Searching for the Origin of Masses
Hiroyuki Iwasaki
KEK, High Energy Accelerator Research Organization
1-1 Oho, Tsukuba-shi, Ibaraki-ken, 305-0801 Japan
Abstract. The origin of masses of weak gauge bosons as well as quarks and leptons is one of
the most mysterious themes in high energy physics. In the standard model, it is explained as a
result of a spontaneous symmetry breaking of the vacuum, which is caused by self-interaction of
an unknown complex scalar field. Three weak gauge bosons acquire their masses by "eating" the
three components of the scalar field. The remaining one component survives and called the
"Higgs boson." Masses of quarks and leptons are generated by an interaction between those
particles and the scalar field. Searching for the Higgs boson is a key step toward deeper
understanding of nature. In the lecture, a brief introduction of theoretical aspects and an
experimental approach based on an ongoing project will be given.
1. INTRODUCTION
There are four fundamental interactions: gravitaional, electromagnetic, weak, and
strong interaction. Indeed classical gravitational interaction is well understood in the
framework of general theory of relativity, but there is not a satisfactory quantum
theory treating gravity. The other three interactions are well formulated in a consistent
manner called "the standard model." Although it is called model, all the experimental
results are consistent with this model with good accuracy.1 Since quantum effect of
gravity becomes significant only when the energy scale is order of 1019 GeV, it can be
ignored as long as we consider phenomena at least below ITeV energy scale.
In the modern picture, an interaction is caused through exchanging particles
between subject particles. The mediating particles are called "gauge bosons," which
are photon for the electromagnetic interaction, weak bosons (W, Z) for the weak
interaction, and gluons for the strong interaction. Elementary particles in the standard
model are categorized into matter fermions, gauge bosons, and a yet-discovered Higgs
boson. The matter fermions are six quarks, three charged leptons, and three neutrinos.
In order to find out a dynamics of an interaction between the elementary particles,
one needs a guiding principle. A requirement of the "local gauge invariance" of the
Lagrangian is such a principle in the standard model. We can deduce the interaction by
demanding the gauge invariance on the Lagrangian for free particles. Its direct
1
Only one exception is neutrino masses observed recently. But it can be incorporated into the
framework of the model. In fact, neutrino masses are very small and have been assumed massless in the
standard model.
CP634, Science of Superstrong Field Inter actions, edited by K. Nakajima and M. Deguchi
© 2002 American Institute of Physics 0-7354-0089-X/02/$ 19.00
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consequence is that the gauge bosons have to be massless. Indeed photon and gluons
are massless, but weak gauge bosons are quite heavy. In addition, the weak interaction
violates parity: the W boson couple to left-handed fermions only. In such a case,
fermions have to be also massless, which also contradicts with the real world. A key
solution of the mass problem is so called the "Higgs mechanism." And the model
predicts a massive spin-0 boson called the "Higgs boson."
In the following section, an overview of the Higgs mechanism is given. In section 3,
a big international project aiming at discovery of the Higgs boson is introduced. It is
called "LHC." A pp-collider and two detectors under construction are briefly reported.
In section 4, it is explained how the experiments reveal the Higgs boson. Discovery
potential of the Higgs mass is studied. A summary of the lecture is given in the last
section.
2. HIGGS MECHANISM
Here we will see only flavor of a basic idea of the Higgs mechanism and not get
into the electroweak theory,2 which is beyond our scope of this short course [1]. We
consider U(l) gauge invariant Lagrangian for a complex scalar field
$ = (fa + i(j)2 ) / 1/2 described by
L =(D^}\D^)-V(^~F^\
(2.1)
where D^ is a covariant derivative,
D^d^+igA^
(2.2)
where g is the coupling constant, A^ is the U(l) gauge field and the last term is the
kinetic energy of A , and
^=3,4-3,4,.
(2.3)
The potential is
#) 2 .
(2.4)
One can see that the Lagrangian is invariant under U(l) local gauge transformation,
0 -» e*"(*> ,
(2.5)
4^4-l^a.
(2.6)
o
If \i2 > 0 and A = 0 , the potential F(0) simply describes the mass term of the
scalar fields. If, however, somehow // 2 < 0 and A > 0 , the potential looks like a
bottom of a wine bottle as shown in Fig. 1 . In this case, the point (fa , <j)2 ) = (0, 0) is no
2
In the electroweak theory, we have to deal with SU(2)xU(l), instead of U ( l ) . We will briefly
comment on the electroweak theory at the end of this section.
