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High Energy Physics

by: Horace McClure

High Energy Physics PHY 245A

Horace McClure
GPA 3.82


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This 11 page Class Notes was uploaded by Horace McClure on Tuesday September 8, 2015. The Class Notes belongs to PHY 245A at University of California - Davis taught by Staff in Fall. Since its upload, it has received 28 views. For similar materials see /class/191841/phy-245a-university-of-california-davis in Physics 2 at University of California - Davis.


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Date Created: 09/08/15
Physics 245A Winter 2006 Joe Kiskis Notes Lie groups Classical groups All the nonsingular ale invertable linear transformations of an ndimensional complex vector space form a group called the general linear group GLnC These are Lie groups The classical groups are Lie groups that are subgroups of the GLnC Special groups Let 6 with j 1 n be n basis vectors for an n dimensional vector space A general vector 3 with components 37 in this basis is written 3 ij implied sum on j n such vectors 3 i 1n determine a volume v c After a transformation there are n new vectors ekDfo and a new volume v cadn D233D To get v v for any set of vectors 313 we need 621 chm or detD 1 Transformations satisfying this condition form a group the special linear group SLnC Metrics For a more re ned structure a metric is introduced g3 3 E C or R for two vectors 3 and 3 g is nonsingular and linear on 3 g3 a31 b32 ag3 31 bg3 32 Bilinear metrics are also linear on 3 ga3i b3 3 ag3 13 bg3 3 Sesquilinear metrics are still linear on 3 but on 3 ga3 1 b3 3 ag3 3 9093299 Bilinear symmetric bilinear and g3 3 g33 Sesquilinear symmetric sesquilinear and g3 3 g33 Bilinear antisymmetric bilinear and g3 3 g33 Orthogonal groups The eld is the reals For a bilinear symmetric metric we can nd a basis with h39j MW 6 diag11 1 1With 13 1 s and n 1 s The orthogonal transformations On n R are those that preserve that form of the metric so that 927 57leng The further requirement that detD 1 gives the special orthogonal groups 50nnR Unitary groups Now the eld is the complexes A sesquilinear symmetric can also be put in the same diagonal form Then the transformations satisfying 97 glefD form the unitary group Unn and with detD 1 the special unitary group SUnn Symplectic groups The eld can be R or C The bilinear antisymmetric metrics can be nonsingular for even n and satisfy g gj The symplectic groups 512m B or C preserve a metric with this property A common choice for g is 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 Physics 245A Winter 2006 Joe Kiskis Lie groups and Lie algebras Topologically the discrete groups are of points that can be labeled with integers Now we study continuous groups like 302 and 303 or the classical matrix groups These groups are manifolds topologically locally they look like pieces of Rquot for some n The elements of the group can be labeled with a set of n real numbers For 302 n 1 for 303 n 3 for 30N n NN 12 We want to relate group structure and manifold structure ie algebra and topology The local manifold structure is seen in the tangent vector space 3ince any region of the group can be mapped to a region around the identity 6 by a group multiplication we can concentrate on the area around 6 The tangent space at the e is the Lie algebra of the group It is a vector space Consider a matrix group and put on coordinates i 1 n The ele ments of the group are then matrices Da There are many ways to choose the coordinates Let the identity have all coordinates zero D0 6 Look at the region around the identity 51 1 8D Y f 739313971 1 173 hawk 92 63an D Da e iTj1j at2 2 For different choices of coordinates there will be different T s Different will be linear combinations of each other The Lie algebra is the n dimens1onal vector space with basis Tm 1 n and vectors T539 Each element of the group near the identity an element of the algebra A change coordinates on the group G gives a change of basis in the algebra 9 So far 9 is just a vector space What makes it an algebra That is some multiplication G x G gt G For 303 Ti J and the multiplication of basis vectors is 3915 M i ijk 3 For general vectors a Jiai and b Jib ab 15Jj1aibj Jkieijkaibj ia x b J 4 Overview of particle physics The big questions of particle physics are 1 What is the universe made of 2 How is it held together We can start at ordinary distances and work our way down Following that we can also start again from ordinary distances and work up Macroscopic stuff is made of molecules Molecules are made of atoms An atom has a cloud of electrons e surrounding a nucleus Nuclei are made of protons p and neutrons n which are called nucleons The electrons are held to the nucleus by electromagnetic EM forces