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Ann. Henri Poincaré 4, Suppl. 1 (2003) S15 – S30
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/01S15-16
DOI 10.1007/s00023-003-0903-4
Annales Henri Poincaré
Probing Physics at the Planck Scale
G.G. Ross
Abstract. We review the ideas for extending the theory of the strong, electromagnetic and weak interactions to the Planck Scale and a unification with the gravitational interaction.
1 Introduction
One of the triumphs of modern physics has been the construction of the “Standard
Model”, the theory of the strong, electromagnetic and weak interactions [1]. The
Standard Model is a quantum field theory based on the principle of local gauge
invariance. It describes essentially all the observed phenomena of the fundamental
interactions as proceeding through the exchange of the elementary quanta of the
force carriers; the photon for the electromagnetic interaction, the gluons for the
strong interaction and the W ± and Z bosons for the weak interactions. However
many physicists think that the Standard Model is incomplete and are pursuing
the elusive “Theory of Everything” which they think represents the ultimate unification of the fundamental forces, including gravity. In this talk I will discuss how
such searches have led to a discussion of Physics at the Planck Scale where the
unification of the fundamental forces, including gravity, is thought to occur.
2 The Standard Model
2.1
The states of matter
The Standard Model describes all the matter we observe in the universe in terms
of a small number of elementary particles with interactions described by simple
laws. The protons and neutrons which make up the atomic nucleus are made from
elementary constituents, the (charge 23 ) up and the (charge - 13 ) down type quarks
and the “gluons”, the carriers of the strong force which holds them together. In
addition the Standard Model includes the electron (charge -1) as an elementary
state and its neutral partner, the neutrino. These “leptons”, like the quarks, feel
the electromagnetic and weak forces but, unlike the quarks, do not feel the strong
force. To complete the Standard Model, three families or copies of the quarks and
leptons are needed to describe the states currently observed c.f. Figure 1.
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Figure 1: The matter states of the Standard Model
2.2
The interactions
The laws that govern the strong and weak interactions follow from the same local
gauge principle applied to a relativistic field theory that led to Quantum Electrodynamics (QED), the quantum version of Maxwell’s theory of electromagnetism.
The underlying ingredient is the recognition of a symmetry relating the states of
the theory. Such a symmetry is based on patterns and the patterns of the Standard Model are illustrated in Figure 1. The experimental observation of the weak
interactions of the left-handed electron and its neutrino suggest that these states
are intimately related and have similar properties. In the Standard Model these
states are assigned to doublets of the group SU (2) as is shown in Figure 1. The
weak force is due to the exchange forces generated by with the gauge bosons of
SU (2), the W ± and Z bosons. The charged weak interaction arises through the
exchange of a W boson which changes a left-handed electron into a left-handed
electron neutrino. The neutral weak interaction arises through the exchange of a
Z boson which couples an electron. The representation is “chiral” in the sense
that the transformation properties of the left- and right- handed states differ. The
right-handed components of the electron, muon and tau are singlets under SU (2),
hence do not change under an SU (2) rotation. In turn this means they carry no
weak charge and do not couple to the weak gauge bosons and do not participate in
the weak nuclear force. Note that we have not observed right-handed components
of neutrinos and they are not included in the Standard Model particle content. As
may be seen from Figure 1 the same pattern is repeated for the quarks with the
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left-handed components assigned to doublets and the right handed states singlets
under SU (2).
The strong interactions result from the gauge bosons of an SU (3) local gauge
symmetry, with each quark coming in three “colours” and belonging to a triplet
of states related by SU (3) rotations as shown in Figure 1. There are eight gauge
bosons, “gluons”, associated with SU (3) corresponding to the different ways they
can couple to the three colours of quarks.