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longer the minimum. It has a circle of minima3 of the potential in the ^ -02 plane of
radius v, such that
' +022 = v2
with v2 = -—.
(2.7)
We can take any point in this circle as an absolute minimum of energy, which
corresponds to a "vacuum." We choose a point (^1,^2) = (t>,0) as the vacuum and
introduce new real fields, rj and £ as
(2.8)
Circle of minima
radias — \)
FIGURE 1. The potential V(0) for a complex scalar field.
Once a certain vacuum is chosen, the vacuum no longer possesses a global gauge
symmetry. It is called the "spontaneous symmetry braking." Now if we fix the gage as
(2.9)
we get
(2.10)
By substituting (f>'(x)
rewrite it as,
into the Lagrangian (2.1) and neglecting a constant, we can
L' = (
i-—F F^v .
+—js
(2.11)
There are only two fields left, a scalar 7] and a vector gauge boson A^. The third
term shows that the gauge boson A^ has acquired a mass,
The value I) is called the "vacuum expectation value."
414
mA=gv.
(2.12)
The spurious % -field has disappeared4 and its freedom is turned into a longitudinally
polarized (helicity zero) state of the massive gauge boson A^. This is called the
"Higgs mechanism." The second term is a mass term of 77 with,
m^=^2^v2 .
(2.13)
The massive 77 is called the Higgs boson. We can deduce V from the coupling
constant g and the mass of the gauge boson mA by using (2.12). Nevertheless,
since we don't know the parameters A, we cannot estimate the Higgs mass mn. On
the other hand, we know how the Higgs boson couples to the gauge boson from the
sixth (two Higgs bosons and two gauge bosons) and the seventh term (a Higgs boson
and two gauge bosons) of (2.11). Once we discover the Higgs boson and measure its
mass, we can deduce the parameter A . Then the self-couplings of the Higgs boson are
known from the fourth and fifth terms of (2.11).
This is the mechanism of how the gauge boson acquires the mass. Then one may
ask why the photon and the gluons are still massless. The reason for the gluon is
simple. The gluons couple to "color charges" which represent three degrees of
freedom.5 Since the scalar field 0 does not have a color charge, and hence does not
couple to gluons. The gluons are free from the spontaneous symmetry breaking of the
vacuum and it can be massless.
In case of the photon, it is not so simple. In the standard model, the electromagnetic
interaction and the weak interaction are unified in a single framework called the
"electroweak theory," where the gauge group is SU(2)LxU(l)Y. There are three
gauge fields, W19 W2, W3 for SU(2)L and one gauge field, B for C/(l)7. These
four fields couple to four scalar fields 0. (/ = 1,2,3,4) which belong to SU(2)L.
Three scalar fields out of the four disappear when SU(2)L gauge is fixed. After the
spontaneous symmetry breaking, Wl and W2 become massive charged W± (80
GeV); properly chosen linear combinations of Wz and B make massive
neutral Z (91 GeV) and massless photon; one real scalar field remains, which is the
Higgs boson.
Masses of the fermions are generated through the couplings of fermions with the
scalar field (j>. The mass of a fermion is simply proportional to the coupling. If a
fermion is heavy, it is because its coupling to the scalar field is strong. Due to the
same reason, the Higgs boson couples to a fermion in proportional to its mass. Now
that we know how the Higgs boson couples to all the other particles (as well as
self-couplings), we can calculate the production cross sections of the Higgs boson and
its decay modes with good precision as a parameter of its mass.
4
If the gauge is not fixed and (2.8) is simply substituted into (2.1), there appear additional terms which
contain the kinetic term of the | -field without mass term, m^212.11 means there appears additionally
an unwanted massless scalar boson, called a "Goldstone boson."
5
The gauge group of the strong interaction is SU(3).
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3. LHC PROJECT
The Large Hadron Collider (LHC) is an accelerator under construction at European
Organization for Nuclear Research (CERN) near Geneva in Switzerland [2], It is a
circular machine assembled in a tunnel approximately 100m deep underground,
circumference of which is 27 km. Protons of 7 TeV collide on protons of the same
energy (pp collider), namely 14 TeV in proton-proton center-of-mass energy (ECM)-6
Since the Higgs mass is not predicted from the theory, the machine have to be capable
to produce Higgs particles over the wide range up to 1 TeV mass scale. Protons are
accelerated with several steps to 7 TeV: up to 0.05 GeV with a linear accelerator
(Linac), up to 1.4 GeV with Proton Synchrotron Booster (PSB), up to 26 GeV with
Proton Synchrotron (PS), up to 450 GeV with Super-Proton Synchrotron (SPS), then
up to 7 TeV with LHC. There are two general-purpose detectors called ATLAS and
CMS. A conceptual figure of the LHC complex is shown Fig.2.
