Phys 557 – Lecture 2

Units and Sizes –

Recall from freshman physics that one of the most confusing issues in the introductory course is the question of units. For quantities with units (which I will call “dimensionfull” quantities) the specific size will depend on the choice of units. For example, in (old) English units I am nearly six feet tall, while in by now standard (except in the US) MKS units I am just 2 meters tall. This is clearly a confusing situation. Further, the MKS system exhibits three fundamental dimensionfull quantities, length (m), mass (kg) and time (s). The actual magnitudes of the standard units were chosen to correspond to human scales (e.g., the size of a king). These choices are, of course, unnatural for particle physics applications. For example, the mass of a proton is 1.67 x 10^-27 kg while the “size” of a proton is measured in fermi’s (1 fm = 10^-15 m), not meters. Likewise the lifetime of a typical particle that decays via the strong interactions is of order 10^-23 s, the time for light to travel across a particle of size a fm.

Further, since particle physics is “naturally” relativistic and quantum mechanical, we would like to choose units so that the speed of light c, which is large in MKS units, and h, which is small in MKS units, are both of order 1. It turns out we can address all of the above issues by defining a new set of “particle physics units” such that both c and h are exactly to 1!!!! In the process we have reduced the number of types of dimensionfull quantities to 1. We have

Thus time now has the same units as distance. Likewise mass and energy have the same units and both go like 1/distance or 1/time. In these new units the mass of the proton is essentially 1 GeV (0.938 GeV/c²). We also have one fm equal to 1/(197 MeV) ~ 1/(200 MeV) = 1/(0.2 GeV). It is typical in particle physics to express nearly all dimensionfull quantities in terms of the “natural particle” unit of GeV. A list of useful relationships is provided in the following table (and also on the web).

Units

Sizes (Ignore factors of 2)

As suggested by the table the areas of the particles (the cross sections for scattering) are typically measured in millibarnes (mb).

You will not be surprised to learn that with only one fundamental type of dimensionfull unit it is easy to define dimensionless ratios. In many instances these are the simplest quantities to understand in particle physics. As we will see, the gauge field theories that underlie the Standard Model contain, in fact, no dimensionfull parameters in the classical (and symmetric) limit. Consider, for example, the case of QED. In MKS units the electric charge of an electron has size e ~ 1.6 x 10^-19 C. Thus one often sees the fine structure constant as (e₀ is the electric permittivity of the vacuum with value 8.85 x 10^-12 fm)

We will use Heavyside-Lorentz units where m₀ (the magnetic permeability of the vacuum) and e₀ are both 1. In our units charges are pure (dimensionless) quantities

In our efforts to understand the fundamental interactions of particle physics the really interesting (and difficult to explain) quantities are the small number of dimensionfull quantities. Examples include L_QCD, the fundamental dimensionfull parameter characterizing the strong interaction (and induced by the radiative corrections), G_F (the Fermi constant) or M_W (the mass of the W boson), the dimensionfull parameters that characterize the weak interactions and G_N, Newton’s constant, that characterizes the gravitational interaction.
Vocabulary of the Standard Model (and beyond):

(see also the PDG webpage http://pdg.lbl.gov/2002/contents_sports.html# stanmodeletc)

The big picture: First the interactions -

Interaction	Observed	Strength	Range	Carrier
Strong QCD	nuclear forces	~1	~10^-15 m	Gluon	Standard Model
EM	Atomic systems	~10^-2	1/r^2	Photon	Standard Model
Weak	Decays	~10^-5	~10^-18 m	W^±, Z	Standard Model
Gravity	Astronomy	~10^-39	1/r^2	Graviton	“SUSY Strings”

+ the Higgs sector that is (presumably) responsible for the generation of masses but that has not yet been directly detected.