In order to include electromagnetism in the Standard Model, it is necessary to
add a “U (1)” gauge symmetry corresponding to the local conservation of electric
charge. This gives rise to a single gauge boson B (since there is only one type
of electric charge). The photon is made up of a mixture of B and the W 3 gauge
boson of Figure 3a. The orthogonal combination of the B and W 3 gauge bosons
is the neutral weak boson, the Z. This completes the gauge symmetry structure
of the Standard Model. The full gauge symmetry is SU (3) ⊗ SU (2) ⊗ U (1).
The Standard Model has been found to be amazingly successful in describing
all phenomena associated with the strong and electroweak interactions, the success
culminating with the discovery of the W and Z bosons in 1983 at CERN [3] and
the gluons at DESY [4].
2.3
Partial Unification
The exchange forces associated with these gauge bosons provides us with a partially unified quantum description of the strong, the weak and the electromagnetic
interactions and, to date, essentially all observed phenomena are consistent with
the predictions of the Standard Model. A very important property of these interactions is that the strength of the interactions depends on the length scale being
probed, a property due to the screening effects of the virtual states necessarily
present.
This is illustrated in Figure 2 [5]. It may be seen that the strength of the
strong force reduces as the momentum scale increases (or the distance scale decreases) the effect being dominated by the antiscreening effect of the virtual gluons.
As a result at high momentum scales the strength of the strong force approaches
that of the electromagnetic force which is also shown. Finally it may be seen
that the strength of the weak force also approaches that of the other forces at
high scales. In addition to a small screening effect there is a significant change in
the effective weak coupling due to the weak propagator. The dimensionless coupling characterizing the underlying strength of the weak interactions is given by
GF (Q)Q2 where GF (Q) is the Fermi coupling. In the Standard Model this cou2
2
pling is due
= g 2 /8.Q2 /(Q2 + MW
)
√ to the exchange of a W boson so GF (Q)Q
2
2
where g/2 2 is the gauge coupling constant and 1/(Q + MW ) is the gauge boson
2
and is weak due to the large
propagator. At low scales GF (Q)Q2 g 2 /8.Q2 /MW
2
W mass MW 80GeV. At high scales GF (Q)Q g 2 /8 and, as may be seen from
Figure 2, is intermediate in strength between the strong and weak interactions.
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Figure 2: The experimental measurement of the running of the strong force with
the momentum scale Q at which it is measured. Also shown are the electromagnetic
and weak interactions as a function of the momentum scale.
However the approach of the coupling strengths, while suggestive, represents
only a partial unification in as much as the couplings are not equal and there
is no explanation in the Standard Model why they should even approach each
other because the gauge group, SU (3) ⊗ SU (2) ⊗ U (1), involves three unrelated
factors. Moreover, as may be seen from Figure 1, there is a complicated (chiral)
multiplet structure needed which has been input by hand to generate the observed
structure. Perhaps most importantly, the unification is of only three of the four fundamental interactions and there is no connection with the gravitational force. The
gravitational coupling is proportional to Newton’s constant with inverse mass di19
2
mensions, GN ∝ MP−2
lanck . MP lanck is the Planck mass, approximately 10 GeV /c
2
(the unit of mass used in particle physics is 1 GeV /c , approximately the mass of
the proton). The weak coupling is given at low energies by the Fermi constant,
−2
, where the W boson mass MW is approximately 102 GeV /c2 some 17
GF ∝ MW
orders of magnitude smaller than the Planck mass. The quark and charged lepton
masses range from 12 10−3 GeV /c2 for the electron to approximately 175GeV /c2 for
the top quark while the neutrinos in the Standard Model are massless. For many
the origin of this hierarchy is the most pressing of all the questions raised by the
Standard Model.
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3 Grand Unification
Apart from the inclusion of gravity, an answer to the questions raised by the partial
unification of the Standard Model is provided by Grand Unification. Grand Unified
theories (GUTs) seek to embed the Standard Model in a unified structure which
can relate its multiplet structure and interactions by extending the patterns that
led to the Standard Model. The archetypical GUT is based on the symmetry
group SU (5), chosen because it is the smallest group which can accommodate the
SU (3) ⊗ SU (2) ⊗ U (1) gauge group of the Standard Model [6, 7, 2]. In this the
strong, the weak and the electromagnetic forces are just different facets of the
one underlying (SU (5)) local gauge interaction. The quarks and leptons are also
related in SU (5) because they belong to the common multiplets as shown in eq.(1).






