Luminosity Calculationin Crab Crossing Simulation

Feb., 2018

River and Vasiliy

· 1 ·

· 2 ·

Outline

Introduction

Analytic Methods

Numerical Calculations

Benchmark Result

Future Plans

· 3 ·

• Crab dynamics simulations are in progress usingElegant without Beam-Beam effect

• Beam-Beam effect is important for crab dynamics,e.g. due to synchro-betatron resonances (Yves hadshown it with BeamBeam3D)

• BeamBeam3D has simplified beam dynamics. Wewould like to combine accurate dynamicssimulation with an adequate beam-beam model.

• We develop a code in Python for simulating beam-beam effect and calculation of luminosity during thesimulation process.

Introduction

· 4 ·

Initial Distribution

StartPoint

1St CrabCavity

InteractionPoint

2nd CrabCavity

Elegant process

PythonBeam-Beam Interaction

Introduction

· 5 ·

1) Direct Integral Method (J. E. Augustin, A. W. Chao, …)calculates luminosity for any collision configurations.

2) Bassetti-Erskine-based technique extended to finitebunch length, crossing angle and bunch offset; it isstraightforward to use it to apply beam-beam kick toindividual particles.

Analytic Methods

· 6 ·

Direct Integral Method (in the lab frame)

1 2 1 2c BN N f N dxdydsdtρ ρ= ∫ ∫ ∫ ∫

( , , , ) ( ) ( ) ( ), 1, 2i i i i ix i iy i is ix y s t x y s ct iρ ρ ρ ρ= − =

For the Gaussian beam2

21( ) exp , ,

2 2iwiw iw

ww w x yρσ π σ

= − =

2

00 2

( )1( ) exp2 2is

is is

s ss sρσ π σ

−− = −

*2

* 21 ,i

ii w

ww i

sσβ

σ= + 0 (the beam center to the colliding point)s ct=

Analytic Methods

· 7 ·

Direct Integral Method (Coordinates transformation)

φφ

s

1s

2x1x

2Beam

2Beam

1Beam

1Beam

2s

1 cos sinx x sφ φ= ⋅ − ⋅1 cos sins s xφ φ= ⋅ + ⋅

2 cos sinx x sφ φ= − ⋅ − ⋅2 cos sins s xφ φ= − ⋅ + ⋅

If there is a crossing angle in horizontal (x, s)-plane2φ

Analytic Methods

· 8 ·

Direct Integral Method (after integration over y, s0, x)2

2 21 21 21 2

3 2 22 1 2 1 21 2

1 cos 2 cos 2expcos

2

x xs srev B

x x y ys s

ss sN N f N ds

s s s s

φ φσ σφ

π σ σ

+− + ++ =+ ++

∫

Comparing with the “Head on” 1 20 * 2 * 2 * 2 * 2

1 2 1 22rev B

x x y y

N N f N

π σ σ σ σ=

+ +

( )2 2 * 2 * 2

* 2

cos iw iwiw

iw

ss

φ β σ

β

+=where

* 2 * 2 * 2 * 2 2 21 21 2 1 2 1 2

2 2 1 2 1 21 20

1 cos 2 cos 2exp 22 cos x xx x y y s s

x x y ys s

ss s

dss s s s

φ φφ σ σ σ σ σ σ

π σ σ

+− + ++ + + =+ ++

∫

The luminosity reduction factor with hourglass effect and crossing angle for two bunches

Analytic Methods

· 9 ·

Model for Lorentz Boost Method

(a)

(b)

Beams colliding at an angle:in the original frame (a) and in the boosted frame (b).

The direction of the Lorentz boost is indicated by a dished line.

Analytic Methods

· 10 ·

Assume one IP (the interaction point ) in a ring located at s=0,where s is the azimuthal coordinate.

At the IP, based on Lorentz transformation, the coordinates of aparticle are boosted so that the collision becomes head on, thenthe particle interacts with the other beam in this boosted frame.

Finally, the luminosity calculation for the head-on is carry out inthis frame.

