Lecture 10. September 28, 2017

Transcription

1 Lecture 10 September 28, 2017 The Standard Model s QCD theory Comments on QED calculations Ø The general approach using Feynman diagrams Ø Example of a LO calculation Ø Higher order calculations and running of α(q 2 ) Development of Quantum-chromo dynamics Ø Introduction of color charges Ø Discussion of SU c (3) invariance Ø The QCD Lagrangian 1

2 Comments on QED calculations It is not the purpose of this course to spend much time carrying out matrix calculations. The goal of to develop the SM theory that can be used to do the calculations. That said, I want to review the general procedure used to go from the theory (expressed as a Lagrangian density) to a transition matrix element for a specific decay or interaction. To illustrate this, calculate the cross section for: e + e +! µ + µ + due to a pure QED interaction (photon exchange). There are basically 3 steps to the calculation. 2

3 Comments on QED calculations 1. Start with the Lagrangian density for QED and carry out a a perturbative expansion in terms of the fine structure constant α. 2. Each term in the expansion can be calculated using the rules developed by Richard Feynman to obtain the transition matrix element for that order of approximation (see Feynman rules handout). 3. For the case where a sum and average is done over spin states the (messy) gamma matrix products can be converted into the trace of a product of gamma matrices, which results in expressions in terms of the dot product of 4-vectors This is a trick discovered by Casimir. 3

4 An example LO QED calculation 4

5 Calculation of e + e +! µ + µ + The diagram below is for lowest order contribution to this process. It is adorned with the particle line and vertex quantities specified by the Feynman rules (check it out). P 2 P 4 (P 1 + P 2 ) 2 = W 2 P 1 P 3 T (e e +! µ µ + )= v 2 (ie µ )u 1 ( ig µ W 2 )ū 3 (ie )v 4 5

6 Calculation of e + e +! µ + µ + Making use of the contraction g µ = µ the matrix element is: T (e e +! µ µ + )= Therefore T 2 = T T * is: ie2 W [ v µ 2 2 u 1 ][ū 3 µ v 4 ] T (e e +! µ µ + ) 2 = e4 W 4 [( v 2 µ u 1 )( v 2 u 1 ) ][(ū 3 µ v 4 )(ū 3 v 4 ) ] This is in general a mess. But let s assume the spin states are not observed and we want the prediction for unpolarized e + + e - producing all muon final states. Therefore average over initial spin states and sum over final: T (e e +! µ µ + ) 2 = 1 4 Ps 1,s 2,s 3,s 4 T 2 This actually makes the calculation simpler. 6

7 Calculation of e + e +! µ + µ + T (e e +! µ µ + ) 2 = 1 4 Ps 1,s 2,s 3,s 4 T 2 = e4 4W Ps 2 1,s 2 [( v µ 2 u 1 )( v 2 u 1 ) ] P s 3,s 4 [(ū 3 µ v 4 )(ū 3 v 4 ) ] Now here is where some technical tricks enter that allow you to simply evaluate the above sums. The sum over products of gamma matrices can be converted to a sum over traces of the matrices. This leads directly to expressions for T 2 in terms of the dot product of momentum 4-vectors. For example consider the term: P s 1,s 2 [( v 2 µ u 1 )( v 2 u 1 ) ] 7

8 Calculation of e + e +! µ + µ + Casimir s Trick For a process with initial spin states s, final spin states s 0 X [ū (s) (p A ) 1 u (s 0 ) (p B )] [ū (s) (p A ) 2 u (s0 ) (p B )] =Tr 1 (/p A + m A c) 2 (/p B + m B c) s,s X 0 [ v (s) (p A ) 1 v (s 0 ) (p B )] [ v (s) (p A ) 2 v (s0 ) (p B )] =Tr 1 (/p A m A c) 2 (/p B m B c) s,s X 0 [ū (s) (p A ) 1 v (s 0 ) (p B )] [ū (s) (p A ) 2 v (s0 ) (p B )] =Tr 1 (/p A + m A c) 2 (/p B m B c) s,s X 0 [ v (s) (p A ) 1 u (s 0 ) (p B )] [ v (s) (p A ) 2 u (s0 ) (p B )] =Tr 1 (/p A m A c) 2 (/p B + m B c) s,s 0 where u (s) (p) and v (s) (p) are Dirac spinors, their conjugates are ū (s) (p) and v (s) (p), and 1 and 2 are any complex 4 4 matrices. Factors of mc appearing in the traces on the right-hand side of these relations have implicit factors of the 4 4 identity matrix. Here P = P µ µ and = 0 0 In our case 1 = µ, 2 =, P A = P 2, P B = P 1 and m A = m B = m electron 8