dr
dy
db
e+
−ν c












R
0
−ucb
ucy
−ur
−dr
ucb
0
−ucr
−uy
−dy
−ucy
ucr
0
−ub
−db
ur
uy
ub
0
−e+
dr
dy
db
e+
0






(1)
L
Of course there must be an explanation for the non-observation of the additional
gauge bosons implied by SU (5). Provided the mass of these bosons, MX , is larger
than the current energy of particle accelerators we will not have been able to find
the X bosons directly. However a much stronger bound is available through the
virtual effects of the X bosons since they mediate new processes which have not,
so far, been observed. In particular the new interactions of SU (5) mediate proton
decay. The current experimental limit on the decay lifetime of the proton is an impressive 1032 years giving a limit on the X boson mass of MX > 1016 GeV /c2 ! Note
also that in SU (5) there is no room for a right-handed neutrino component. This
offers an explanation for the different properties of the neutrino for the absence
of this state is what forces the neutrino to be massless in the Standard Model.
In extensions of SU (5) to more unified GUTs such as SO(10) the missing righthanded neutrino appears. However it is expected to acquire a mass at the stage of
spontaneous symmetry breaking of SO(10) to SU (5) which we expect to be very
large and greater than MX . As a result the left-handed neutrino states remain
2
/MX ) with the result that the left-handed neutrino
very light with mass ≈ O(MW
states can only acquire a very small mass, in agreement with recent measurement.
In fact SO(10) is a very attractive extension of SU (5) because a single 16 dimensional representation contains all the states of a single family plus the right handed
neutrino. Thus it enormously simplifies the multiplet structure of Figure 1, leaving
only the family replication to be explained, perhaps by extending the symmetry
to include a family symmetry.
There remains, of course, the question of determining the couplings and
masses in the Standard Model. In a GUT, such as SU (5), with a single group
factor, there is a single gauge coupling constant describing the strong, weak and
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Figure 3: Plot of the (inverse) couplings of the Standard Model gauge group factors
up to the Grand Unified mass scale
electromagnetic couplings. Thus it offers an explanation of the partial unification
of couplings observed in Figure 2.
However the unification of couplings should apply at a scale at which the
differences between the gauge bosons of SU (5) can be neglected. This means we
should continue the couplings above the mass of the X bosons. This is shown in
Figure 3 where it may be seen that the couplings continue to approach each other,
suggesting a true unification [9]. Moreover the scale at which they come together is
very high, in agreement with the scale needed to inhibit proton decay. However, in
detail, the couplings never meet, the discrepancy being 8 or 9 standard deviations.
4 Supersymmetry
Grand Unification, while very attractive, introduces a fundamental problem. This
occurs because in a field theory with pointlike couplings, radiative corrections
probe all virtual momentum scales. As a result they can generate large corrections to particle masses unless the mass is protected by a symmetry. Most of the
states of the Standard Model are protected. The gauge bosons cannot acquire a
mass while the gauge symmetry is unbroken and thus the gluons and the photon remain massless even after radiative corrections are included. The fermions
belong to chiral representations and similarly cannot acquire mass while the Standard Model gauge group is unbroken. The problem occurs in the sector needed
to spontaneously break the SU (2) gauge symmetry. In this spontaneous breaking
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the order parameter is supplied by the vacuum expectation value of a scalar field,
the Higgs boson. However its mass is not protected by a symmetry and so receives
large radiative corrections which drive it up to the Grand Unified scale. This would
mean that the SU (2) symmetry is broken at the scale MX , giving mass of O(MX )
to the W and Z bosons and to the quarks and lepton. To avoid this unacceptable conclusion requires that the symmetry of the Standard Model be extended to
include a new symmetry capable of protecting the Higgs scalar mass from large
radiative corrections.
