1. 16 February 2011Modern Physics III Lecture 61 1 Modern Physics for Frommies III A Universe of Leptons, Quarks and Bosons; the Standard Model of Elementary Particles Lecture 6 Fromm Institute for Lifelong Learning, University of San Francisco

2. 16 February 2011Modern Physics III Lecture 6216 February 2011Modern Physics III Lecture 62 Agenda Administrative Matters Patterns and Symmetries in Nature (con.) Isospin and the Weak Interaction Color and the Strong Interaction Broken Symmetry

3. 16 February 2011Modern Physics III Lecture 63 Administrative Matters Full schedule of colloquia is posted on the Wiki and should be posted in Fromm Hall. Next colloquium is in March A list of popular books pertaining to Elementary Particle Physics is posted on the Wiki. It has been updated this week. There are corrections to the slides for last week’s lecture which are posted.

4. 16 February 2011Modern Physics III Lecture 64Modern Physics III Lecture 54 Rx for invoking a symmetry group at a vertex: 1)Identify or postulate a set of N fermions that are observed, or expected, to act as a ‘fundamental’ multiplet. N is the dimension of a symmetry group G, and the set of fermions is an N-plet under G. 2)An operation from G when applied to a member of the N-plet transmutes it into another member. 3)Every transmutation is interpreted as being due to the emission or absorption of a field boson (aka a gauge boson). e.g. e is a fermion that cannot be changed into something else electromagnetically  it is a singlet under some 1-D group, call it U(1) There is only one boson associated with the group, . Note that  is itself uncharged.

5. 16 February 2011Modern Physics III Lecture 655 How can a symmetry produce a force field? Take a table cloth or a square piece of Al foil. Pretend that the material extends to infinity. Now rotate the cloth thru some arbitrary angle Any piece of the cloth appears the same as before. The cloth is invariant under global rotations Special relativity does not allow the simultaneous rotation of the entire universe. Only local symmetry rotations are allowed, that is a symmetry where the amount of rotation varies from event to event in space time

6. 16 February 2011Modern Physics III Lecture 66Modern Physics III Lecture 56 Secure the edges of the cloth. Place a finger near the center and give an arbitrary twist The cloth that was under the finger still appears the same. Local invariance. A spray of wrinkles radiates outward from the twisted area. The local twist cannot be connected smoothly with the undisturbed cloth at large distances. This, and any other, local symmetry creates a field. The wrinkles are analogous to the field lines or lines of force as they were known in pre-quantum days. In a Feynman diagram, the wrinkles appear as gauge bosons. Back to the EM field: Gauge groups act like rotations. Simplest case is rotation in a plane.

7. 16 February 2011Modern Physics III Lecture 67 What quantity (other than position looks like a rotation angle? How about the phase (  Suppose that phase symmetry is a local symmetry is a general property of Nature, which if applied locally, produces field wrinkles. Reinterpret Feynman diagram for e - moving through space. Direction and → momentum vector, f → energy Absolute  cannot be measured, only  s are observable. Probably  universe is symmetric under global phase changes but special relativity doecn’t allow application of global symmetries. Our e - should be invariant under local  s which produce wrinkles in space-time. Photons carry the wrinkles Gauge twist Gauge untwist Gauge wrinkle

8. 16 February 2011Modern Physics III Lecture 68 In QED Can we try to apply this to other forces? This  is a simple number as it looks like rotation in a plane, group U(1) and the photon  is a simple beast. The gauge symmetry of the photon is exact.  is proportional to the field strength, E  As E   → 0,  → 0 For a massive boson, E  → mc 2. A massive particle cannot be turned off “gracefully”.   is a single number, it is Abelian. It carries no memory of the way the  was created (no information about vertex coupling).   does not carry a charge. Other forces might be harder and more complicated

9. 16 February 2011Modern Physics III Lecture 69 Recall Rx for invoking a symmetry group at a vertex: 1)Identify or postulate a set of N fermions that are observed, or expected, to act as a ‘fundamental’ multiplet. N is the dimension of a symmetry group G, and the set of fermions is an N-plet under G. 2)An operation from G when applied to a member of the N-plet transmutes it into another member. 3)Every transmutation is interpreted as being due to the emission or absorption of a field boson (aka a gauge boson). Can we find sets of fermions that resemble each other very closely in some respects? How about the proton and the neutron? m p  m n (0.14%) and the strong forces between pp, np and nn are practically identical

10. 16 February 2011Modern Physics III Lecture 610 We’ve talked about this before as strong and weak isospin Treat n and p as different states of the nucleon. In “quantum mechanics language” we see the nucleon amplitude, N, as a superposition of p and n amplitudes  Pure p Pure n  is called the mixing angle

