Optimization Parameter Design of a Circular e+e- Higgs Factory

Optimization Parameter Design of a Circular e+e- Higgs Factory D. Wang (王逗)1), J. Gao (高杰), M. Xiao (肖铭), H. Geng (耿会平), Y. Guo (郭媛媛), S. Xu (许守彦), N. Wang (王娜), Y. An (安宇文), Q. Qin (秦庆), G. Xu (徐刚), S. Wang (王生) Institute of High Energy Physics (IHEP), Beijing 100049, China

Abstract: In this paper we will show a general method of how to make an optimized parameter design of a circular e+e- Higgs Factory by using analytical expression of maximum beam-beam parameter and beamstrahlung beam lifetime started from given design goal and technical limitations. A parameter space has been explored. Based on beam parameters scan and RF parameters scan, a set of optimized parameter designs for 50 km Circular Higgs Factory (CHF) with different RF frequency was proposed. Key words: Circular Higgs Factory (CHF), parameter design, optimization, RF technology PACS: 29.20.db

1.

source of heating. Besides, when two bunches undergo

Introduction

collision at an interaction point (IP), every particle in

With the discovery of a Higgs boson on LHC at the

each bunch will feel the deflected electromagnetic field of

energy of about 125 GeV [1, 2], the world high-energy

the opposite bunch and the particles will suffer from

physics community is investigating the feasibility of a

additional heatings. With the increase of the bunch

Higgs Factory, a complement to the LHC for studying the

particle population Ne, this kind of heating effect will get

Higgs. The low Higgs mass makes a circular Higgs

stronger and the beam emittance will increase. There is a

Factory possible. Compared with the linear collider, the

limit condition beyond which the beam emittance will

circular collier as a Higgs Factory has mature technology

blow up. This emittance blow-up mechanism introduce a

and rich experience. Also, circular Higgs Factory has

limit for beam-beam tune shift [9]

potentially a higher luminosity to cost ratio than a linear y 

one at 240 GeV [3]. So, much attention is given to the design of circular Higgs Factory and several proposals

2845 2

T0

 y N

(1) IP

have recently been put forward [4-8]. In order to find the

Where NIP is the number of interaction point (When there

optimized machine parameter design started from the

are NIP interaction points, the independent heating effects

required

have to be added in a statistical way), y is the transverse

luminosity

goal,

beam

energy,

physical

constraints at IP and some technical limitations, we study

damping time and T0 is the revolution time.

a general analytical method for the parameter choice

3.

based

to beamstrahlung

on

the

maximum

beam-beam

tune

shift,

beamstrahlung-driven lifetime and beamstrahlung energy

Beam lifetime limit and energy spread limit due When two head-on colliding electron and positron

spread.

beams penetrate each other, every particle in each beam

2.

will feel the electromagnetic field of the other beam and

Beam-beam parameter limit coming from beam

emittance blow-up

will be defected. This deflection process has some

In e+ e- storage ring colliders, due to strong quantum

undesirable effects. Firstly, the deflected particle will lose

excitation and synchrotron damping effects, the particles

part of its energy due to the synchrotron radiation, called

are confined inside a bunch. The position for each particle

as beamstrahlung, which will increase the energy spread

is random and the state of the particles can be regarded as

of the colliding beams, and hence increase the uncertainty

a gas, where the positions of the particles follow statistic

of the physical experiments. If the beamstrahung is so

laws. Apparently, the synchrotron radiation is the main

strong that particles’ energy after collision is beyond the

1) Corresponding author at: +86-10-88236743; E-mail address: [email protected] (D. Wang).

ring’s energy acceptance, they may leave the beam and strike the vacuum chamber’s walls, and hence beam lifetime is decreased. Secondly, the deflected particles will emit photons, hadrons, etc., which will increase the noise background level in the detector. Additionally, after

second kind. From eqs. (1) and (4) one finds a limit for the luminosity L0  c m 

2

s

1

  0 .7  1 0 

34

1  r 

E 0 G eV

1



cm  y 

 I b  m A  P0  M W  N

*

IP

the collision particles will change their flying direction L m ax  L 0 Fh

with respect to the axis by a certain angle. If this angle is large enough the particles after the collision will interfere with the detection of small-angle events.

(6) (7)

In our method, the goal peak luminosity L0, the energy of the ring E0, the bending radius of the main dipole

In order to control the extra energy spread by

magnets , the synchrotron radiation power P0 (machine

beamstrahlung to a certain degree, we introduce a

technical constraint), the aspect ratio r and the IP number

constraint in this paper as

NIP are the known quantity. From these input parameters  BS 

1 3

(2)

0

one gets first 4

U

where 0 is the nature energy spread and BS is the extra

0

 8 8 .5  1 0

energy spread due to beamstrahlung.