The only possible symmetry capable of keeping a Higgs boson light is supersymmetry (plus a chiral symmetry) . Supersymmetry [8, 2] assigns scalars to
supermultiplets with fermion partners and thus their properties are related :
φ (J = 0) SUSY
−→ ψ (J =
1
) requires mψ = mφ
2
Thus if the fermion mass mψ is forbidden by a chiral symmetry as discussed
above the scalar mass will also vanish. In this case there must be supersymmetric
partners to the Standard Model states. In order to prevent radiative corrections
reintroducing the hierarchy problem, the states must be quite light, less than or
of order 1 TeV.
The extension of the Standard Model to include supersymmetry requires the
addition of many new states. These include the “squarks” and “sleptons”, spin
zero partners of the spin-one-half quarks and leptons respectively. Also needed are
the “gluinos”, the “Wino”, the “Zino” and the “photino”, spin-one-half partners
of the Standard Model gauge bosons, the gluon, the W, the Z and the photon
respectively. Although these new states are thought to be considerably heavier than
their partners, and hence should not yet have been found directly in laboratory
experiments, they will contribute as virtual states to the radiative corrections
discussed above. The exciting observation that solves the hierarchy problem is that
the new SUSY radiative corrections have the same magnitude as the Standard
Model contributions but opposite sign. Thus, when SUSY is exact, there is no
correction to the Higgs mass. When SUSY is broken at a scale MSUSY the new
supersymmetric states acquire mass of O(MSUSY ) and this cancellation is spoilt.
However the resultant contribution is of O(MSUSY ) and provided this is not too
large (i.e. ≤ O(1T eV )) it is consistent with the observed electroweak breaking.
In summary the supersymmetric solution to the hierarchy problem requires new
physics beyond the Standard Model in the form of the spectrum of new SUSY
states at a scale accessible to the next generation of accelerators.
5 Unification of couplings
Although there is no direct evidence for a new supersymmetric state, the is strong
indirect evidence for the existence of such states coming from the unification of
gauge couplings. This follows because the new supersymmetric states must be light
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Figure 4: Evolution of the gauge couplings in the supersymmetric extension of the
Standard Model
to protect the Higgs from receiving large radiative corrections to its mass. These
light states also contribute to the screening of the gauge couplings and thus affect
the gauge unification prediction. In Figure 4 I show the evolution of the couplings in
the Supersymmetric Standard Model. One may see that that the supersymmetric
prediction is quantitatively better than the non-supersymmetric case shown in
Figure 3. This only works if the new supersymmetric states are “light” on the
Grand Unified scale with masses less than 1T eV /c2 just as is required if one is to
keep the Higgs boson light.
A better way to illustrate this may be obtained by using the assumed unification to predict the value of the strong coupling in terms of the weak and
electromagnetic couplings[11]. This is shown in Figure 5 where one may see that
the predicted region in the sin θ W − αs plane occupies only a very small part of the
a priori allowed region. This is quite impressive already as it appears to be better
than one part in one hundred. If one adds the constraint that MX must be large
to avoid rapid proton decay this region shrinks even further. Using the measured
value of the strong coupling one may use this plot to determine the prediction for
sin2 θW . One finds
sin2 θW = 0.2314 ± 0.002 − 0.25(αs − 0.119))
to be compared with the experimental value sin2 θW = 0.2312 ± 0.0002.
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Figure 5: Graph (bottom left) illustrating the precision of the test of gauge unification. The area between the two curves is the predicted region allowing for the
uncertainties in the supersymmetric particle masses with no constraint on the unification scale. The graph (upper right) indicates the prediction for the unification
scale that results from the requirement that the gravitational strength should also
unify.