mainly just the Coulomb force The EM force can be described by saying that the charged particles exchange the quanta of the EM field photons y The nucleons are held together in the nucleus by strong interactions The associated eld is called the color eld It is a generalization of the EM field The quanta of the color field are called gluons g You must be wary of the jargon in high energy physics For example the word color here is just a name it has nothing to do with real colors like red and blue that we perceive directly Also some of the jargon is imaginative and some of it is just plain dumb We will see why gluons have their name shortly At this point it might seem that a satisfactory universe could be built from e p n y and g Only the strong force would need more discussion Things did not work out that way at all For reasons that are not known the universe is far richer and more interesting than that Perhaps it s to keep us doing physics and away from mischief that could be much worse Here are some things that exist but are not obviously necessary for a nice universe Antiparticles For example the antiparticle of the electron is the positron e It is just like the electron except for having the opposite charge Similarly the antiproton 7 and the antineutron IT exist Since the neutron has no charge how do the neutron and the antineutron differ The neutron as well as the proton has another kind of charge called baryon number B which is not zero The n has B21 and the i has 2 1 For the electron B20 Some particles such as the photon are truly neutral and are their own antiparticle Antiparticles seem to be required by relativity and locality causality There is also a nice particle antiparticle symmetry called TCP for time reversal charge conjugation and parity Since these things are pleasing to us we can accept antiparticles without too much worry But then we must ask why there are so many particles around and so few antiparticles There is a definite asymmetry in the quantity of matter vs antimatter in the universe There is much more hydrogen with ep than antihydro gen with e There are some interesting ideas on this but no de nite answer yet This general situation where there is a symmetric underlying law but asymmetric phenomena is a recurring theme In certain other contexts it is called spontaneous symmetry breaking 2 decay of the neutron A neutron that is not bound in a nucleus is unstable and decays with a lifetime of 886 sec via n gtpe78 Associated with this there is a new particle the neutrino In this case it s the electron s antineutrino and a new force the weak interaction 3 Quarks The strong interaction is not a simple force between nucleons The nucleons are not fundamental Each nucleon is made of three quarks So far as we know today the quarks are fundamental spin 12 fermions Two kinds of quarks are needed to make ordinary matter The label for the different kinds is called avor Of course this has nothing to do with ordinary flavor The two avors are up u and down d I told you that some of the jargon is dumb A proton is uud and a neutron is udd It is the quarks that have the color charge of the color field Each avor of quark comes in three colors red blue and green The theory of the colored quarks and the color field is called quantum chromodynamics QCD It is a generalization of the Maxwell equations which are the basic equations of electrodynamics This color force acts to glue the three quarks together to make the nucleon That is why the quanta of the color eld are called gluons Now for a little bit on units Every specialty has its own convenient units In high energy physics we tend to measure many things in energy units The convenient energy units are MeV GeV or TeV millions billions or trillions of electron volts One electron volt is the energy that an electron gains by falling through an electric potential drop of one volt 160x10 19J Now introduce the speed of light c Notice that MeVc has momentum units and MeVc2 has mass units The next step is to choose length and time units so that the speed of light c1 Then energy momentum and mass are all measured in energy units This brings to mind the famous Einstein relation Emc2 To get lengths into the picture recall another famous relation from de Broglie this time phhk In this p is the momentum h is Planck s constant k2rc is the wave number and hzhZcrc The units of h are momentum length Now choose length units so that 711 That leaves length with units of inverse momentum or equivalently inverse energy This is all very weird but very convenient once you get used to it Thus energy momentum and mass are all in MeV and length is in lMeV To get back to standard units you need to know two numbers c3x108ms and hc197MeV f ffe1mifm10 15m For example for mass 1 GeV1 GeVc2178x10 27kg and for length 1GeVhcGeV0 197x10 