Accelerator frame Lab frame Boosted frame Luminosity

The coordinate system is called theaccelerator coordinate. Here x and y are horizontal and verticalcoordinates, are three-momentums, (c is thelight velocity, t is the arrival time at the position s), and h is the“Hamiltonian” (Details can be found in our technical report)

( , , , , , ; , )x y zx p y p z p h s=x

, ,x y zp p p z s ct= −

Analytic Methods

· 11 ·

Accelerator frame Lab frame

For the laboratory frame, the Cartesian system is given by

( , , , , , ; , )X Y ZX Y Z P P P H T=X

where H is the true Hamiltonian, which is the energy, and T is thetime. The relations between the accelerator coordinates are

1 0 1 00 1 0 00 0 1 00 0 0 1

A

cT zX xZ sY y

− =

0

00

/1 0 1 00 1 0 00 0 1 00 0 0 1

z

x X

Z

y YB

p H c Pp P

Ph P Pp P

− = −−

and

where P0 is the absolute value of the three-momentum P of the reference particle.

and2

z zs + −−=

Analytic Methods

· 12 ·

Based on this model, after Lorentz boost, the collision becomeshead-on in other words, in Lorentz Boost Frame, we can easilycalculate the luminosity via the head-on formula, which is wellstudied. Without offset, the Lorentz boost yields

and

****

cT cTX X

LZ ZY Y

=

* / /***

X X

Z Z

Y Y

H c H cP P

LP PP P

=

where

1/ cos sin tan sin 0tan 1 tan 00 sin cos 00 0 0 1

L

φ φ φ φφ φ

φ φ

− − − = −

Analytic Methods

· 13 ·

and

Accelerator Lorentz Boosted frame

1

*( ) (0) 1/ cos 0 0 0 (0)*( ) (0) tan 1 0 0 (0)

* 0 0 sin cos 0 0*( ) (0) 0 0 0 1 (0)

z s z zx s x x

A LAs

y s y y

φφ

φ φ−

− = = −

2

12

* 1 tan tan 0* 0 1/ cos tan / cos 0

* 0 0 1/ cos 0* 0 0 0 1/ cos

z z z

x x x

y y y

p p pp p p

B LBh h hp p p

φ φφ φ φ

φφ

−

− − − = =

***

ii

hhp∂

=∂

where and 0 021*( *, *, *; *) ( , , ; ) ( *, *, *; *)

cosx y z x y z x y zh p p p P h p p p P h p p p Pφ

= =

Analytic Methods

· 14 ·

Finally, *

*

*

* tan [1 sin ]

* sin

* / cos sin

x

y

z

x z h x

y y h x

z z h x

φ φ

φ

φ φ

= + +

= +

= +2

* ( tan ) / cos* / cos

* tan tan

x x

y y

z z x

p p hp p

p p p h

φ φφ

φ φ

= −

=

= − +

and( )( )( )

2* 2 2 2

2* 2

2* 2 2

tan

cos

x x z

y y

z z

σ σ σ φ

σ σ

σ σ φ

= +

=

=

Accelerator Lorentz Boosted frame (continue…)

Analytic Methods

· 15 ·

e.g., the effective beam size

then

where, is the luminosity reduction factor due to the crossing angle.

2 21_ 2x xx eff σσ σ+=

x

After Lorentz boost, we can calculate the luminosity by using “head-on” formula,

( ) ( ) ( ) ( )1 2

2 2 2 2* * * *1 2 1 2

02

c B

x x y

x

y

N N f N

π σ σ σ σ= =

+ +⋅

( ) ( )2 2* * 2 21 2 1 _2y y y y y effσ σ σ σ σ+ = + =

2 21_ 2y yy eff σσ σ+=

( ) ( ) ( )2 2* * 2 2 2 2 21 2 1 2 1 2

2 22 2 21 2

1 2 2 2

_

1 2

tan

1 tan

x x x x z z

z zx x

x

x eff x

x

σ σ σ σ σ σ φ

σ σσ σ

σ

φσ σ

+ = + + +

++ +

+

= ⋅

=

Analytic Methods

· 16 ·

In general, following the integration strategy, consider the luminosityreduction effects caused by crossing angle and offset

Luminosity 0 ⋅= ⋅ ⋅

where2 2 2 2

2 21 2 1 22 2 2 21 2 1 2

1

1 tan tanz z z zx y

x x y y

σ σ σ σφ φσ σ σ σ

=+ +

+ ++ +

( ) ( )22

2 2 2 21 2 1 2

exp2 2

yx

x x y y

δδ

σ σ σ σ

= − − + +

22

2 21 2

2 2 2 21 2 1 22

tantanexp y yx xz z

x x y y

δ φδ φσ σσ σ σ σ

+ = + + +

Analytic Methods

· 17 ·

Hourglass Effect: For real case, the β-function in a drift spacevaries and depends on:

where β* is the β-function at the interaction point

2

2( ) * 1*

ssβ ββ

= +

The name “hourglass effect” comes from the shape of β(s).