5.1
Unification with gravity
Given the success of the unification prediction one may ask whether the unification
can extend to include gravity as well. Following our discussion of the weak force
where the relevant dimensionless coupling was GF (Q)Q2 . In this case the relevant
coupling is GN (Q)Q2 where GN (Q) is Newtons constant. In Figure 5 I show that
this coupling also comes close to unification with the other couplings provided
the unification scale is vey high, close to the Planck scale, In my opinion this is
a very strong hint that the four fundamental interactions do unify and motivates
the search for the underlying theory capable of doing so. The only candidate is
the (super) string.
6 String Unification
Since others at this meeting will talk about string theories themselves, I will concentrate on the question how is unification changed in superstring theories? String
theory is the only candidate we have for a unification of all the fundamental in-
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Ann. Henri Poincaré
teractions including gravity [12]. Of course at low energies one is looking at the
four dimensional effective field theory which results from string theory defined
at the Plack scale. Often this involves a stage of compactification of some of the
space-time dimensions and in the low energy theory these act as internal degrees
of freedom. The resulting 4D theory has several promising properties :
• The symmetries are specified by the underlying string. For consistency the
string theory should be supersymmetric and this symmetry may survive in
the low energy effective theory offering an explanation for the hierarchical
structure of masses observed. In the heterotic string case there is an E8 ⊗ E8
gauge symmetry before compactification leading to an E8 ⊗ E6 on compactification on a Calabi Yau manifold. The resulting gauge group can easily
accommodate the Standard Model but also can be larger. For example it
can incorporate the Grand Unified groups SU(5), SO(10) or E6 . The only
symmetries that are known to descend from the string are gauge symmetries,
either continuous or discrete. Unfortunately there are many possible vacua
in the compactified 4 dimensional theory and we do not know how to select
between them.
• The multiplet structure is specified in a given string vacua. Usually this includes some number ng of chiral families + some number nV of vector like
states which come in complex conjugate pairs. The latter are expected to
acquire mass at a high scale through a stage of symmetry breaking below
the compactification scale. Many 3 generation examples are known. Moreover, if the string theory is built from level-1 Kac Moody level theories the
representation content of the theory is restricted offering an explanation to
the question why only low lying representations of quark and leptons are
observed.
• String theory has only one fundamental parameter, the string scale which
can be related to the Planck scale. All other parameters are determined in
terms of the vacuum expectation values (vevs) of moduli fields, ΦM . For
example the string coupling constant is determined by
1/2
gi
(Mstring ) =
ki
2
gstring
(2)
where ki is a parameter associated with the particular string construction
and
1
2
.
(3)
=
gstring
<S>
Here S is a moduli field known as the dilaton. In the absence of supersymmetry breaking the moduli fields have no potential and any value for their
vacuum expectation value gives a viable string theory. Once supersymmetry
is broken, the moduli’s vev will be fixed and the coupling will be determined
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as in eq 3. Similarly the Yukawa couplings which determine the quark and
lepton masses are determined as functions of additional moduli known as
complex structure moduli. (Unfortunately we are still far from an understanding of the string vacuum structure and hence from a prediction of the
string coupling.)
6.1
M-theory determination of the gauge unification scale
The prediction for the gauge unification scale in the weakly coupled heterotic string
follows from the general form of the 4D Lagrangian [15]
4
1
√
2
Lef f = − d10 x gα−1
10 ( 4 R + 3 T rF + · · ·)
α
α
√
4
1
2
= − d4 xV gα−1
(4)
10 ( 4 R + 3 T rF + · · ·)
α
α
In this we may see that Newton’s constant, GN , and the value of running gauge
couplings at the unification scale, αGUT , are given in terms of the 10D string
1
coupling α10 , the string tension α ∼ Mstring
and the volume of the 6D compactified
space V by
α10 α4
α10 α3
, αGUT =
(5)
GN =
64πV
16πV
−6
For the case that α10 is small the volume V is approximately Mstring
and one
obtains eq(6) eliminating V between the two equations.
Mstring ≈ gstring × (5.2 × 1017 GeV ) ≈ 3.6 × 1017 GeV
(6)
which, c.f. Figure 3, is only a factor of 20 above the “observed” gauge unification
scale [16].