9 Calculation of e + e +! µ + µ + Using the Casmir rule (with electron mass =0) P s 1,s 2 [( v 2 µ u 1 )( v 2 u 1 ) ] = Tr[ µ ] P 1 P 2 and using a gamma matrix identity (see handout): then P s 1,s 2 [( v 2 µ u 1 )( v 2 u 1 ) ] = 4[P µ 1 P 2 g µ P 1 P 2 + P 1 P µ 2 ] So you get the idea. Repeat this for the other spinor product involving the muon final state. Take the product of the two, and contract various 4-momenta products. 9

10 Calculation of e + e +! µ + µ + The result is: T (e e +! µ µ + ) 2 = 1 4 Ps 1,s 2,s 3,s 4 T 2 = 8e4 W [(P 4 1 P 4 )(P 2 P 3 )+(P 1 P 3 )(P 2 P 4 )+m 2 µ(p 1 P 2 )] This reduces the calculation to kinematics of the dot product of momentum 4-vectors, most easily calculated in the cm of the e + e +! µ + µ + collision. 10

11 Calculation of e + e +! µ + µ + P 1 =[E,0, 0,E] P 2 =[E,0, 0, E] P 3 =[E, ~p 0 ] P 4 =[E, ~p 0 ] Neglect electron mass. Muon mass = m. E = p = W/2 p 0 = p (W, m, m)/(2w )= p W 2 (2m) 2 /2 P 3 p p θ -p P 1 e- e+ P 2 µ - µ + P 4 -p P 1 P 4 = P 2 P 3 = E 2 (1 + p0 E cos ) P 1 P 3 = P 2 P 4 = E 2 p (1 0 E cos ) P 1 P 2 =2E 2 11

12 Calculation of e + e +! µ + µ + Substitute the 4-momenta dot products into the equation on page 11 for the matrix element and use e 4 =( p 4 ) 4 =(4 ) 2 2 T (e e +! µ µ + ) 2 = 1 4 Ps 1,s 2,s 3,s 4 T 2 =(4 ) 2 2 [1 + cos 2 +(2m/W ) 2 (1 cos 2 )] or if W >> m simply T (e e +! µ µ + ) 2 =(4 ) 2 2 [1 + cos 2 ] 12

13 Calculation of e + e +! µ + µ + This can be converted as usual to cross sections: d 2 (e + +e!µ + +µ ) dcos d = (~c)2 q (2W ) 2 1 ( 2m µc 2 2 W )2 [(1+cos 2 )+( 2m µc 2 W )2 (1 cos 2 )] (e + + e! µ + + µ )= 2 (~c)2 3W 2 2 q 1 ( 2m µc 2 W )2 [2 + ( 2m µc 2 W )2 ] The above applies to any un-polarized e - e + production of fermion pairs with the required scaling by f q 2 and colors for quarks N c = 3. 13

14 Higher order QED corrections and running of α(q 2 ) 14

15 Higher order contributions to e + e +! µ + µ + The lowest order contribution is (e e +! µ µ + ) 2 To include the next order correction requires collecting all matrix element amplitudes that, when summed and squared, have contributions up to α 3. For example: T 2 =(A 0 + A 1 )(A 0 + A 1)=A A 2 1 +(A 0 A 1 + A 0A 1 ) where A 2 0 2, A and (A 0 A 1 + A 0A 1 ) 3 Due to interference effects, cross section contribution up to α 3 requires matrix element amplitudes up to order α 2. See other examples in class. 15