FIGURE 2. An image of the LHC complex [6].
In addition to the beam energy, another important parameter of a collider is
"luminosity." It is defined as
N=£<J,
(3.1)
where, N is a number of events, £ is the luminosity, and a is a cross section. Cross
sections for interesting processes are usually small. Therefore, the luminosity has to be
large enough to get statistically significant events. The luminosity is proportional to
6
Accelerators being used for energy frontier particle physics are collider type machines. The present
world highest record of E^ is 2 TeV. The beam energy is most effectively used in collision processes.
In the fixed target case, on the other hand, most of the beam energy is just spent to boost the whole
system. If we need ECM of 14 TeV in a pp fixed target experiment, the beam energy has to be
10Q,QQQTeV!
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each beam current and a collision frequency, and inversely proportional to the beam
cross sections. Protons11in a beam are grouped into many clusters called bunches, each
of which contains 10 protons. Bunch spacing is 7.5m and beam collision occurs
every 25 ns (40 MHz). The beam shape at the interaction point is round and its
transverse size (1 sigma) is 16 |Hm. The design luminosity is 1034 cm'V1 at each
collision point. The main machine parameters are summarized in Table 1.
TABLE 1. Main parameters of LHC
Parameter
Unit
Circumference
Collision type
Beam energy
27
Proton on proton
7.0
1034
0.54
25
2.8
8.4
1232
Design luminosity
Circulating current/beam
Bunch spacing
Bending radius
Dipole field
Number of dipole magnets
km
TeV
cm-Y1
A
ns
km
T
Although the LHC ring is huge, a magnetic field has to be 8.4 Tesla in order to
bend the 7 TeV protons. Such a high field can be achieved with a NbTi
superconducting magnet operated at 1.9 K. Since the colliding beams are both protons,
two beam-channels are necessary. These two are embedded in a single cryostat. Each
coil aperture is 56 mm, and the magnet length is 14.2m.
m^mm^A
FIGURE 3. Conceptual bird's-eye view of the ATLAS detector. Approximate dimensions are 24m in
diameter, 42m in total length. The overall weight is about 7,000 tons. Air-core toroid magnets are used
in barrel and endcap regions for the muon spectrometer.
417
There are two general-purpose detectors for the LHC experiment, ATLAS (A
Toroidal LHC Apparatus, Fig.3) [3] and CMS (Compact Muon Solenoid, Fig.4) [4].
Both are gigantic detectors having cylindrical structure7 and are hermetic to capture
almost all the emerged particles.
The innermost detector surrounding the interaction region is a tracking system with
light materials to measure momenta of charged particles. Outside the tracker, there
exists an electromagnetic calorimeter system with high-atomic-number materials to
measure energy of electrons and photons. Then a hadron calorimeter system comes
next. It is made of heavy materials to measure energy of hadronic partcles, such as
pions, kaons, protons, neutrons. The outermost device is a muon spectrometer to
detect muons and measure their momenta. Since the muon interacts with materials
electromagnetically and much heavier than the electron, it can easily penetrate heavy
materials. A neutrino escapes whole detector system and cannot be detected directly,
A
i
'$$$$$$$$$13$$$$$$%
i ff|Jiiii
FIGURE 4. Conceptual bird's-eye view of the CMS detector. Approximate dimensions are 15m in
diameter, 22m in total length. The overall weight is about 12,500 tons. It is much compact compared
with ATLAS, but is heavier because of using iron yoke outside of a solenoid magnet The magnet is
superconducting one with inner diameter of 5.9m and length of 13m. It generates a 4 Tesla magnetic
field inside the coil.
7
In a collider type experiment, particles produced at the collision emerge in all directions. In a fixed
target experiment, on the other hand, the particles are boosted to the forward direction.
418
but energy carried by the neutrino causes energy imbalance in the transverse plane.
Since the detector is hermetic, total sum of the transverse energy has to be zero. In this
way, we can identify its existence and even measure its transverse energy.8 A high
energy quark or gluon is observed as a bundle of particles, called "jet."
While cross sections for interesting processes are small, typically 1 pb or less,9
proton-proton total cross section is 100 mb. It means non-interesting underling events
occur with a rate of 1 GHz at the normal luminosity of 1034 cm'V1. In addition, about
100 particles are emerged from each of such an event. Therefore radiation
environment is harsh and the detectors, including readout electronics, have to be
radiation hard. In order to select only interesting events, a sophisticated event
triggering and data-taking system is necessary. Furthermore, the large volume data
flow (500 Gbits/s) requires a huge mass storage (1 PB/year) system and gigantic
computing power (5TIPS).