· Matter particles (fields) - The basic building blocks of matter are spin ½ fermions. As we learn from relativistic quantum mechanics (see chapter 4 in Rolnick) spin ½ particles (and antiparticles) satisfy a matrix form, linear equation of motion.

leptons – electron (e), muon (m), tau (t) and their corresponding neutrinos (i.e., the ones they are coupled to by the weak interaction, n_e , n_m , n_t) Þ 6 total plus antiparticles (recall that the existence of antiparticles is another result of requiring a relativistic theory of spinor particles). Leptons interact via EM, weak interactions and gravity. Since the neutrinos are electrically neutral and the weak coupling is only to the left-handed component (right handed for the anti-neutrino), the simplest version of the SM has massless neutrinos only in the left-handed chiral state. There is now evidence that at least two of the neutrinos have nonzero masses.

quarks – up, down, charm, strange, top, bottom Þ 6 “flavor” types. All come in 3 “colors” Þ 18 total plus antiparticles. Quarks interact via strong interactions, EM, weak interactions and gravity. All quarks have nonzero masses and the top quark is VERY heavy (~ 175 times the mass of a proton). The quarks bind together to form the directly observed strongly interacting particles – the hadrons. These fall into two classes: ½ integer spin baryons, integer spin mesons. The baryons are composed of quark triplets (), while the mesons contain quark-antiquark pairs (). These multi-quark bound states are color singlets, i.e., zero total color charge.

The quarks and leptons seem to aggregate naturally into three families or generations – (u, d, e, n_e), (c, s, m, n_m), (t, b, t, n_t), which seem to be identical reproductions of each other except for their masses. But why 3 generations? This seems to be a deep question, perhaps related to our very existence (in the sense that it seems to be related to CP violation and thus the abundance of baryons over anti-baryons).

· Gauge particles (fields) – as we will learn (see chapters 7 & 8 in Rolnick), the existence of a local gauge invariance requires the theory to have an associated massless vector (spin 1) field that couples to the charge associated with the symmetry. These particles mediate the interactions and are the “glue” that holds the matter fields together in the universe.

Photon (g) – the gauge boson that couples to the electric charge of electromagnetism. The corresponding U(1) symmetry is unbroken and so the photon is massless. All charged particles correspond to 1-D fundamental (singlet) representations of U(1). U(1) is an Abelian symmetry group and the photon itself is electrically neutral.

Weak bosons (W^±, Z⁰) – the 3 gauge bosons that couple to the weak charge (responsible for radioactive decays). The corresponding symmetry is a spontaneously broken SU(2) and the observed weak bosons all have masses of order 100 GeV. Pair-wise the quarks and leptons are members of 2-D fundamental (doublet) representations of SU(2), e.g.,

The weak bosons are in the 3-D adjoint representation (modulo issues of mixing with the U(1) of EM). SU(2) is a non-Abelian symmetry group and the weak bosons have weak charge (they couple to each other). Strictly, the W bosons couple only to the left handed components of the quarks and leptons, with the right handed components appearing as SU(2) singlets.

Gluons (the Quantum ChromoDynamics bosons – g) – the 8 gauge bosons that couple to the “color” charge of the QCD strong interaction. The corresponding SU(3) symmetry is unbroken and so the gluons are massless. The three colors of each flavor quark form 3-D fundamental (triplet) representations of SU(3), while the gluons are in the 8-D adjoint (octet) representation. SU(3) is also a non-Abelian symmetry group so that the gluons carry color charge and couple to each other.

• Higgs Boson (h) – in the simplest version the Higgs boson is described by a complex doublet (under the weak SU(2)) of scalar (spin 0) fields. Of the 4 degrees of freedom, one of the electrically neutral components acquires a “vacuum expectation value”. Thus in the ground state of the universe this field has a (constant) nonzero value everywhere. Its couplings to the weak bosons, quarks and leptons provide masses (think of trying to walk through water) and “spontaneously” breaks the SU(2) symmetry. The other three Higgs degrees of freedom (of electric charge ±1 and 0) are “eaten” during the symmetry breaking process to “become” the longitudinal components of the W^± and Z⁰. In the unbroken phase of the theory, the massless weak bosons, like the photon, exhibit only transverse polarizations. There were hints of a Higgs boson at the LEP machine CERN at ~ 114 GeV two years ago, before the accelerator was shut down to make way for the LHC. We must now wait for confirmation from either Run II at the Tevatron (now in progress at Fermilab) or from the experiments at the LHC (at CERN), which will require more patience as physics will begin only in about 2007.
  