It is difficult to overemphasize the potential importance of this result. Our
belief that there is a stage of Grand Unification of the strong, weak and electromagnetic interactions rests largely on the quantitative success of the unification of
the associated couplings. The prediction of the unification scale would be the first
indication of unification with gravity.
While the prediction within a factor of 20 is encouraging, the residual discrepancy raises some doubts. One promising explanation comes from the string
itself. If one uses the “measured” value for V = O((1 − 3).1016 GeV −6 ) one may
obtain the value of α10 instead. This gives an enormous value, quite inconsistent
with the assumption of weak coupling that went into the derivation of eq(5). Thus
the failure of the prediction of eq(6) is not surprising - it was the wrong calculation.
Instead one should go to the strongly coupled case.
That it is now possible to discuss strong coupling relies on an understanding
on relations between strong coupling and weak coupling regimes known as dualities. It has long been known that there are five distinct classes of string theory, a
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fact that has caused concern because one might hope that the “Theory of Everything” is unique. However developments in string theory have shown that there
is a rich class of duality symmetries relating various string theories [13, 14] so
now the expectation is that all string theories are related to each other and to an
underlying theory known as “M-theory”. This theory lives in eleven dimensions
and the various string theories may be viewed as different limits of this underlying
theory. In addition it is now realised that eleven dimensional supergravity also
belongs to this web of inter-related theories. A feature of the dualities relating
different theories is that they often relate one theory in the weak coupling (perturbative) limit to another theory in the strong (nonperturbative) limit or one theory
in the small compactification radius limit to another in the large compactification
limit. By making use of these dualities one may gain insight into non-perturbative
physics by performing calculations in the dual theory in which the perturbation
series makes sense.
In the strongly coupled heterotic string case Horava and Witten [15] showed
that the theory is equivalent to M-theory. Although it is not possible to construct
M-theory explicitly it is possible to write down the long-distance effective field
theory coming from M-theory. It is just 11 dimensional supergravity compactified
on a six-dimensional Calabi-Yau space times a line interval, CY3 ⊗ S1 /Z2 . The
effective Lagrangian in this case is given by [15]
Lef f
1
=−
2πκ211
√
d x gR −
11
1
8π(4πκ211 )2/3
√
d10 x gT rF 2 + · · ·)
(7)
giving
GN =
κ211
(4πκ211 )2/3
, αGUT =
2
16π V R11
2V
(8)
The reason this form changes is that the gravitational fields now propagate in the
bulk of the 11th dimension while the gauge fields live on the end points of the line
integral (one E8 on each fixed point) and propagate only in 10D. One may see
that this changes the relation between Newton’s constant and αGUT through the
appearance of the radius of the 11th dimension, R11 . Given the appearance of a
new parameter it is clear one may adjust it to eliminate the discrepancy between
scales found in the weakly coupled case. This gives [16]
πR11 = (5.1015 GeV )−1
(9)
While it is encouraging that the discrepancy between scales has been eliminated,
apparently the predictive power has been lost. In order to determine the size of the
compactified dimensions it is necessary to understand the vacuum structure of the
theory which determines the magnitude of the moduli field setting this scale. At
present this is beyond our ability so the unification with gravity, while suggestive,
remains unproven.
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7 Probing the Planck scale
In this talk I have described some very promising ideas for the complete unification of the fundamental forces, including gravity. Will these ideas be tested in
the foreseeable future? As we have discussed there is already evidence for unification coming from the unification of the gauge couplings of the theory. Additional
predictions of relations between the masses of the fundamental matter fields, the
quarks and leptons are also in good agreement with measurement. In addition
there is a good explanation for why the SU (2) symmetry is broken to give masses
to the W and Z bosons leaving the gluons and photon massless. These indirect
tests of the underlying theory are encouraging. What about more direct tests of
unification? One of the central ingredients is the need for supersymmetry with its
associated states with a mass in the T eV range. Such states will be accessible to
the next generation of accelerators, the LHC at CERN and the Tevatron at Fermilab in Chicago. It may be that the new supersymmetric states provide a source
of the missing dark matter in the universe and searches for supersymmetric dark
matter are being actively pursued.
