Let s move to Bound States

Transcription

1 Let s move to Bound States When we disuss bound states of two objets in entral-fore potential, kineti energy and potential energy are ~the same. How does this ompare to the rest energy of the objets? Hydrogen ionization energy:.6 ev vs 0.5 MeV rest mass Masses of b and quarks are ~relatively large, so we an also onsider them non-relativistially (whih makes them muh easier). NOT true for uds quarks

2 Briefest of overviews of hydrogen atom Energy levels ~ -/n Fine struture from relativisti orretions and spinorbit oupling s(e) = / so j=l +/- / Griffiths

3 Lamb shift Lamb shift: Led to development of quantum eletro-dynamis! QED orretions to the eletronproton interation break degeneray of two levels with same n, j but different L (so S/ and P/ are not fully degenerate) 4 Griffiths Willis Lamb

4 Finally, hyperfine splitting 5 Spin-orbit oupling is prinipally due to spin of eletron interating with B field from nuleus (fine struture). Muh smaller is spin of nuleus interating with B field from eletron. Goes as (me/mp) 4 hene hyperfine (and not fine) splitting. For n=, the differene in energy states of proton (e and p spins aligned vs anti-aligned, whih is lower) is 5.9 µev = 40 MHz = m. Famous m line (penetrates dust!)

5 On the m line Lifetime of m is millions of years! Thankfully, enough hydrogen an provide this transition. Long lifetime = narrow width, so this is exellent for spetrosopy (Doppler shifts) Used extensively in radio-astronomy, studying the early Universe, galaxy formation, measuring distanes to objets, osmology 6 Pioneer Plaque: m line defines distane and unit of time

6 On to quarkonium Differenes between quarkonium and hydrogen/ positronium: Don t really know the potential (strong fore!) Also, interation between quarks is large. Doesn t work for two light quarks, either Instead of onsidering different states as energy levels of an atom, onsider different bound states as different partiles, eah with a different mass. Start with mesons (muh easier than baryons) Q---Q 7

7 What sorts of potentials an we use? 8 At short distanes, we know that QCD is not a strong fore. Reasonable to start with /r potential. At large distanes, we know that fore grows exponentially. Try V~kr (others ould also work). Of ourse, k an be a funtion of r too!

8 Labeled (onfusingly, to me!) as n (s+) LJ (L=S,P,D,... for 0,,... and s=0 or for anti-aligned or aligned spins), with J=L+s Charmonium 9 Griffiths

9 Charmonium 0 Griffiths Rih set of states! J/Psi state disovered in 974

10 Why was J/Psi disovered first? From PDG Due to C-parity!

11 Bottomonium Griffiths Even riher set of states! Upsilon disovered in 976

12 How to aount for spin in hadron masses? Reall magneti moment formula: µ e m S Spin-spin interations in hadrons have two omponents: µ S e m S S µ S e m S S e m S S Sum is then e m ` m m m ps S q Apm ` m q m m ps S q

13 What is the spin term? 4 ps S q, S S ` S S S ` S ` ps S q ps S q `S S S S and S = ±/ S = S = (/)(/+) = /4 S = ()(+) = (spins aligned) or S = (0)(0+) = 0 (spins anti-aligned) So: S S = /4 (spins aligned) or: S S = -/4 (spins anti-aligned)

14 How to aount for spin in hadron masses? 5 pm m q m ` m ` Apm ` m q Mass M of meson omposed of quarks with mass m and m then generially looks like this m m ps S q S S = /4 (spins aligned) or: S S = -/4 (spins anti-aligned) Can try something simpler, and assume A is a onstant (it surely is not, but maybe that is a reasonable approximation)

15 Fits for masses (from Griffiths) 6 Very nie agreement! But need to be areful... For example: uu ` dd ss? 6

16 On to baryons 7 A lot more ompliated - have three quarks, and thus three spins to add together. Most importantly, mesons are always omposed of a quark and an anti-quark, ie never ontain two of the same partile. In baryons, however (example: proton = uud), this no longer has to be true. Regardless, though, baryons have half-integer spin (three quarks with s=+/- / an ombine to give s= +/- / or +/- / only)

17 How to add three spins To add three spins together, we first start by adding two of them together. Bak to those C-G tables from the PDG... 8 Combining two / x / partiles ` ` 0 ` 00 ex: J= ` ` 0 00 ex: m,m= -/,+/-

18 Let s rearrange ` ` 0 ` 00 9 ` ` 0 00 These are the easy ones ` `

19 Let s rearrange (an also use tables for this) ` ` ` ` 0 ` Add together here 40? 0 0? ` ` ` `? ` `

20 Let s rearrange (an also use tables for this) ` ` ` ` 0 ` Subtrat these 4? ? ` `? ` `

21 Putting it together 4 0 0? 0? ` ` `?? ` ` ` ` When we add the third quark we will have to add spin / with either spin 0 or spin