  For a picture of the spontaneous symmetry breaking process think of a potential with the shape of the bottom of a wine bottle (or a Mexican hat – or a hill surrounded by a circular valley). Put a ball at the center of the configuration at the top of the central hill. This situation is rotationally symmetrical about the axis of the hill, which passes through the position of the ball, but is not the lowest energy state. If a tiny earthquake shakes the hill, the ball will tend to role down into the valley. A prior, the ball is equally likely to role in any direction because of the initial symmetry of the problem. When the ball stops rolling (there is tall grass that slows the ball down), it will be somewhere in the valley. We are now in a lower energy configuration but we have spontaneously broken the underlying rotational symmetry!

Summary table (PDG numbers) for the particles:

Particle	Spin	QEM	SU(2)L Weak*	SU(3)QCD	Mass
Electron, e	½	-1	LH-, RH-		0.510998902 ±0.000000021 MeV
ne	½	0	LH- (RH-)**		< 2.8 eV
Muon, m	½	-1	LH-, RH-		105.6583568±0.0000052 MeV
nm	½	0	LH- (RH-)**		< 0.17 MeV
Tau, t	½	-1	LH-, RH-		MeV
nt	½	0	LH- (RH-)**		< 18.2 MeV
u quark	½	2/3	LH-, RH-		1.5 to 4.5 MeV
d quark	½	-1/3	LH-, RH-		5 to 8.5 MeV
c quark	½	2/3	LH-, RH-		1.0 to 1.4 GeV
s quark	½	-1/3	LH-, RH-		80 to 155 MeV
t quark	½	2/3	LH-, RH-		174.3 ± 3.2 ± 4.0 GeV
b quark	½	-1/3	LH-, RH-		4.0 to 4.5 GeV
Photon, g	1	0	***		0 (< 2x10^-16 eV)
Z^0	1	0	***		91.1876 ± 0.0021 GeV
W^±	1	±1			80.451 ± 0.033 GeV
Gluon, g	1	0			0
Higgs, h	0	0			> 107 GeV

* To understand the multiplets of the SU(2) of the weak interactions we should discuss the concept of “mixing”. If we have two basis sets, e.g., one defined by the mass eigenstates and one defined by the SU(2) interactions, we can define mixing between the two. This is just what happens for the weak interactions. The lower member of the SU(2) doublet, in which the u quark is the upper member, is not the d quark mass eigenstate but rather a mixture of the d and s quarks (, in terms of the Cabibbo angle q_C). This allows the strange quark to decay into the u quark by emitting a virtual W^-. The full mixing structure is described by the 3x3 CKM (Cabibbo-Kobayashi-Maskawa) quark mixing matrix. Current experiments indicate that neutrinos exhibit oscillations (e.g., n_m® n_t). Thus there must be similar mixing for the neutrinos and at least two of the neutrinos (and probably all) must have nonzero masses (in order to define the second basis set).

** If the neutrinos are ordinary massive Dirac fermions, but with extremely small masses, then they will have RH components like the charged leptons that do not participate in the Weak interactions. The neutrinos could also be Majorana fermions, i.e., the neutrinos are their own antiparticles. In that case there is no extra Weak singlet component. There are just the LH neutrino and the RH “antineutrino”, both of which are members of weak doublets.

*** Strictly speaking the physical Z⁰ and photon are mixtures of the neutral gauge bosons of the underlying unbroken U(1) and SU(2) symmetries.

To understand several features of the structure of the Standard Model, especially the strong interactions, we must understand at least qualitatively the ideas of renormalization, running couplings, dimensional transmutation, infrared slavery and asymptotic freedom. As suggested in chapter 4 of Rolnick (see especially Fig. 4.3) the way we evaluate the various contributions that arise in relativistic quantum mechanics and relativistic quantum field theory essentially guarantees that individual contributions, defined by a specific Feynman diagram, will often be infinite. However, the sum over all contributions (Feynman diagrams) will be finite if we consider a real physical quantity. We view the (finite) physical quantities as being calculated as the result of subtracting two infinite quantities, or multiplying infinity times zero, finding a finite answer in both cases. The formal structure of this remarkable procedure is typically codified by the “renormalization group”. The resulting behavior of quantities like coupling constants, i.e., the effective charge of a particle for a specific interaction is somewhat surprising and it is essential that we establish some understanding of this issue. We will come back to this in more detail later but for now we need some of the results.