7.1
Epilogue
In this talk I have concentrated on the “classic” unification prediction which applies
close to the Planck scale some fifteen orders of magnitude above the energy scale
accessible to present accelerators. In this case it is impossible to produce the string
states which would directly verify an underlying string unification. The best we
can hope for is that these states may leave an imprint on the development of
the universe after the Big Bang which is visible today in the structure of the
universe and the microwave background. However recent developments in the study
of string theory have raised another interesting possibility that may make the
string states more accessible to experimental test. As discussed in the previous
Section it is possible that the relation between the string scale and the Planck
scale gets modified through the appearance of a new space dimension with radius
R11 much larger than the Compton wavelength of the string states. As a result the
underlying unification scale can be reduced. This has raised the question how large
can such new dimensions be and how low can the string scale be? Surprisingly it is
possible to have new space dimensions as large as 0.1mm [17]! The first indication
of such a new space dimension would be new massive “Kaluza Klein” excitations
of the graviton, perhaps as light as fractions of an electron volt! However these
states couple with gravitational strength and so would not yet have been observed.
Associated with a large new dimension is the reduction in the string scale and the
extreme possibility is that the string scale should be as low as a T eV. In this
case the Planck scale reflects the existence of a very small coupling and not the
existence of very heavy states. In this extreme case one might even do without
supersymmetry because the virtual momenta involved in radiative corrections are
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Ann. Henri Poincaré
cut-off at the T eV scale. In such schemes the next generation of accelerators would
even be able to directly produce string excitations.
While an interesting possibility, I consider such low scale unification schemes
to be much less likely than the “classic” picture. If the gauge couplings are to unify
at a low scale there must be additional light states which change the running of
the couplings. As a result the remarkably precise prediction for the unification of
gauge couplings discussed above would then have to be considered to be a complete
accident.
8 Conclusion
The 20th century has seen the construction of a relativistic quantum field theory
of the strong, electromagnetic and weak interactions. This theory is remarkably
successful and its predictions are consistent with all observed phenomena. However the theory leaves unanswered several important questions, perhaps the most
important being what is the origin of the mass of the fundamental states and how
does the unification extend to the fourth fundamental force, gravity? Remarkably
there is already indication that there is an underlying “Theory of Everything” capable of providing answers to these questions. Provided there is a new symmetry,
supersymmetry, the observed strengths of the interactions become equal at a scale
close to the Planck scale. In string theories the unification scale is related to the
string or Planck scale offering a simple explanation for high scale unification. The
small mass of the neutrinos is a natural outcome of the large scale of unification.
The parameters determining the quark, charged lepton and neutrino masses fit well
with an underlying GUT combined with a spontaneously broken family symmetry.
If these ideas are correct we should be on the threshold of observing completely
new phenomena associated with the new supersymmetric states. The very real
hope is that the 21st century will see our understanding of the origin of mass and
the construction of a fully unified theory of all the fundamental interactions.
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References
[1] For introductory reviews of the Standard Model see : M. Veltman, “Gauge
Theories,” In Fraser, G. (ed.): The particle century, Bristol, UK: IOP (1998)
46–56;
G. Altarelli, “How The Standard Model Works,” In Fraser, G. (ed.): The
particle century, Bristol, UK: IOP (1998) 127–141.
[2] G. Ross, ”The Standard Model and Beyond,” In Fraser, G. (ed.): The particle
century, Bristol, UK: IOP (1998) 155–168.
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Ann. Henri Poincaré
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G.G. Ross
Department of Physics
Theoretical Physics
University of Oxford
1 Keble Road
Oxford OX1 3NP
U.K.
and
Theory Group
CERN
1211 Geneva 23
Switzerland