16 Higher order contributions to e + e +! µ + µ + The other feature of QED calculations is the running, or varying, of the strength of the EM coupling constant α(q 2 ) where Q is the 4-momentum probing the vertex. At low Q 2, α(q 2 ~ 0) = 1/ ! (Q 2 ) 16

17 Higher order contributions to e + e +! µ + µ + An approximate expression for the running EM coupling is: (Q 2 )= o P i f 2 i [Q2 (2m i ) 2 ]ln[q 2 /(2m i ) 2 ] where the sum is over leptons and quarks with mass 2m i c 2 < Q, f i is the fractional fermion charge and o = (Q 2 = 0) 1/137. ( see Barger and Phillips, page 202) Using this with loops of electron, muon, tau leptons and u,d,s,c,b quarks (x3 for color), using reasonable masses, and evolving to Q 2 = (M W c 2 ) 2 with M W c 2 = 80.4 GeV (M 2 W ) 1/128 è a slow increase of EM coupling with energy. 17

18 Introduction to the theory of Quantum Chromo Dynamics 18

19 QCD vs QED theory A. Goshaw Physics 846 I choose to follow up the discussion of QED with QCD rather than the weak interaction. The main reason is: Ø both QED and QCD the forces are spin-independent Ø the topics of quark doublets and mixing does not enter Ø there is no need of the Higgs mechansim Some obvious differences: Ø quarks carry the color charge, leptons do not and therefore QCD ignores the lepton sector Ø in QCD the boson force carriers (gluons) carry a bi-color charge and are therefore self-interacting Ø the coupling strength a s (Q 2 ) increases with decreasing Q 2, making the tool of perturbation expansions unusable for many application (bound quark states forming mesons and baryons). 19

20 Getting started: some experimental facts Measurements confirm that each flavor of quark comes in three varieties, which we artistically call color. The example we studied is the measurement of quark pair production from e - e - collisions where the measured QED cross section requires a factor of 3 to agree with data. There are many other examples from meson and baryon decays that require 3 colors for each quark flavor. The data also requires the 3 colored quarks have the same mass. Let s label each quark color state (e.g., red, green, blue) by a subscript j = 1,2,3 much easier to write equations. 20

21 Introducing color triplets A. Goshaw Physics 846 Since quarks are fermions we start with the same free particle Lagrangian density used for the development of QED. For a quark of a specific flavor q and color j : L qj = i(~c) q j µ q j -(m q c 2 ) q j q j where the q j are 4 dimensional Dirac fermion spinors. All the m j are equal to the mass of this quark flavor = m q. Introduce a 3 dimensional quark 2 color vector Ψ for each quark flavor. Remember that in this notation = 4 q 3 r q g 5 = the q j are 4-dimensional Dirac spinors. q b 2 4 q 1 q 2 q

22 Introducing color triplets The Lagrangian density for quark flavor q can be compactly written in terms of the 3-dimevsional color vector Ψ : L q = i(~c) µ -(m q c 2 ) Now return to the procedure used to develop QED theory. 2 Require that the Lagrangian is invariant under a transformation of = 4 q 3 1 q 2 5 in the 3-d color vector space. q 3 To get a transformatio in a 3-d color space we can use SU(3), now written as SU c (3) to indicate color space. 22

23 Introducing SU c (3) invariance A. Goshaw Physics 846 In analogy to the notation used for U Q (1) in QED write SU c (3) as:! 0 =exp[ ig s a (x)t a ] where the β a (x) are real functions of (ct,x,y,z) and we use a convention where the 3x3 matrix with λ a the eight 3x3 Gell-Mann matrices. (see L8 p13-14 and next page). T a = 1 2 a The constant g s will control the strong interaction coupling strength just as e did for QED. In QED e = p 4 em In QCD g s = p 4 s 23