3



Ib 

V. I. Telnov [10] pointed out that at energy-frontier

(8)

E 0 (G eV )

(9)

P0 U

0

e+e− storage ring colliders, beamstrahlung determines the Cq

0  

beam lifetime through the emission of single photons in

(10)

J

the tail of the beamstrahlung spectra. Unlike the linear collider case, the long tails of the beamstrahlung energy

where U0 is the energy loss per revolution due to

loss spectrum are not a problem because beams are used

synchrotron radiation, Ib is the average beam current,

only once. If we want to achieve a reasonable

Cq=3.83210-13 m is a constant and J is the longitudinal

beamstrahlung-driven beam lifetime of at least 30

damping partition number (In general case, J=2.).

minutes, we need to confine the relation of the bunch population and the beam size as follows [7, 11] N

 0 .1

e

 x *

z

Then the vertical beta function at IP can be got according to eq. (6)



(3)

3  re

y  *

2

where x* and z are the horizontal and longitudinal beam

y 

2845

structure constant (1/137). 4.

The luminosity of circular collider is expressed by 2

s

1

]  2 .1 7  1 0

34

1  r   y

(1  r )

E 0 I b P0

N

L0

(11)

IP

2

T0

 y N



2845

U

2

2 E 0 N

IP

(12)

0 IP

Recalling the original definition of the beam-beam tune

Beam parameters calculation

L[cm

34

And the maximum beam-beam tune shift is [9]

size at IP, re is the electron classical radius (2.81810-15m),

 is the energy acceptance of the ring and  is the fine

0 .7  1 0

eE 0 (G eV ) N b N T0  y ( c m ) *

e

shift, for the flat beam, it can be expressed by Fh

(4) y 

where r=y*/x* is the aspect ratio of the bunch, y* is the

N e re 

*

(13)

y

2    x *

* y

beta function value at the interaction point, y is the

where x* and y* are the bunch transverse dimensions

vertical beam-beam tune shift and Fh is the luminosity

after the plasma pinch effect (two colliding bunches are

reduction factor due to hour glass effect which is

fully overlapped).

expressed as follows Fh 

From eq. (13), one finds 

* y



 y   y  exp  K0  2  2   2 z   2 z  *2

z

*2

(5)

where K0 is the zero order modified Bessel function of the

N

e

 x *

* y



2  re 

*

y

y

Combining eq. (3) with eq. (14), one has

(14)

0 .2     y

2

Ne



 x  y 2

3 re  3

z

4

(15)

U

*

0

 8 8 .5  1 0

Ne

0



 x y

(16)

U

3

z

So, according to eq. (15) and (16), we get 



x

5 .7 7  0 

0

Cq

0  

2 .6 re  r

2845

    y r

*



y



y

 r



y



y

x





y

 r

*

(19) 

*

y



y

x 

(20)



x 



2

(21)

x

x

2   y re 

*



y

Fh 

3  re N e 2

z



0 .1  

(23) x

Finally, in order to calculate the total bunch number, eN eN

(T0 

b

T0

C0

)



y



y



y

2

(33)

*

(34)

 

2

(35)

x

x

*

 x

(36) y

y

z



3  re N e

*

b

(37) x

 y   y  exp  K0  2  2  2  z    2 z 

(38)

I bT 0

(39)

*2

y



0 .1  

z



eN

*2

e

L  L0 Fh

(24)

5.

(40)

Optimized design for a 50 km Higgs Factry

c

where T0 is the revolution time which is decided by the circumference of the ring C0.

5.1. Parameter scan Using the method above, we scan the goal luminosity

Then, having the bunch population eq. (22), it’s easy

(L0) with different bending radius , IP number NIP and energy acceptance  (All the input parameters including

to get the bunch number N

b



I bT 0 eN

(25)

e

As a summary, we obtain a set of machine parameters with luminosity goal L0,



N

we have to refer the expression for the average current Ib 

(31)

(32)

re 

y

And also from eq. (3) one gets 

* y

2

(22)

 x

IP

x

2   y

Ne 

From eq. (13) one gets Ne 

(30)

    y r

*

x 

N

5 .7 7  0 



E 0 I b P0

2



x 

(1  r )

(29) IP

(18)

x



34

L0

example) and the aspect ratio r, one can get the vertical beam size and horizontal emittance:

2 E 0 N

0 .7  1 0

y 

With certain given coupling factor  (0.005 for

U0

2

(17)

y

(28)

J

 y ,m ax 

*

(27)

P0

Ib 

eq. (2), one finds 2



y

From the constraint of beamstrahlung energy spread in

(26)

E 0 (G eV )