2

2( ) ( ) * 1*

ss sσ β ε σβ

= = +

Analytic Methods

· 18 ·

For the symmetric-collider and the two identical flat beams,* * *1 2 ,x x xσ σ σ= = * * *

1 2 ,y y yσ σ σ= = 1 2 ,s s sσ σ σ= = * *y xσ σ<<

Including the hourglass effect and the beam-tilt effect, K. Hirata and A.W. Chao suggested using following approximate solution

Analytic solution: symmetric-collider case with “Flat Beam”

00

2 ( )bae K bπ

=

where*

*2y

za

β

σ=

2*2

*1 tanz

xb a σ φ

σ

= +

*2 2 *

2

0 2

2

2 *cos si2

exp2nx z

xy

y

δδ

σ φ σ φ σ

= − −

+

0 00

2 ( )bae K bπ

⋅=

and

If the beams are, in addition, offset transversely,

and

where

Analytic Methods

· 19 ·

In the numerical calculation processes, two colliding bunched beams are cut intomany slices whose normal direction is parallel to the longitudinal direction. theluminosity will be calculated by the summation of each individual discrete part.

0 , , ,,,2

B ci j i j i ji j

i j

N fπ

= ∑

1, 2,0 , 2 2 2 2

1, 2, 1, 2.

i ji j

x x y yi j i j

N N

σ σ σ σ=

+ + , 2 2 2 2

2 21 2 1 22 2 2 21, 2, 1, 2,

1

1 tan tani j

z z z zx y

x x y yj j i j

σ σ σ σφ φσ σ σ σ

=+ +

+ ++ +

( ) ( )22

, 2 2 2 21, 2, 1, 2,

exp2 2

yxi j

x x y yj j i j

δδ

σ σ σ σ

= − − + +

where,

22 21 2,, 2 2 2 2

1, 2, 1

2

, 2,

tantanx2

e p y yx xz zi ji jx x y yj j i j

δ φδ φσ σσ σ σ σ

+ = + + +

Numerical Calculations

· 20 ·

Result

Case 1: Test of the Crossing Angle Factor (Symmetric Collider,Flat Beam without Hourglass Effect and offset)

Electron_Mass 0.000511 GevElectron_Beam_Energy 10 GeVCollision_Frequency 1.19E+08 HzElectron_Bunch_Length_RMS 1.0 cmNumber_of_Electron_per_bunch 3.70E+10Number_of_Slices_for_e_bunch 20Normalized_emittance_x_for_e 4.32E-02 cmNormalized_emittance_y_for_e 1.72E-05 cmbeta_star_x_for_e 4.00E+02 cmbeta_star_y_for_e 8.00E+01 cmCrossing_Angle_x 0.0~6.0 DegreeCrossing_Angle_y 0.0 DegreeOffset_x 0.00E+00 cmOffset_y 0.00E+00 cm

The analytic results vs. the numerical results for the case 1. The x-axis means the crossing anglewhich unit is degree, the y-axis shows the reduction factor, where L_0 is the luminosity withouthourglass and tilt effect, L_A and L_N are the luminosities of the analytic and the numericalrespectively. The numerical result matches the analytic result very well.

50x yσ σ= *z xσ β<< *

z yσ β<<

φ(deg.)

⁄ 𝐿𝐿𝐿𝐿 0

· 21 ·

Case 2: Test of Crossing and Hourglass Effect together(Symmetric Collider, Flat Beam without offset)

The analytic results vs. the numerical results for the case 2. The x-axis means the crossing anglewhich unit is degree, the y-axis shows the reduction factor, where L_0 is the luminosity withouthourglass and tilt effect, L_A and L_N are the luminosities of the analytic and the numericalrespectively.

Electron_Mass 0.000511 GevElectron_Beam_Energy 10 GeVCollision_Frequency 1.19E+08 HzElectron_Bunch_Length_RMS 1.0 cmNumber_of_Electron_per_bunch 3.70E+10Number_of_Slices_for_e_bunch 20Normalized_emittance_x_for_e 4.32E-02 cmNormalized_emittance_y_for_e 1.72E-05 cmbeta_star_x_for_e 4.0 cmbeta_star_y_for_e 0.8 cmCrossing_Angle_x 0.0~6.0 DegreeCrossing_Angle_y 0.0 DegreeOffset_x 0.00E+00 cmOffset_y 0.00E+00 cm

50x yσ σ=

⁄ 𝐿𝐿𝐿𝐿 0

φ(deg.)

Result

· 22 ·

Case 3: Test of Hourglass Effect (Symmetric Collider,Head-on Flat Beams without offset)

The analytic results vs. the numerical results for the case 3. The x-axis means the electron bunchlength (RMS), the y-axis shows the reduction factor, where L_0 is the luminosity withouthourglass and tilt effect, L_A and L_N are the luminosities of the analytic and the numericalrespectively.

