Comments by Prof.Chen after the paper was turned in Asisceier.you neeoprovide thesom show the ditrrenees o th om of the B-Wohe if you energ hatUniot Min 中华中中中中华华中中中中水卡中中中中中中中中中中中中华中中中中中年中中中中办中华中中中中 Hell Prof Chen e to your M(.e) M=5Gwew小+t@4 hic(r.小+6re小Ia2 g,grzar+carAcreHeTGrel M,= GM this form,it is casy to evaluate cross-scctions for interactionsith known chiralities Thus you get fourrot of the form: ee→)-l+cosl+rc+c7cif where
Comments by Prof. Chen after the paper was turned in: ******************************************** Tom – As discussed earlier, you need to provide 1. the explicit formulas for FB F B A F B − ≡ + , A_LR, etc, in terms of masses, widths (with and without energy dependence due to higher order electro-weak corrections, as derived in class) of the Z's at some energies s, and show the differences of these formulas (which sort of indicate the theoretical uncertainties of the B-W form when the interaction energy is far away from the resonances!). 2. measurement and systematical errors on A_FB, etc. in order to see if you can discover a high mass Z' at low energies. 3. use the s dependence of A_FB, A_LR, etc. to solve your problem of that "Unfortunately, this can only measure a ratio of coupling to mass until the Z' mass is reached." - Min ******************************************** Hello, Prof. Chen, I have attached a response to your comments. Thanks Tom ******************************************** The three relevant Feynman diagrams for e+ e - → µ+ µ- are annihilation through the photon, Z, and Z’. Their (simplified) matrix elements are: 2 - ( )( e M e e) s ν γ µ ν = γ µ γ (1.1) ( ) ( ) ( ) ( 2 2 - 2 [ ][ - Z e e R L Z R R L R L L R R L L Z Z Z GM M c c c e e c e )] is s M M µ ν µ ν ν ν = + µ γ µ µ γ µ γ γ Γ + + e (1.2) 2 ' ' 2 ' ' ' - 2 [ ' ( ) ' ( )][ ' ( ) ' ( )] - Z e e R L Z R R L R L L R R L Z Z Z GM M L c c c e e c e is s M M µ ν µ ν ν ν = + µ γ µ µ γ µ γ γ Γ + + e A A (1.3) where and . With this form, it is easy to evaluate cross-sections for interactions with known chiralities. Thus you get four cross-sections of the form: R V c c = − c L V c c = + c 2 2 2 2 ( ) (1 cos ) 1 ' 4 e L R L R L L L L d e e rc c r c c d s σ α µ µ µ µ θ − + → = − + + + + Ω ' e (1.4) where
r=- 1.5 4周 s-M2+M 1.6) A sy (+cos+cos0 17 =1+2G[Re(r)+Re((++2Re(r) tion Ac-F+B 1.10) 4=站 0.2 A-n 0.25 50100502025030 MEM you find be Taylor expandod as
2 2 2 2 Z Z Z Z GM s r is e s M M ⎛ ⎞ = ⎜ ⎟ Γ ⎝ ⎠ − + (1.5) and 2 ' 2 2 2 ' ' ' 2 ' Z Z Z Z G M s r is e s M M ⎛ = ⎜ Γ ⎝ ⎠ − + ⎞ ⎟ (1.6) To measure the Forward Backward Asymmetry, you need to use unpolarized beams, and do not measure the polarizations of the resulting muons. Thus you need to average those four cross-sections to get the actual cross-section: 2 2 0 1 [ (1 cos ) cos ] 4 d A A d s σ α = + θ + θ Ω (1.7) In general, A0 and A1 are complicated factors involving all of the cross-terms from 1.4. However, if c c ' ' these terms become e e i i i i i c c c µ µ = = = = 2 2 2 2 2 2 0 2 2 1 2 [Re( ) Re( )] ( ) [ 2Re( )] A V V A 2 = + c r + r + c + c r + r + r r (1.8) 2 2 2 2 2 1 2 2 4 [Re( ) Re( )] 8 [ 2Re( )] A A V A 2 = + c r r + c c r + r + r r (1.9) With F = the rate of µ+ going in the forward direction (0<θ<π/2) and B = the rate of µ+ going in the backward direction (π/2<θ<π), FB F B A F B − ≡ + (1.10) A simple integral shows that 1 0 3 8 FB A A A = (1.11) Fig. 1: AFB plotted with MZ’ = 2TeV, ΓZ’ = 2 GeV, and G’ = G 0 W / 1000 Fig. 1 shows this function plotted with some possible Z’ parameters. The main features of the plot are the two interference patterns at the Z and Z’ masses. In the region where MZ ECM MZ’ you find that 2 2 2 ' G s r e ≈ − and AFB can be Taylor expanded as
+2eRe(s)-1-25G 4.12 hee tond t of ces of ntesinaccp can 0 E! 0.1 The co r in the yrcctna the the m uncertainty in momentum.Even have no μfe very not be
( ) 2 2 2 2 ' 1 1 2 Re 2 2 2 2 A A G c r c e ⎛ ⎞ + = − ⎜ ⎟ ⎝ ⎠s (1.12) Thus in this region, it is possible to find the strength of the interaction without knowing the mass of the Z’. Fig.2 shows a sample data set in this region with some estimated error bars. The errors in the y-direction come from uncertainties in measuring the flux in the forward and backward directions. The main sources of this error would be uncertainties in acceptance, uncertainties in reconstructing the positive versus negative muons, and uncertainties in momentum reconstruction near θ = π/2. Uncertainties in acceptance can be minimized by having a rotating detector, allowing that error to be calibrated to 1-2%. The other errors can be minimized by having good position resolution in the tracker. These errors should also be at 1-2%. Fig. 2 shows 5% error bars in the y-direction, and the signal is still clearly visible. Fig. 2: A sample data set for discovery of the Z’. The flat line is the Standard Model prediction, while the curved line is the prediction from fig.1. The error bars are a worst-case scenario, and should be twice as small in a wellrun experiment. The errors in the x-direction are due to uncertainties in the beam energies. Beambeam interactions give a spread in the beam energy of between 1% and 10%. Reconstructing the center-of-mass energy of each collision could allow that spread to be taken into account in a maximum-likelihood fit. However, there are errors associated with the momentum reconstruction. At high energies, these errors should be on the order of 1- 2%. In fig. 2, I plotted the worst-case scenario of a 10% uncertainty in momentum. Even with that worst-case scenario, the signal is still visible. Some sources of error, such as Poisson statistics and cosmic-ray muons have not been included here because they should be small. The e+ e - → µ+ µ- reaction has a very large cross-section even without a Z’, so gathering enough statistics should not be a problem. For that same reason, the flux of cosmic-ray muons should not be large enough to cause problems. Even if there were many cosmic-ray muons, their only effect would be to shrink AFB by a constant factor, so the decrease as a function of energy would still be evident
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Thus even below the Z’ mass, it is possible to show the existence of a Z’. In a worst-case scenario, the signal looks good, and is probably at least a 2σ deviation from the standard model. With more work to reduce the errors and a better analysis method, this experiment should be able to show the existence of a Z’ at the 5σ level required for discovery