3

the peak luminosity and the technical limitation are listed in table 1). We get some meaningful results which are shown From Fig. 1 to Fig.8. Fig. 1 shows that larger

beam energy E0, ring

luminosity needs smaller vertical IP beta function, and

circumference C0, IP numbers NIP, bending radius ,

larger bending radius and less interaction point can lose

synchrotron radiation power P0, aspect ratio r, coupling

the IP beta, while the energy acceptance will not affect IP

factor  and energy acceptance  as input.

vertical beta function. Fig. 2 shows smaller bending radius and less interaction point give larger vertical beam-beam

tune shift, while the parameter y has no relation with

design the low emittance lattice with much high energy of

peak luminosity and energy acceptance. Fig. 3 shows that

120 GeV (same conclusion as [7]). Also we see that larger

larger luminosity needs smaller bunch population and

bending radius, more IP and smaller energy acceptance

larger energy acceptance will decrease the bunch

will relax the limit for emittance. Fig. 6 tells us that the

population, while the interaction number and bending

bunch length has no relation with the peak luminosity and

radius will not affect the bunch population. Fig. 4 tells us

IP number. While smaller bending radius and smaller

that we need more bunch number to get higher luminosity,

energy acceptance can help to increase the bunch length.

and also smaller bending radius and smaller energy

Finally, Fig. 7 and Fig. 8 shows less interaction point,

acceptance can reduce the total bunch number. Meanwhile

larger bending radius and larger energy acceptance

the bunch number has no relation with the IP number. For

produce larger hour glass factor and hence larger

the single ring collider, bunch number should not be too

luminosity. So if we want to increase the luminosity we

large due to the parasitic beam-beam effect. Fig. 5 shows

have to increase the bending radius and energy acceptance

that

while reduce the IP number.

higher

luminosity indicate

smaller

horizontal

emittance (a few nanometer) which suggest a difficulty to TABLE I. Input parameters for machine design Energy

Circumference

E0

C0

120GeV

50 km

Goal luminosity L0 34

-2 -1

1610 cm s

IP number

SR power

Bending

aspect

Couplin

Energy

NIP

/beam P0

radius 

ratio r

g 

acceptance 

12

50 MW

56.2 km

200

0.005

5%12%

FIG. 3. Bunch population as the function of peak luminosity FIG. 1. Vertical beta at IP as the function of peak luminosity

FIG. 4. Bunch number as the function of peak luminosity

FIG. 2. Vertical beam-beam tune shift as the function of peak luminosity

minimum value for IP number. Assuming the maximum fill factor of the dipoles is 80%, 6.2 km bending radius will be a limit for the 50 km ring. Then what we need to consider about is how large the energy acceptance can reach and which parameter constraints the enlargement of energy acceptance. 5.2. Constraints from RF system As long as a set of beam parameters is determined, we FIG. 5. Horizontal emittance as the function of peak luminosity

need to check the RF system to see if the bunch length and energy acceptance can be achieved. Firstly, considering the synchrotron radiation energy loss have to be compensated by the RF cavities, one finds U

0

(41)

 e V r f s in  s

where Vrf is the total voltage for the RF cavities and s is the synchrotron phase. According to eq. (41), one gets  U

 s    a r c s in 

0

 eV rf 

FIG. 6. Bunch length as the function of peak luminosity

   

(42)

The nature bunch length is expressed by 

z





2  E 0

p

f rf T 0 e V rf c o s  s

(43)

R 0

where p is the momentum compaction factor, frf is the RF frequency and

R

is the average radius of the ring. Then,

the expression for the energy acceptance is  

FIG. 7. Hour glass factor as the function of peak luminosity

where

q 

e V rf U

 0  f rf T 0 E 0 

2U



p

q  1  a rc c os( 2

1  ) q 

(44)

. Combining the eqs. (43) and (44), we

0

can get the RF frequency frf and the momentum compaction p for given RF voltage Vrf and energy acceptance . In order to see how large the energy acceptance we can get, we make a scan of energy acceptance with different RF voltage (The bending radius is fixed to be =6.2 km). The results are shown in Fig. 9 and Fig. 10. From Fig. 9, one finds that larger energy acceptance FIG. 8. Real luminosity as the function of peak luminosity

need higher RF frequency and lower RF voltage indicates

Overall speaking, we should decrease IP number,

lower RF frequency for the fixed energy acceptance. Fig.

increase bending radius and energy acceptance in order to

9 shows a linear dependence of the RF frequency to the

achieve higher luminosity. Obviously NIP=1 is the

energy acceptance. If one wants to choose 350 MHz RF

frequency like LEP2 the corresponding energy acceptance is about 3%, and if one prefers 1.3GHz RF technology the energy acceptance will be about 8%. In other words, the maximum luminosity which we can obtain is closely related with the RF technology (frequency). From the beam dynamics point of view, lower RF frequency is a better choice because the cavities with lower frequency have larger aperture and hence lower impendence which is a favor for the collective instabilities. Also considering there are still technical difficulties to directly use ILC 1.3

FIG. 10. Momentum compaction factor as the function of energy

GHz SC technology on storage rings [12], it’s better to

acceptance

choose the frequency lower than 1GHz (700 MHz for example).

