Lecture 19. November 2, Comments on HW3. Introduction to EWK symmetry breaking. Hydro Model of a Quark-Gluon Plasma Tianyu Dai

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1 Lecture 19 November 2, 2017 Comments on HW3 Introduction to EWK symmetry breaking Hydro Model of a Quark-Gluon Plasma Tianyu Dai 1

2 Class Plans Date Lecture Essay presentation (15 minutes) Nov Tianyu Hydro model of QGP Nov Long Detection of axions Nov Baran Neutrino mass from tritium decay Nov Sourav The strong CP problem Nov Michael Composite Higgs models Nov Ping Family/flavor symmetry in the SM Nov. 23 Thanksgiving break Nov Nov (last class) 2

3 Comments on HW3 P2 Problem 1 (4 th generation quarks): Just an example of calculating decay branching ratios in the context of searches for a 4 th quark generation. Problem 2 (HERA): From 1992 to 2007 the electron-proton collider HERA operating at the laboratory DESY in Germany was at the frontier of high energy physics. Using 28 GeV electrons and 820 GeV protons a cm energy of 320 GeV could be reached. This opened the window for some unique new physics searches and provided and ideal probe of proton structure with electrons 3

4 Comments on HW3 P2 Problem 2 (HERA): A search was made for lepto-quarks carrying both lepton number and color charge. These were products of some grand unified theories combining the QCD and EWK interactions and a hot topic of experimental research for many years. q e LQ e,ν q λ 1 SCALAR LEPTOQUARKS WITH F=2 H1 CI p 10-1 EXCLUDED S 0, L H1 single prod. L3 indir. limit W cm =2 p xe e E p where x = the fraction of the proton momentum carried by the quark q D0 pair prod M (GeV) 4

5 Comments on HW3 P2 Problem 2 (HERA): One of the lasting contributions from research at HERA were measurements of the parton distribution functions (PDF s) of the proton. These are essential for any precise SM tests at the LHC since the PDF s enter directly in predictions. See arxiv: v1 [hep-ex] 2008 for areview of HERA physics. xp(x) xu xu v xu xp(x) Q 2 =10 GeV 2 H1 PDF 2000 MRST2001 CTEQ6 xu xu xd xd v xg xd H1 PDF 2000: Q 2 = 4 GeV 2 experimental errors model uncertainties x H1 Collaboration xd xg xd x parton distribution Figure 12: Determination of the sum of up, anti-up, down and anti-d and of the gluon distribution in the proton based on the H1 neutral an section data. Left: the parton distributions with their experimental a 5

6 Comments on HW3 P3 Problem 3 (QCD field equations for quarks and gluons): The exercise is to start from the QCD Lagrangian and use the Euler-Lagrange equation to get the field equations for quarks and gluons. The QCD Lagrangian given was discussed in L11-12: L QCD ==- 1 4 F µ a F a µ + i(~c) q j µ D µ jk q k -(m q c 2 ) q j q k jk F a µ µ G G µ a g s f abc G µ b G c D µ jk = jk@ µ + i g s ~c [T a] jk G µ a 6

7 Comments on HW3 P3 Problem 3 (QCD field equations for quarks): For a quark field q i (flavor q color i ) it is q i ] q i =0 i(~c) µ q i -(m q c 2 )q i = g s µ [T a ] ik q k G µ a This is analogous to the QED quark field equation (see L9, p30): with g s! ef,[t a ] ik! ik, G µ a! A µ =) QED 7

8 Comments on HW3 P3 Problem 3 (QCD field equations for qluons): For a gluon field G g α the Euler-Lagrange @[@ G g ] - g =0 Here the derivation is not trivial and involves manipulations with the properties of the SU(3) structure constants f abc. See HW3 solutions page 10 for the result, and also the (brief) discussion in section 16.1 in the text. I include here some observations showing where the QCD gluon field differs from the QED photon field due to the non-abelian character of SU(3) (i.e. the generators due not commute resulting in some non-zero structure constants). 8

9 Comments on HW3 P3 Problem 3 (QCD field equations for qluons): Define the field tensor G g βα for the gluon field G g α to be G g G g G g for the eight gluon fields G g just as for QED where: F A A for the single vector @[@ G g ] - Applying the E-L eq. to the QCD Lagrangian results in 3 source terms for the current associated with the field G g α : g G g = J g = J g (1) + J g (2) + J g (3) 9

10 Comments on HW3 G g = J g (1) + J g (2) + J g (3) The three current terms specify different sources of the gluon G g α field coupling. The first one is from quarks: G g J g (1) = g s [ q j [T g ] jk q k ] G g q k q j g s G g This term is analogous to QED with g s! ef,[t a ] ik! ik, G µ a! A µ =) QED 10