4. HIGGS SEARCH
A proton is not an elementary particle, but composed of quarks and gluons. Its
static picture is a system made of three valence quarks, namely two up-quarks and one
FIGURE 5. Left: Conceptual picture inside a proton. Quarks are shown with small balls, gluons with
springs. Sea quarks as well as valence quarks exist inside a proton. These quarks and gluons are not
stable, but are crated at a certain time then annihilate soon after. Right: Parton distribution function at
Q2=10 GeV2. It shows probability density of a certain parton carrying momentum fraction x of proton,
•^ ~~ Pparton ' Pproton '
8
Why only the transverse energy? Along the beam direction, most of the energy is carried by proton
remnants and these particles escape into the beam pipe. Therefore, the longitudinal energy generally
does not balance when we use detected particles even in an event without neutrino.
9
The unit "pb" means pico (10"12) barn (10~24 cm2), namely 10"36 cm2. Similarly, "mb" is mill (10~3)
barn and "fb" is femto (1Q~15) barn.
419
down-quark. However, other types of quark-antiquark pairs, as well as gluons, are also
crated and annihilate continuously inside the proton. The quark and gluon are called
"parton" as a collective name. Each type of parton carries momentum fraction of the
proton. It is not constant but depends on Q2, which is the momentum-transfer squared.
This dynamical image of the proton and a parton distribution function at certain Q2 is
shown in Fig.5. One should note that an elementary process is a parton-parton
collision, and usually only a small fraction of the center-of-mass energy (14 TeV) is
used.
iP.
i
FIGURE 6. Main diagrams of Higgs production.
iff1
•
FIGURE 7. Higgs production cross section as a function of its mass for various processes. The unit
of Ipb (left vertical scale) is 10~36 cm2. The number of events for 105 pb"1 (right vertical scale)
corresponds to the data accumulated for about 120 net days with the nominal luminosity, 1034 cm'V1.
420
Since Higgs boson couples strongly with the heavier particles, it is produced
through top-quark (174 GeV) or weak gage bosons, W (80 GeV) or Z (91 GeV). Main
diagrams for the Higgs production are shown in Fig.6. The production cross section
through each process is shown in Fig.7 as a function of Higgs mass. For any of the
Higgs mass, "gluon-gluon fusion" is the largest cross section, and "WW and ZZ
fusion" is the next except for the Higgs mass of below 100 GeV. If the Higgs mass is
around 400 GeV, millions of events will be produced within a year.
Branching ratios of the Higgs boson is shown in Fig.8 as a function of its mass.
Above 180 GeV where real WW or ZZ decay is kinematically allowed, these decay
modes are dominant. Below 100 GeV, it predominantly decays to a b-quark pair, and
about 8% to a tau pair. Around 100 to 150 GeV, it decays to a gamma pair with about
0.1% level. Although it is a small branching ratio, this decay mode is promising
because of its clean signature.
FIGURE 8. Branching ratio of the Higgs particle as a function of its mass. Above 180 GeV, it
dominantly decays to a gauge-boson pair. The gorge at 160 GeV is due to the fact that the real WW pair
becomes possible, while the real ZZ is still not open.
We can study whether a signal process stands out or is hidden by background
processes by simulating both processes. Since background events usually dominate
over signal events, various event-selection cuts are applied in the event analysis which
are effective to reject the backgrounds while accepting the signals. Although the
signal-to-background ratio improves after the event selection, number of signal events
is also inevitably reduced. As a measure of observability of a signal process, we define
"signal significance" as Ns I ^JNB , where Ns is the number of the signal events and
NB is that of the background events. If it exceeds five, the signal process is expected
to be visible. Since the significance is approximately proportional to the square-root of
421
an integrated luminosity, we can estimate how much integrated luminosity is
necessary and how long it takes to accumulate.