11. 16 February 2011Modern Physics III Lecture 611Modern Physics III Lecture 511 Mixing angle: Consider the p, n doublet, Isospin mixing angle 0°0° 60° 120° 180° Superposition allows us to write any quantum state as a linear combination of other states To maintain normalzation and to ensure that superposition of the combination state with itself yields the combination itself, we choose to write the combination as Thus the symmetry behaves like a rotation with a mixing angle  A symmetry acting on an N-plet behaves like a rotation in some abstract N-space

12. 16 February 2011Modern Physics III Lecture 612 We don’t observe any other particles resembling p and n as closely as they resemble each other.  N has precisely 2 states: it is a doublet under the isospin group.  I-spin group is 2-D  the group is non- Abelian. Recall rotations of dice and Rubik’s cube. (?) internal degree of freedom of gauge boson. A, B, C symmetry rotations (?) is associated with the charge of the gauge boson. Yang-Mills Fields: Ignore  q and  m so isospin symmetry is exact. Again relativity allows only local symmetry. Global would be unobservable anyway For EM: e →(twist) → e + 

13. 16 February 2011Modern Physics III Lecture 613 For the nulcleon: 4 possible transformations (1) p → (twist) → p + b 1 (2) n → (twist) → n + b 2 (3) p → (twist) → n + b 3 (4) n → (twist) → p + b 4 b for boson Could be caused by same boson 3 different field quanta: general property for N > 1 N-plet with local symmetry  (N 2 – 1) gauge bosons Yang-Mills field or special unitary group of 2 dimensions, SU(2) Non-Abelian means order of twists counts  memory of vertex interaction  Y-M boson can carry isospin charge. n, p have  q  Some Y-M bosons carry electric charge as well Y-M bosons can interact amongst themselves

14. 16 February 2011Modern Physics III Lecture 614 A RED HERRING: Set of 3 candidates: the pions, an I-spin triplet p p  n  n  p   n pn Or by inverting the time order of the   p n   Nucleon state changes  has right mass for Compton wave length to match range of nuclear forces

15. 16 February 2011Modern Physics III Lecture 615 Problems: Recall, exact local U(1)  massless, spin 1 (vector) boson [the table cloth is actually an analogy to 4-D space time so the simplest possible field quantum is spin 1] We assumed exact local SU(2) symmetry   should be a massless vector boson. Unlike  is not stable, Long enough for nuclear force by factor of 10 7 but why the 0,  difference Answer:  is not truly “elementary”, it is composed of a pair

16. 16 February 2011Modern Physics III Lecture 616 The Color Field: If the  is composite of 2 components, one spin up and one spin down 12 Composite => decay no longer unexpected Still have  vs. 0 difference to explain Suppose  composite => baryons are also composite What is the fermion multiplet for baryons? Why can’t we see the constituents? Turns out N = 3 is simplest possible case and luckily it appears to work. Basic triplet of fermions are of course the quarks and the symmetry group is SU(3)

17. 16 February 2011Modern Physics III Lecture 617 EM, U(1): 1 charge (-) [thanks Ben] 1 anticharge (+) Su(3): (N 2 -1) = 8 massless vector bosons (gluons) Non-Abelian => gluons interact amongst themselves => gluon emission can change quark color just like SU(2) bosons shange isospin charge. We have never seen color directly => Gluons are exchanged in such a way as to paint the world white Simplest ways to make white (color singlets):

18. 16 February 2011Modern Physics III Lecture 618 Now the  exchange nuclear force diagram looks like time If we stop here we can only make one baryon, q R + q B + q G Even at low energies we have 2 (p,n). How do we make more baryons?

19. 16 February 2011Modern Physics III Lecture 619 We need to add an additional degree of freedom or quantum number, call it flavor. p charged n neutral  u and d have different electrical charges Baryons being a white combination of 3 quarks  quark electrical charges are integral multiples of 1/3. Suspicion: Does ‘whiteness’ and integral electric sharge being the same  a deeper unity between color and electric charge? What hadrons can we expect to make? Depends on the number of quark flavors. We know of 6: u, d, s, c, b and t. There are arguments that these 6 are all the quarks there are.