11 Comments on HW3 G g = J g (1) + J g (2) + J g (3) The second term describes gluon triple gauge couplings: G cµ G g J g (2) = g s [ f gbc G µ b G cµ ] G g where G cµ µ G c G cµ G µ b g s G g and the third term describes gluon quartic gauge couplings: G µ e G f G g J g (3) = - g 2 s[ f gbc f cef G µ b G eµg f ] G g g 2 s G µ b G g 11

12 Comments on HW3 P4 Problem 4 (tests of SU L (2) gauge invariance): This shows how the weak gauge invariance can be used to select terms allowed in the Lagrangian. For example tensors terms are not invariant. µ 1 2 (1 5 ) can be shown to = R µ L using properties of gamma matrices. Since SU L (2) transforms only L this term will not be invariant. 12

13 Comments on HW3 P5 Problem 5 (test of U Q (1) gauge invariance): L = 1 4 Gµ G µ + i µ - m f m2 b B µb µ - c 1 µ B µ - c 2 µ 5 B µ - c 3 B B µ where G µ µ B B µ The test here involves making the following substitutions in the above Lagrangian and checking for invariance: 0 =exp[ ie (x)q] [1 ie (x)q] =[1 ie (x)f] with Q = f 0 =[1+ie (x)f] B 0 µ = B µ µ 13

14 Comments on HW3 P5 This is then an exercise in following through the transformation of each term testing for invariance, neglecting terms µ The invariance requires: m b = c 2 = c 3 = 0 and c 1 must = ef. Therefore the result reduces to the SM QED Lagrangian. 14

15 Electroweak symmetry Breaking 15

16 Introduction and Preview The minimal Standard Model has one scalar, electrically neutral Higgs boson H o. This is the version introduced by a group of theorists in 1964 as a mechanism to generate mass terms into the theory. For each massive particle m x there is a coupling constant G x. This introduces at first glance a whole new set of parameters into the Standard Model. But these coupling strengths are fixed by the particle masses, so you can either count the masses or the Higgs couplings as the free parameters. Therefore the parameter counting we did early in the course is correct. 16

17 Introduction and Preview For fermions of mass m f we will show that the required coupling strength to the Higgs field is: gf = gw p 2 m f M W With g w = and M W c 2 = GeV some couplings are: particle Mass GeV g f electron x 10-6 tau x 10-3 top quark Remarkably, the Higgs boson couples to the top quark with unit coupling strength. There is nothing in SM theory that requires this. 17

18 Mass terms in Lagrangians Since the Higgs mechanism generates mass terms, lets review what we will be looking for. We need to identify the mass terms appearing in the Lagrangian for spin 0, 1/2 and 1 particles. 1. For a spin 0 real scalar field φ(x): L = 1 µ m For a spin 0 complex scalar field φ(x) = [φ 1 (x) + i φ 2 (x)]/ 2 L µ 1 - m 2 where - m 2 =- 1 2 m m For spin 1/2 fermions described by the spinor ψ: L = i(~c) µ - m 18

19 Mass terms in Lagrangians 4. For a spin 1 abelian field B µ : L =- 1 4 F µ F µ m2 B B 5. For spin 1 non-abelian fields. The one of interest is 3-component field W i µ entering the weak interaction Lagrangian L weak =- 1 4 W aµ W µ a m2 i W iµw iµ with W µ a µ W a W µ a - g w f abc W µ b W c In our past discussion of the weak interaction the mass term did not appear as it was required to be zero by the exact gauge symmetry. Now we need to resurrect the mass terms for the charged W and neutral Z bosons. 19

20 Mass terms in Lagrangians 5. For spin 1 non-abelian fields (con.) As in constructing the weak interaction theory introduce W ±µ = 1 p 2 (W µ 1 iw µ 2 ) W 0µ = W µ 3 Here the mass m 1 and m 2 must be equal. Call the mass of the charged W bosons m c and of the neutral boson m 0. Substituting the above W fields into the Lagrangian on the previous page: L weak =- 1 4 W aµ W µ a + m 2 c W µ W +µ m2 0 W 0 µw 0µ Keep a record of the mass terms for these 5 fields as they will be used to identify the masses generated via the Higgs mechanism. 20

21 Next Lecture 1.Generating fermion masses using a real scalar field 2. Generating boson masses using a complex scalar field (the Higgs mechanism) 21

22 SM prediction for W! e + e decay width v e P e e e ū e P e Z P Z 22