1. Lorentz Invariant kT smearing
Timo Alho, Jan Rak and Sami Räsänen
Note:
This presentation is based on (parts of) Timo Alho’s Research Training report, i.e. erikoistyö, FYSZ470
+ some spices

2. Motivation: Assume that kT has 2D Gaussian distribution in
Direct gamma - trigger pTt
xEz
pout kT
away-side fragments - associated particles pTa
-h: direct gamma does fix Energy scale if no kT
Assume that kT has 2D Gaussian distribution in
(kTx,kTy) in LAB frame (CMS of pp). How this is
reflected to the pTt and pTa, if the back to back
momentum of pair is pT at the pair CMS frame?

3. Some kinematics
Note: only transverse degrees of freedom “active” when y1=y2

4. Kinematical limits Solve k2T: Write M2 in LAB and pair CMS
to obtain dot product of transverse
vectors
and kinematical limits
Solve k2T:
You need to solve components of kT vector in order
to make a change of variables (kTx,kTy) -> (pTt,pTa)
=> Lorentz transformation from pair CMS to LAB

5. Lorentz boost

6. Solve kTx and kTy +- signs of momentun vectors partly cancel
Using results from previous slide
Then k2Ty = k2T - k2Tx and you get, up to signs
Jacobian

7. Main result: Where 0 < p2T  pTtpTa Normalization is checked by:
analytic integration
Mathematica

8. “Spices”
Probability to have trigger
pTt for given pT and <k2T>

9. Analytic solution Mathematica is able to calculate
analytically from later form (but not from original). Result:

10. Results 1 P(pTt) pT = 2 GeV/c (<k2T>)1/2 = 3 GeV/c pTt
Figure shows
from analytic formula + numerical integration of original
and modified expressions; all three are the same
Is the analytic result nothing but academic curiosity?

11. Results 2 Triangles: Monte Carlo simulation
Solid (red) curve: Timo’s formula
Dotted line: Gaussian
Note: Gaussian only in the limit pT >> (<k2T>)1/2

12. Summary & Outlook