Running Couplings – we know from relativistic quantum field theory that what we think of as the vacuum is not really empty but filled with virtual particle – antiparticle pairs that are constantly forming and then annihilating. In the presence of a real charge these pairs will be polarized and influence the effective charge that we observe. (Recall classical screening effects in EM and the permittivity of the vacuum, e₀, mentioned above.) Thus when we talk about the charge of a particle, or better, the strength of its coupling to the appropriate gauge field, we must be careful to specify the “size” of the particle we are talking about (i.e., the effective volume inside of which we are measuring the net charge). This size will be determined by the Compton wavelength of the gauge particle that is to be scattered by the charge (essentially the volume over which the interaction is coherent). In our units this is just 1/momentum of the gauge particle. Since we will typically work in momentum space, we select the “renormalization scale” m to have units of GeV and refer to the coupling (squared) at the scale m, a(m). In the gauge theories of interest the coupling varies with m - it runs! The dimensionless value of the coupling is thus related to a dimensionfull value for m. This process of relating a dimensionless quantity with a dimensionfull one is called dimensional transmutation.

With the picture to the left in mind, it is reasonable that the short distance (large m) EM coupling is larger (less screened) than the coupling observed at long distance. In fact, the coupling is so weak at long distances that it essentially stops changing. The “infrared fixed point” result is the usual 1/137.035999…. we are accustomed to in classical EM. Less well known is the result that the EM coupling becomes arbitrarily strong at short distances.

This remarkable behavior is actually observed in the data. The linear relationship between 1/a and ln(m) suggested in the figure is correct for small coupling.

For the case of the strong interactions, QCD, there is a similar renormalization effect complicated by the facts that there are 3 colors (and 3 anti-colors) carried by the quarks (and anti-quarks) and that the gluons themselves carry color charge. The vacuum polarization due to the virtual quarks-antiquark pairs again serves to screen color charges, just like in the U(1) case, but the effect of the virtual gluons is to anti-screen (i.e., to enhance the charge inside the volume rather than hide it)!! (I don’t know any slick way to draw a picture illustrating this.) As long as there are not too many kinds of quarks (and 6 flavors is small enough) the gluons determine the sign of the renormalization effect and the strong coupling decreases as the scale m increases (asymptotic freedom) and increases as m decreases (infrared slavery). Note also that the latter point means that there is no saturation effect at long distances and the strong interactions are confining in the soft limit momentum, long distance limit – the quarks never get out of hadrons! The particles separated by long distances must be color singlets (i.e., no net color charges). As we will learn, we can make a color singlet from a quark and antiquark or 3 quarks, which explains the observed mesons and baryons, respectively. The fact that a_QCD reaches the value 1 around a scale of 250 MeV connects nicely to the observation that hadrons are of size 1 fermi.

The concept of running coupling constants suggests the related concept of Grand Unification (see chapters 22.2 in Peskin and Schroeder or 17.2 in Rolnick). Maybe at large scales (short distances) an even larger symmetry group obtains that includes the observed U(1), SU(2) and SU(3) symmetries, e.g., SU(5) or SO(10). Thus, as suggested by the figure, the 3 couplings will converge to a single value at this large scale. (As we will discuss later, we know that something like this does happen in the unification of the electro-weak theories. Note that the physical EM coupling is a linear combination of the couplings 1 and 2 in the figure.) In the Grand Unified Theory (GUT) the leptons and quarks are in the same representation of the underlying symmetry group. Thus there must be a symmetry generator that transforms quarks into leptons and allow protons to decay (into mesons and leptons). This effect has not yet been observed (hence the lower limit on the GUT scale) but the idea of Grand Unification remains extremely attractive. It also serves to illustrate the very important idea that the physics can (will?) change as we reach larger energies with our accelerators. We should view the Standard Model as an effective field theory that is only known to be valid up to 1 TeV. New degrees of freedom are likely to be relevant at larger energies.

And Beyond

The Standard Model does not explain the value of ~ 1 TeV for the vacuum expectation value of the Higgs (i.e., the quadratic divergences in the scalar sector suggest that the appropriate scale should be the Grand Unification scale or above) nor the role of gravity (i.e., how does it work quantum mechanically). The fact that gravity seems to have a natural scale (unlike the U(1), SU(2) and SU(3) gauge theories of the EM, weak and strong interactions, which exhibit no intrinsic scale – although renormalization introduces one), the Planck scale at 10¹⁶ TeV that is very much larger than the Higgs scale is referred to as the “hierarchy problem”.