Here we show some examples of how the Higgs boson can be seen for several
processes [5]. An expected invariant mass distribution of two gammas for H -» 2y is
shown in Fig.9 in a case that the Higgs mass is 120 GeV and an integrated luminosity
is 100 fb"1. It takes 120 net days and corresponds to about one calendar year to
accumulate 100 fb"1 with the nominal luminosity. A clear peak can be seen on the
smooth background slope. The signal significance becomes 6.5. For the heavier Higgs,
the decay mode, H -» ZZ —» 4£, namely Higgs decays to a ZZ pair and each
1
120
135
mw(GeV)
FIGURE 9. Expected invariant mass distribution of two gammas for MH=120 GeV, and for an
integrated luminosity of 100 fb"1. A clear peak can be seen on the smooth background slope (left). The
peak after subtraction of the background (right).
|Ldt=10fb" 1
(no K-factors)
1000
m4l (GeV)
1500
mlvjj (GeV)
FIGURE 10. Left: Expected invariant mass distribution of four leptons coming from H —» ZZ —>•
4£ for MH=300 GeV and an integrated luminosity of 10 fb-1. Right: Expected invariant mass
distribution of one lepton, one neutrino, and two jets coming from H —> WW —» Ivjj for MH=600
GeV and an integrated luminosity of 100 fb-1.
422
Z decays to a charged-lepton pair, is the most promising channel. Such a case is shown
in Fig.10 (Left) for Mn=300 GeV and an integrated luminosity of 10 fb"1. If the Higgs
mass is much heavier, we can use H —» WW -> £vjj. Distribution of invariant mass
for this mode is shown in Fig.10 (Right) for Mn=600 GeV and an integrated
luminosity of 100 fb"1.
A summary plot of the signal significance for various decay channels is shown in
Fig. 11 as a function of Higgs mass for an integrated luminosity of 100 fb"1. After one
year of LHC running with the nominal luminosity, we can discover the Higgs boson of
any mass range below 1 TeV with a high significance. We can then measure the decay
width of Higgs boson, as well as its mass, and its couplings to fermions. Through the
consistency check, we will confirm the particle as the Higgs boson predicted in the
standard model.
; ; ; : H -» yy + WH,ttH(H -» yy)
» ttH <H -> bb)
4 H -> ZZe) -» 41
a
;i;
1
.
H -*• WW° -*- Ivlv
T H -» ZZ -»- Uvv
« H -j- WW -> Ivjj
—— Total significance
10
ATLAS
|Ldt = 100fb"1
(no K-factors)
10
10
mH (GeV)
FIGURE 11. Statistical significance for various decay modes with the ATLAS detector for an
integrated luminosity of 100 fb"1.
5. SUMMARY
The origin of masses of weak gauge bosons as well as quarks and leptons is one of
the most mysterious themes in high energy physics. In the standard model, it is
explained as a result of a spontaneous symmetry breaking of the vacuum, which is
423
caused by a self-interaction of unknown complex scalar field. Three degrees of
freedom of the scalar field out of the four disappear when SU(2)L gauge is fixed, but
appear as the longitudinally polarized states of the three massive gauge bosons. The
remaining one freedom survives, which is the Higgs boson. Masses of quarks and
leptons are generated by an interaction between those fermions and the scalar field.
The LHC project is to discover the Higgs boson. Since the Higgs mass is not
predicted, the machine and detectors were designed capable of covering full rage of
the Higgs mass below 1 TeV. The machine is a pp-collider operated at ECM = 14
TeV with a nominal luminosity of 1034 cm'V1. The energy is approximately 7 times
larger than the present world record. There are two general-purpose detectors, ATLAS
and CMS. The machine and the detectors are under construction at CERN, and its first
collision is scheduled in April 2007.
Once the machine is operational as designed, the Higgs boson will be discovered
within a year. Then its property will be studied and checked whether it is consistent
with the standard model in the following years. If the Higgs is not discovered, by any
chance, it also opens a new interesting scenario, "beyond the standard model." In
either case, the LHC project will bring us deeper understanding of nature.
ACKNOWLEDGMENTS
The author greatly appreciates Prof. Nakajima for giving him an opportunity to
have a short lecture to young and motivated students. Prof. Nakajima also kindly
invited the author to attend the symposium, "Science of Super-Strong Field
Interactions" being held just after the "Shonan lectures." It covered variety of fields
from basic theoretical aspects to high-energy cosmological phenomena. Most of those
talks were new to him and were very interesting. The author appreciates Prof.
Nakajima's effort to lead the symposium as well as the Shonan lectures so successful.
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in Modern Particle Physics," John Wiley & Sons, Inc. New York, 1984, pp. 311-354.
2. LHC study group, "The Large Hadron Collider," CERN/AC/95-05(LHC), 20 October 1995.
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Large Hadron Collider at CERN," CERN/LHCC/94-43, LHCC/P2, 15 December 1994.
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6. One can get information on the LHC project via CERN Web site,
http ://welc ome.cern.ch/welcome/gateway.html.
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