15m The proton and neutron masses are both close to 1 GeV The proton mass is 9383MeV and the neutron is 13MeV heavier at 9396MeV Let s return to the big questions As we understand it today the ordinary stuff of the universe is made of four matter particles There are the electron e and its neutrino ve which are called leptons and the two quarks u and d For each quark there are the three colors and for each particle there is also its antiparticle so we are really talking about 1133x216 particles This is called the first family The universe is held together by four forces For each of these forces there is an associated field and an associated quantum for that field The field quantum appears as a particle I will explain later why some of the masses are in quotes And now for the deep mystery In some extravagant impulse shopping Nature bought two more families of quarks and leptons These have the same structure as the first family The difference is that the masses are larger No one understands the reason for more than one family or the values of the masses These are active research areas Neutrino mases There is now convincing evidence that neutrinos are not massless but the masses may be as small as about 005 eV Since the six quarks and their antiparticles can be combined into bound states in many ways many particles in addition to the p and n are possible Hundreds of these have been observed One of the great accomplishments of the quark model has been to give a unified description of these many states Here is some more jargon These are names for classes of particles 1 ermions are particles for which you are allowed to put only one of them in a given state This is called Fermi Dirac statistics All the leptons and quarks are fermions 2 Bosons are particles for which you are allowed to put any number in the same state In fact putting more in the same state is favored This is called Bose Einstein statistics The photon the gluon and the other quanta of the force fields are bosons 3 sztons are the fermions that do not have strong interactions These are the e u 39c and their neutrinos 4 Hadrons are the strongly interacting particles Masons are the hadrons that are also bosons eg 313 p n and K b Baryons are the hadrons that are also fermions e g p n A A and 2 Even though there are just four interactions particle physicists dream of and work on a more unified description There has been no great progress so far However some important work has been done The Glashow Weinberg Salam theory of the electroweak interaction sort of unifies the EM and weak interactions in a way that is not particularly pretty except that it accounts for a huge body of experimental results The electroweak theory combined with QCD is referred to as the Standard Model It contains one more fundamental particle that was not listed above It is called the H iggs boson It is closely related to the W and Z and to the large masses that they have It has not yet been observed One of the biggest activities in high energy physics these days is looking for the Higgs There is no experimental result that forces us to look beyond this theory Nevertheless considerable effort is devoted to looking for more unified theories Those that unify QCD and the electroweak theories are called grand unified theories GU Ts The simplest of these is based on the gauge group SU5 which is big enough to contain the SU3xSU2xU1 of the standard model The SU5 model has many nice features It is unique in making predictions that are difficult to wiggle out of Unfortunately one of these is wrong Most GUTs including SU5 predict an unstable proton SU5 gives an actual number for the decay rate which would have been seen by now in the current ambitious experiments It has not been seen However there is new hope for the SU5 GUT in a supersymmetric version With the addition of supersymmetry there are new particles which alter the way the running couplings run The interactions unify at a higher energy scale and the proton lifetime increases enough to be out of range of current experiments Even more ambitious theories try to get gravity into the picture Recent attempts along this line are characterized by a new symmetry called supersymmetry This symmetry transforms fermions into bosons and visa versa Early versions were called supergravity models Later versions are in the context of string theory and are called superstring models In string theory the fundamental objects are not particles or fields they are strings with extent in one dimension Decays and lifetimes The truly stable not yet observed to decay particles are y v s e and p All others decay by weak EM or strong interactions Those that are truly stable and those that decay by weak or EM interactions have much longer lifetimes than they would if they could decay strongly They are