From Fig. 10, we can see that the requirement of enlarging energy acceptance is translated to design a low momentum compaction lattice and also larger RF voltage will relax this tolerance and lose the difficulties of lattice design. So we need to make a reasonable choice for the total RF voltage while balancing the constraints from the RF frequency and momentum compaction. 5.3. Optimized machine parameters Combining the discussions in 5.1 and 5.2, we get a set of new designs for the 50 km Circular Higgs Factory with three typical RF frequencies corresponding to different RF technology (Table 2). For these designs, we choose

FIG. 9. RF frequency as the function of energy acceptance

=6.2 km to get the maximum luminosity and each time the peak luminosity L0 is raised to a highest value until the minimum y (confine y at IP will not smaller than 1 mm) is reached.

TABLE II. Optimized parameters of Circular Higgs Factory (CHF) with different RF technology 350 MHz (LEP2-like)

700 MHz

1.3 GHz (LEP3-like)

technology

technology

technology

Number of IPs

1

1

2

1

Energy (GeV)

120

120

120

120

Circumference (km)

50

50

50

50

SR loss/turn (GeV)

2.96

2.96

2.96

2.96

Ne/bunch (1012)

1.61

0.79

1.12

0.33

Bunch number

11

22

16

53

16.9

16.9

16.9

16.9

50

50

50

50

0.065

0.065

0.065

0.065

Bending radius (km)

6.2

6.2

6.2

6.2

Momentum compaction

0.43

0.38

0.38

0.21

Beam current (mA) SR power /beam (MW) B0 (T)

(10-4)

IP x/y (m)

0.2/0.001

0.2/0.001

0.2/0.001

0.2/0.001

Emittance

x/y (nm)

29.7/0.15

14.6/0.073

29.1/0.15

6.1/0.03

Transverse

IP (um)

77/0.38

54/0.27

76/0.38

35/0.17

x/IP

0.103

0.103

0.073

0.103

y/IP

0.103

0.103

0.073

0.103

VRF (GV)

4.1

6

6

9.3

f RF (MHz)

350

704

704

1304

z (mm)

4.6

2.2

2.2

0.95

Energy spread (%)

0.13

0.13

0.13

0.13

Energy acceptance (%)

3.5

5

5

7.7

BS (10 )

9.7

13.8

13.8

21.3

n

0.86

0.6

0.6

0.39

BS (10-4)

4.3

4.3

4.3

4.3

30

30

30

30

0.49

0.68

0.68

0.87

2.2

3.1

2.2

4.0

-4

Life

time

due

to

beamstrahlung (minute) F (hour glass) 34

-2 -1

Lmax/IP (10 cm s )

suggestions from Professor Yifang Wang. This work was

6. Conclusion

supported by the National Foundation of Natural Sciences

In this paper, a general method of how to make an

Contract 11175192.

optimized machine parameter design of a circular e+eHiggs Factory by using analytical expression of maximum

1

beam-beam tune shift and beamstrahlung beam lifetime started from given luminosity goal, beam energy and

G. Aad et al. [ATLAS Collaboration], Phys. Rev. Lett.108, 111803 (2012).

2

technical limitations was developed. By using this method,

S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B710, 403 (2012).

one reveals the relations of machine parameters with goal

3 A. Blondel, A. Chao, W. Chou, J. Gao, D. Schulte and

luminosity clearly and hence give an optimized design in

K. Yokoya, IHEP-AC-2013-001, February 15, 2013.

an efficient way. Also, we point out that the highest luminosity which we can get is closely related with the RF technology (frequency) and higher luminosity favors higher RF frequency. So the maximum luminosity that is

4

A.

Blondel

and

F.

Zimmermann,

CERN-OPEN-2011-047, arXiv:1112.2518 [hep-ex]. 5 A. Blondel et al., CERN-ATS-NOTE-2012-062 TECH, 2012.

realizable is subject to the detail RF technology. Finally a

6 K. Oide, KEK Seminar, 13 February 2012.

series of optimized designs with different RF frequency

7 Yunhai Cai, HF2012, November 15, 2012.

for 50 km Circular Higgs Factory was proposed based on

8 Q. Qin, et al., HF2012, November 15, 2012.

beam parameters scan and RF parameters scan. Up to now,

9 J. Gao, Nucl. Instr. and methods A533（2004）p.

the luminosity we got is the highest one among the exist designs.

270-274. 10 V. Telnov, arXiv:1203.6563v, 29 March 2012.

Acknowledgments The authors would like to thank the support and

11 V. Telnov, HF2012, November 15, 2012. 12 Andy Butterworth, HF2012, November 15, 2012.