· Gravitons & SUSY – the expected quanta of quantum gravity are spin 2 particles (and will be hunted at LIGO in eastern Washington and Louisiana). Including spin 2 fields in the theory provides a natural motivation for supersymmetry (SUSY), the symmetry that mixes particles of different spin. This symmetry is invoked to try to relate the graviton to the particles we do see (all with lower spin) and also to offer a possible explanation of how to get from the Planck scale to the TeV scale. In the supersymmetric limit every particle we see has a degenerate partner differing in spin by ½ - scalar squarks and sleptons, spin ½ photinos, winos (charginos), zinos (neutralinos), gluinos and Higgsinos, and the spin 3/2 gravitino. We ain’t seem them yet (supersymmetry must be a broken symmetry) but maybe soon!!?? [Note that the divergences from boson loops cancel the divergences from the SUSY related fermion loops “fixing” the divergence problems in the scalar Higgs sector notes above.]

· Strings – since there are so many “fundamental” particles it would be more elegant if the many particles we see are excitations of the same fundamental object – the string. Just like a violin string the fundamental string can become excited and correspond to a state with greater mass and different quantum numbers, the various fundamental particles of the Standard Model. Since the fundamental objects in string theory are extended objects (in 1-D), the short distance (UV) properties of string theory are much less singular than usual field theory (of point particles). Also, since we want both fermionic and bosonic excitations, it is not surprising that the desired string theory (SuperString theory) exhibits supersymmetry. Even though it is not yet known how to calculate scattering amplitudes, masses, etc. in the context of string theory, string theory does tell us some things (assuming that it is the correct underlying theory). For example, we know that supersymmetry must be relevant at some scale for string theories to work and that there are more than 3 spatial dimensions.

· Extra dimensions – If string theories are correct, the universe really has 1 time and 10 (or 9) spatial dimensions (depending on some details of interpretation). The extra dimensions are thought to be currently unobserved either because they are compact (curled up) on a tiny scale (e.g., 10^-35 m) and our microscopes cannot see into them, or because our usual probes do not propagate in the extra dimensions, even if they are extended. This second remarkable scenario can be motivated by the following observation. The usual matter fields and gauge fields are open strings (i.e., with ends), while the gravitons are closed strings (loops). String theory suggests that the ends of the open strings (where the usual charges are located) are confined to regions of lower dimensional – called branes (from membranes) or more carefully D-branes (for Dirichlet-branes). So maybe the matter and interactions of the Standard Model are confined to the 4-D brane (a 3-brane) we know and love and only gravity probes the rest of space.

Recently interest (see the August 2000 Scientific American, page 62) has focused on the possibility that there might exist n extra compact (size R), but not truly tiny (perhaps as large as R ~ 1 mm) dimensions in which only gravity propagates. The usual gravitational potential would then take the form

Thus it might be that the usual Newton’s constant, , appears to define a very large scale (the Planck scale ~ 10¹⁹ GeV) while the true gravitational constant, , defines a scale as small as 1 TeV for the correct values of R and n. This would eliminate the hierarchy problem! Our experimental colleagues, Adelberger, Heckel and Gundlach are avidly pursuing this possibility (see the September 2000 Physics Today, page 22).

Question: Since the gravitational potential defined above is more singular at short distances than the usual potential, is it possible that such gravitational interactions would be evident in atomic or particle interactions? To truly analyze the situation at short distances requires a quantum theory of gravity. However, we can check semi-classically to see at what interaction distance gravity of this form becomes numerically relevant. To proceed we will compare the short distance form above with the usual electrical potential applied to the system of a proton and an electron (a hydrogen atom). Thus we want to find r₀ such that

Solving this in the case n = 2, R = 1 mm (take m_p ~ 1 GeV, m_e ~ 0.5 x 10^-3 GeV), yields

I interpret this to suggest that gravity will still not be significant in existing atomic or particle physics measurements, which are not sensitive to physics at this scale. However as we heard from Scott Thomas at the 9/30/02 Colloquium, gravity may be relevant at the LHC. To properly analyze the physics at this small scale will require something better than our semi-classical theory of gravity.