called stable particles The AEAt uncertainty relation tells us that lifetimes and widths are inversely related 39ET 1 Weak decays Typical is 1 a vMeVe with a mean life 17 E 22 X 10 6 sec The picture for this is V M M e W v 13 For the 39c 17 29 X 10 sec and there are a number of poss1b111t1es T gt VTMVM 17 T gt vTeVe 18 39E gt va and others Notice that the 39c lepton can decay into the p hadron The picture is VT 17 L7 W p d Some hadrons have weak decays only Examples 8 rt auxM 1726gtlt10 sec u M n d W v M A0 a prf 17 26gtlt1010 see u u d d S u W L7 d The classic weak decay is neutron beta decay u u d d 7 886sec d u W e Ve The W and Z bosons of the weak interaction can be produced and their properties can be observed Since the W and Z are so heavy there is a large phase space and many available final states The lifetirnes are short and the widths are large W39eeve WM rW21Gev Zeee1 My rZ 2495200026GeV Note that Z 9 VV contributes to the Z width for each neutrino with mV lt m Z 2 In the three known families even though the quarks and charged leptons get heavier and heavier the neutrinos are all light A precise measurement of FZ tells the number of light neutrinos If there were more families with quarks and charged leptons too heavy to have been discovered but with light neutrinos like the known families FZ would reveal them The experiments on the Z width at SLACSLC and CERNLEP observe the process 6 e Z on resonance and show that the number of light neutrinos is three Thus if there is a fourth family it must have heavy neutnnos EM decays Here the rates are generally faster other things being equal Some examples El Vlt rt aW 17087gtlt1016 sec 20 ery 1774gtlt1020 sec 7 n n lt 77 0 18 n a 371 17 z 10 see Note that there is no final state photon in this case Strong decays These are faster still Examples perm F 153MeV A strong baryon decay is AH a 771 With F 110M6V If a particle can decay strongly it will and will be gone before any possible EM or weak decays have a significant chance to work If no strong decays are possible because the particle does not have strong interactions or because the only available nal states have a different I spin then an EM decay may happen If that is not possible then finally the weak decay may be seen Strong decays conserve everything EM decays violate I spin Weak decays can violate I spin all other flavor quantum numbers strangeness cha1m C CP and probably T but not CPT Some things like EM charge and energy momentum seem to be conserved for good reason Some others like baryon number and lepton number seem to be conserved but for no apparent reason Cross sections These follow a similar pattem Strong interaction cross sections are large For n1 0T 30mb 1b lbarn 10 24 EM cross sections are smaller For 14 OT 100111 2 Cm Weak cross sections are very small For vp O T 10 41 ml at EV lGeV Probing sh01t distances requires high energy and large expensive accelerators The ultimate high energy experiment has already been done It was the big bang Our universe is the final state from that event Cosmology and general relativity tell us that the temperature T of the universe is related to the time t after the big bang by T 0C l at early times So at sufficiently early times pa1ticle energies were arbitraIily large For example at l 10 12 sec CT 5 lTeV There is now a lot of interesting work combining particle physics and astrophysics Dark matter and dark energy Return now to ordinary distance scales and look outward to larger astrophysical and cosmological scales much larger that our ordinary experience From relatively recent observations several shocking conclusions have been drawn 1 Most of the ordinary matter protons neutrons electrons etc in the universe is not in stars and is hidden from us Altogether the ordinary matter makes up only 5 or so of the energy in the universe The part that we see in stars is only a small fraction of that 2 Most of the matter in the universe is not ordinary and we do not know what it is By matter we mean stuff that clumps up due to gravitation However there are some plausible candidates among the postulated but undiscovered particles A leading contender is a neutral particle in a supersymmetric theory The non ordinary dark matter is about 25 3 Most of the energy in the universe about 70 is not matter but some kind of stuff that is spread uniformly over space like a vacuum expectation value of a quantum field or a cosmological constant To fit the observations of an accelerating expansion of the universe it needs to have negative pressure Thus we now know that we know far less than we though we did only a few years ago about the answer to the question What is the universe made of At the moment we can account for less than 1 of it There is plenty of work to be done


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