22 Now we add the third one 4 Combining spin x / partiles We want the inverse of what we have been reading off. Can also use the tables for that! ex: J=/ m=/ ex: m,m= +,-/

23 Now we add the third one 44 Combining spin x / partiles? ` ` ` ` ` ` 0 ` 0 ` 0 ` 0 ` ` ` ex: m,m= +,-/ ex: J=/ m=/

24 Now we add the third one 45 Combining spin 0 x / partiles is trivial 00 ` 00 `

25 So in total? ` ` ` ` ` ` 0 ` 0 ` 0 ` 0 ` ` ` 0 0? 0? ` `? `? 46 ` ` ` ` From first ombination 0 0 ` 0 0 `

26 Let s introdue some nier notation? ` ` ` ` ` ` 0 ` 0 ` 0 ` 0 ` ` ` 0 0? 0? ` `? `? 47 ` ` ` ` From first ombination Don t forget: order matters! 0 0 ` 0 0 ` pòq póq

27 Using the notation `pòq? p `póqq ` p 0 `pòqq p `póqq p 0 `pòqq p 0 `póqq ` p `pòqq p 0 `póqq p `pòqq `póq 00 `Ò 00 `Ó ? pòóq? póòq 0? pòóq `? póòq ÒÒ ÓÓ

28 Putting it together Phew ÒÒÒ? pòòóq `? pòó ` ÓÒq Ò pòòóq? pòó ` ÓÒq Ò? pòó ` ÓÒq póq ` póóòq? pòó ` ÓÒq Ó póóòq ÓÓÓ? pòó ÓÒq Ò? pòó ÓÒq Ó 49

29 How to interpret ÒÒÒ? pòòóq `? pòó ` ÓÒq Ò pòòóq? pòó ` ÓÒq Ò? pòó ` ÓÒq póq ` póóòq? pòó ` ÓÒq Ó póóòq ÓÓÓ Spin / states are easy to interpret: symmetri if we interhange any two quarks? pòó ÓÒq Ò? pòó ÓÒq Ó 50 ÒÒÒ? pòòó ` ÒÓÒ ` ÓÒÒq? pòóó ` ÓÒÓ ` ÓÓÒq ÓÓÓ

30 How to interpret ÒÒÒ? pòòóq `? pòó ` ÓÒq Ò pòòóq? pòó ` ÓÒq Ò? pòó ` ÓÒq póq ` póóòq? pòó ` ÓÒq Ó póóòq ÓÓÓ Two of spin / states are asymmetri under interhange of first and seond quarks? pòó ÓÒq Ò? pòó ÓÒq Ó pòòóq 5? pòó ` ÓÒq Ò? pòó ` ÓÒq Ó póóòq

31 How to interpret ÒÒÒ? pòòóq `? pòó ` ÓÒq Ò pòòóq? pòó ` ÓÒq Ò? pòó ` ÓÒq póq ` póóòq? pòó ` ÓÒq Ó póóòq These last two ÓÓÓ spin / states are symmetri under interhange of first and seond quarks? pòó ÓÒq Ò? pòó ÓÒq Ó pòòóq 5? pòó ` ÓÒq Ò? pòó ` ÓÒq Ó póóòq

32 Bak to baryons 5 We need our quarks to satisfy Fermi-Dira statistis (must be anti-symmetri under exhange of any two quarks) For ground state (l=0), spae wave funtion is symmetri. Left off with wave funtions for spin, olor and flavor. We will see that olor wave funtion is neessarily anti-symmetri. That means that flavor x spin ombination must be symmetri

33 You don t have Chapter 5 HW, but 54

34 Something new to think about I d like for you to begin to think about your final projet. You should put in some effort to prepare it, so time to start now. You need to pik a single analysis or result to present (something publi in the past ~year) Minos ATLAS CMS LHCb ALICE SNO Belle AMS LIGO VIRGO LUX Dark Energy Survey Pierre Auger Fermi Gamma Ray Telesope 55

35 Sine you don t have Chapter 5 HW and Chapter 6 is long 56 Pik an analysis/paper and it to me for approval no later than week from today. I ll ask for some progress reports in the future, but for now I just want you to pik something interesting (and not what you work on for researh, and not what someone else will be talking about). Talk to me if you need help piking a topi!

36 Some more news on your presentation 57 Want a 5 minute presentation on the topi! You should be inluding theory bakground if possible, as well as information on the detetor, the analysis, the bakground estimation, and the signifiane of the result. We will all disuss the presentation for 0 minutes after you re done (aka ask you some questions)