1. Tilt Correction to Geometric Hourglass Effect
Motivation: Get a better understanding on how beams’ coupling affect luminosity and some luminosity based measurements. (i.e. Beam-Beam scans and BaBar’s luminous region measurement.)
William Colocho, May

2. Overview Review hourglass effect and ‘zero bunch length’ tilt effect.
Describe how to combine these two.
Use MIA/LEGO model for lattice functions at IP.
Show how to transport IP values.
Simulation results.

3. Hourglass Luminosity Formula:
(“The Hourglass Reduction Factor for Asymmetric Colliders”, Miguel A. Furman, ABC-21/ESG- technote-161)
Allows for s dependence in drift space:
Does not include tilt of beams due to coupling.
Tilt: Rotation angle of beam’s projection onto XY plane.
Twist: Tilt along the length of the bunch.

4. Zero Bunch Length Tilt Dependence
“LUMINOSITY OF ASYMMETRIC e+e-
COLLIDER WITH COUPLING LATTICES”
Y. Cai SLAC-PUB-8479
2-D Gaussian
2X2 covariance matrix includes tilt angle information.
2D Luminosity with tilt.

5. Method
Start with zero bunch length luminosity formula, including tilt dependence.
Zero bunch length Cap sigmas at IP are calculated from MIA/LEGO model run.
Then generate shape of luminous region (dL/ds) by allowing the covariance matrix (sigmas) to change with s.

6. Covariance (Sigma) matrix S dependence
The S dependence can be computed from a linear map of the beam sigma matrix in the presence of BaBar’s 1.5 Tesla solenoid field.
R(s) is the R Matrix for a solenoid.

7. Results
S dependence of beams’ ellipses near IP.

8. dL / ds and Vertical Luminous Region

9. Summary, Questions, Future Work.
Hourglass and Tilt effects can be combined with S dependence of the covariance matrix (beam sigma) transported with a linear map.
Simulate Beam-Beam scan with this formalism. Including scan at different settings of collision phase.
Include dispersion.
Include phase and bunch length along the train to luminosity calculation and compare with measurements.