
The dynamical equations of motion d20 = d$V(Q, H) + % % % and d2H =
d#y(0, H) + % % % now have the left-hand side proportional to k~3;
therefore, the right-hand side also must be proportional to k~3. Combined
with Eq. (6.2), this requires V(0, H) to be purely quartic. If any
additional terms, such as quadratics are included in V(0, H), they have to
vanish on their own since they will have a different power of k.
For this reason the dynamical breakdown of the SU(2) X U(l) electroweak
symmetry cannot be accomplished with a tachyonic mass term for the Higgs
field H, since this is forbidden in the 4 + 2 formulation. Instead, a
coupling to the dilaton as given in Eq. (6.1) generates the nontrivial
vacuum configuration in Eq. (3.14). The equations of motion for the
scalars (assuming all other fields vanish at the vacuum) are
d2H = AH(H'H -- a2<J>2),
(6.3) d20 = -- 2a2&(H^ H -- a202) + V'(O).
At the vacuum configuration (H^H -- a202) = 0, the Higgs field must
satisfy d2H = 0 while H(X) is also homogeneous of the form Eq. (6.2). We
have seen already that d2H(X) = k~3 d2H(x); therefore, H(x) must be a
constant at the vacuum, but H(X) still depends on the k coordinate
. , .. v0\
\H(k, a, x^)) = -- ,
(6.4)
v = electroweak scale -- 100 GeV.
The electroweak scale of about 100 GeV is determined by fitting to
experiment.
Returning to the equation for O, at the Higgs vacuum, it reduces to the
form d20 = V'(O). But it must also satisfy the vacuum value (H^H -- a202)
= 0 and be homogeneous as in Eq. (6.2); therefore,
. , . v
(<p(a)) = ±-- . (6.5)
tea
Hence, 32(0) = 0, which requires V'((0)) = 0 at the vacuum. If V(O) is
also taken as a quartic monomial V(O) = 4^04 (as required by the b
symmetry), then the only solution is A' = 0, or V(O) = 0 identically (at
the classical level) to fit phenomenology.
Actually, the above discussion for the dilaton O is incomplete. To fully
understand the interactions of the dilaton one must include the
gravitational fields with which it naturally interacts. For example, in
string theory the vacuum expectation value of the dilaton plays the role
of the string coupling constant that controls all string interactions.
There is no full understanding yet how our four-dimensional vacu