24 Introducing SU c (3) invariance A. Goshaw Physics 846 The generators of SU(3) are T a = 1/2 λ a where the λ a are the Gell-Mann matrices: The other f abc = 0 and f abc = - f acb etc The commutation relation are: [T a,t b ] = i f abc T c where the f abc are the structure constants of the SU(3) group. 24

25 Introducing SU c (3) invariance A. Goshaw Physics 846 The strong interaction symmetry is introduced by demanding that L q = i(~c) µ -(m q c 2 ) be invariant under the local SU c (3) transformation! 0 =exp[ ig s a (x)t a ] =[1 ig s a (x)t a +...] where repeated a indices is a sum from 1 to 8. The eight functions β a (x) introduce a transformation in color space that can vary at each space-time point (a local gauge transformation). The assumption is that the strong interaction can be obtained by demanding invariance under this SU c (3) gauge transformation. 25

26 The Lagrangian for the gluon field Before going further we have to introduce one more free-field Lagrangian density. One that describes self-interacting spin 1 boson fields. These are needed for both QCD and the weak interaction. A reminder that the Lagrangian for a non-interacting spin 1 4-vector field A µ is: L = 1 4 F µ F µ (mc/~)2 A A where F µ µ A µ The change to make the fields interacting is to add to the field tensor F µ a term of the form ga µ A where g is the field self-coupling strength. 26

27 The Lagrangian for the gluon field Consider spin-1 4-vector field with self interactions. The a subscript anticipates there may be multiple fields. For SU(2) (the weak interaction) there are 3 fields a=1,2,3. For SU(3) (QCD) there are 8 fields a = 1,2 8. G µ a The Lagrangian density is (sum over all repeated indices): L =- 1 4 F µ a F a µ (mc/~)2 G µ a G a µ F µ a µ G G µ a gf abc G µ b G c where g = the field self-coupling and the f abc are the structure constants for either SU(2) and SU(3). Note that with a =1, and g = 0 this reduces the spin 1 non-interacting boson Lagrangian we used for the QED development. 27

28 Parallels between QED and QCD Pause a minute to consider that the above discussion emphasizes how the approach to developing QCD shadows the approach to developing QCD, with the technical complication of introducing self-interacting fields and the SU c (3) transformation. As for QED, the next step is to postulate that the interactions between the fermions (quarks) and bosons (gluons) can be obtained by demanding that the free-field Lagrangians be modified to obtain invariance under a SU c (3) gauge transformation: L QCD ( 0,G 0µ a )=L QCD (,G µ a) where! 0 =exp[ ig s a (x)t a ] and as for QCD the fields G a µ must also be modified to G a µ by some generalized gauge transformation. 28

29 Obtaining SU c (3) invariance of L QCD (,G µ a) A. Goshaw Physics 846 As for QED let s go directly to the answer. The minimum modification in the free particle QCD Lagrangian that leads to invariance under the SU c (3) transformation is: 1. When transforming Ψ by! 0 =exp[ ig s a (x)t a ] also transform each of the eight G µ a fields by: G µ a! G 0µ a = G µ a +(~c)@ µ a(x)+g s f abc b (x)g µ c 2. Modify the fermion part of the free particle Lagrangian by replacing where: jk@ µ by D µ jk = jk@ µ + i g s ~c [T a] jk G µ a µ = 0,1,2,3 = the 4-vector indices a = 1-8 = the indices for the 8 SU(3) generators j and k = the 3 color indices 1,2,3 for r, g, b 29

30 The QCD Lagrangian from SU c (3) invariance A. Goshaw Physics 846 Collecting all the above together, QCD Lagrangian is: L QCD =- 1 4 F µ a F a µ + q j [i(~c) µ D µ jk -(m qc 2 ) jk ] q k where: q j = the Dirac spinor for a quark of flavor q, mass m q and color j. G µ a = the 4-vector for one of the eight gluons, all of mass 0 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 Note: I have left out one term that survives the test of the required SU c (3) invariance. This violates CP and not observed. More latter. 30

31 Next Lecture: Exploring the QCD interaction 31