Electron_Mass 0.000511 GevElectron_Beam_Energy 10 GeVCollision_Frequency 1.19E+08 HzElectron_Bunch_Length_RMS 0.1~10.0 cmNumber_of_Electron_per_bunch 3.70E+10Number_of_Slices_for_e_bunch 1~100Normalized_emittance_x_for_e 4.32E-02 cmNormalized_emittance_y_for_e 1.72E-05 cmbeta_star_x_for_e 4.0 cmbeta_star_y_for_e 0.8 cmCrossing_Angle_x 0.0 DegreeCrossing_Angle_y 0.0 DegreeOffset_x 0.00E+00 cmOffset_y 0.00E+00 cm

50x yσ σ=

σz(cm)

⁄ 𝐿𝐿𝐿𝐿 0

1 (Slides )

100 (Slides )

Result

· 23 ·

The analytic results vs. the numerical results for the case 4. The x-axis means the number ofslices for electron bunch, the y-axis shows the reduction factor, where L_0 is the luminositywithout hourglass and tilt effect, L_A and L_N are the luminosities of the analytic and thenumerical respectively.

Electron_Mass 0.000511 GevElectron_Beam_Energy 10 GeVCollision_Frequency 1.19E+08 HzElectron_Bunch_Length_RMS 10.0 cmNumber_of_Electron_per_bunch 3.70E+10Number_of_Slices_for_e_bunch 5~100Normalized_emittance_x_for_e 4.32E-02 cmNormalized_emittance_y_for_e 1.72E-05 cmbeta_star_x_for_e 4.0 cmbeta_star_y_for_e 0.8 cmCrossing_Angle_x 0.0 DegreeCrossing_Angle_y 0.0 DegreeOffset_x 0.00E+00 cmOffset_y 0.00E+00 cm

50x yσ σ=

Case 4: Test of Hourglass Effect (Symmetric Collider,Head-on Flat Beams without offset)

Number of Slices

Result

· 24 ·

The analytic results vs. the numerical results for the case 4. The x-axis means the offset on x-direction which unit is σx, the y-axis shows the reduction factor, where L_0 is the luminositywithout hourglass and tilt effect, L_A and L_N are the luminosities of the analytic and thenumerical respectively.

Electron_Mass 0.000511 GevElectron_Beam_Energy 10 GeVCollision_Frequency 1.19E+08 HzElectron_Bunch_Length_RMS 10.0 cmNumber_of_Electron_per_bunch 3.70E+10Number_of_Slices_for_e_bunch 10Normalized_emittance_x_for_e 4.32E-02 cmNormalized_emittance_y_for_e 1.72E-05 cmbeta_star_x_for_e 4.0 cmbeta_star_y_for_e 0.8 cmCrossing_Angle_x 0.0 DegreeCrossing_Angle_y 0 DegreeOffset_x 0.0~3.0 σx

Offset_y 0.00E+00 cm

50x yσ σ=

Case 4: Test of Hourglass Effect (Symmetric Collider,Head-on Flat Beams with offset)

δx / σx

⁄ 𝐿𝐿𝐿𝐿 0

Result

· 25 ·

Case 6: Head-on collision with Hourglass Effect for JLEIC

Crossing_Angle_x 0 DegreeCrossing_Angle_y 0 DegreeOffset_x 0 cmOffset_y 0 cmLuminosity 5.75 (x1033) cm-2s-1

Hourglass reduction 0.75

Result

CM energy GeV 21.9 (low) 44.7 (medium) 63.3 (high)

p e p e p eBeam energy GeV 40 3 100 5 100 10

Collision frequency MHz 476 476 476/4=119Particles per bunch 1010 0.98 3.7 0.98 3.7 3.9 3.7Beam current A 0.75 2.8 0.75 2.8 0.75 0.71Polarization % 80% 80% 80% 80% 80% 75%Bunch length, RMS μm 30000 10000 10000 10000 22000 10000

Norm. emittance, hor / ver μm 0.3/0.3 24/24 0.5/0.1 54/10.8 0.9/0.18 432/86.4Horizontal & vertical β* cm 8/8 13.5/13.5 6/1.2 5.1/1.0 10.5/2.1 4/0.8

Ver. beam-beam parameter 0.015 0.092 0.015 0.068 0.008 0.034Laslett tune-shift 0.06 7x10-4 0.055 6x10-4 0.056 7x10-5

Detector space, up/down m 3.6/7 3.2/3 3.6/7 3.2/3 3.6/7 3.2/3Hourglass(HG) reduction 1 0.87 0.75

Luminosity/IP, w/HG, 1033 cm-

2s-12.5 21.4 5.9

· 26 ·

• Extract all parameters from tracking data

• Apply Bassetti-Erskine kick to individual particles and feed it back into the simulation

• Do crab dynamics studies with beam-beam interaction

Future Plans

· 27 ·