20. 16 February 2011Modern Physics III Lecture 620 The pion revisited:  Need 2  s to conserve momentum u  10 -16 sec EM time scale. Nuclear time scale is  10 -23 sec. What about   ? We need a small amplitude for processes like (small so that   10 -8 sec )

21. 16 February 2011Modern Physics III Lecture 621 Weak Decay:    need small amplitudes  Small mixing angle, i.e, u acts like d small part of the time   long What symmetry?SU(2) if no mixing at all, like n, p doublet. Apply as local gauge symmetry → triplet of gauge bosons Then we could have a vertex like and the pion can decay as (helicity suppressed)

22. 16 February 2011Modern Physics III Lecture 622 u  W + l+l+ l Again branching ratio  1.2 x 10 -4 ____________________________________________________ New SU(2) doublets Quark-Lepton Families:

23. 16 February 2011Modern Physics III Lecture 623 Broken and Hidden Symmetries: Claim: Weak interactions (WI) are a force due to local SU(2) symmetry Pro: Suitable fermion multiplets, Suitable gauge bosons Circumstantial evidence – other 3 known forces arise from local symmetries. EM U(1) Color SU(3) Gravity Lorentz group (not a quantum field theory) Con: We were deceived before when trying to apply isospin to SI --------Weak bosons are not massless.

24. 16 February 2011Modern Physics III Lecture 624 We can make a massless particle disappear gracefully as follows m = 0 For a massive particle, we have a minimum energy mc 2. The particle does nor disappear gracefully. Lorentz symmetry and exact local gauge symmetry can only apply simultaneously in the case where the gauge bosons are massless. A massive particle always has energy, even when motionless

25. 16 February 2011Modern Physics III Lecture 625 Exact Lorentz invariance => no preferred direction in space-time. But, don’t we provide a preferred direction when we “fix the gauge”? A choice must be made or we can’t calculate anything., e.g. We must specify North on a blueprint or the builder can’t build the house where we want it Once we fix the gauge, we have specified a preferred vector and violated Lorentz symmetry ‘though the back door”. This is an apparent Lorentz violation, the exact symmetry still exists, hidden by gauge fixing.

26. 16 February 2011Modern Physics III Lecture 626 Another approach: Our old friend the photon m = 0 => EM range is infinite OK, we can see light that has travelled billions of light years. This, of course is in a vacuum. Suppose we make the light travel through a material, in particular a conductor. The light is quickly attenuated and becomes indetectable. Its range is very short. The  acts as if it has acquired a mass The cumulative screening of the conduction electros acts to make the  appear massive  An “effective mass” This effective mass depends on the photon’s environment. Now suppose the whole universe is a conductor. Then, the photon would always appear to behave as though it had a mass. We would never know that the photon is, in truth, massless. Reminiscent of the Yukawa potential

27. 16 February 2011Modern Physics III Lecture 6 27 Can we introduce something into the vacuum to explain the observed masses of the W and Z bosons? A field which interacts with the W and Z but not with  or the gluons, i.e. this field must interact via the week isospin charge. Regardless of whether the masses are real or effective (generated by screening) we still have troubles with our gauge theories. Hidden Symmetry: Suppose several hundred cold, naked, teen aged boys are presented with a choice of four different styles of free clothing. The situation is symmetric with respect to the choice of styles and none of the boys, fearing uncoolness, dares to pick a style, They just stand around and freeze.

28. 16 February 2011Modern Physics III Lecture 628 Finally, one brave soul, on the edge of hypothermia, grabs a particular style and puts it on. His buddys quickly follow suit. The symmetry has been broken Repeat the exercise with a different group of boys. A different brave soul might very well grab a different style triggering a different final result.. The overall symmetry of the choice of clothing styles was not broken by the capricious selection of one style but was hidden by the choice. Is there a way to introduce the masses of the W and Z bosons in such a way that the mass terms reflect a hidden rather than broken symmetry There are additional compensating terms that allow this to be done

29. 16 February 2011Modern Physics III Lecture 629 Recall that we want to patch up the wrinkling caused by our local twist at large distances. We must use all degrees of freedom to do this. Massless particle: Moves at c, transverse polarization is forbidden, helicity = + 1 Massive particle: Velocity < c, transverse polarization allowed, has 2 more degrees of freedom. Transverse polarization ( helicity = 0) has 2 possible spin orientations at right angles. We can make a massive gauge boson from a massless one if we can borrow 2 degrees of freedom from somewhere We want to arrange things so that with every gauge boson we produce 2 scalar (spin = 0) particles. (massless spin-1) + 2 x (spin-0) = (massive spin-1)

30. 16 February 2011Modern Physics III Lecture 630 Scalar Higgs doublet Pick one of the 2, say lower, and let it pervade the cosmos as a sea of virtual fluctuations. We can’t see this uniform presense just as we can’t see EM vacuum fluctuations Interact with W and Z and each other Hypothesize that this pervasive background of  lower exerts a “drag” force on anything it interacts with, giving mass to the W and Z We have hidden the symmetry of the doublet by picking lower over upper. The symmetry appears to be broken. Actually, the symmetry is not broken, it’s just that upper Isn/t present everywhere the way lower is.