Crab Crossing in a Large Hadron Collider

CRAB CROSSING IN A LARGE HADRON COLLIDER

Jie Wei Brookhaven National Laboratory, Upton, New York 11973, USA

ABSTRACT eﬁts of a relatively small bunch spacing, the beams must cross at an angle of about 70 r to avoid more than one bunch-bunch Since its invention by Palmer[1] in 1988, crab crossing has collision in each experimental straight section. Such a non- been explored by many people for both linear and storage ring zero crossing angle causes a degradation in luminosity. Con-

colliders to allow for an angle crossing without a loss of lumi-

2 sequently, it demands a short bunch length ( cm) which nosity. Various crab crossing scenarios have been incorporated can only be achieved with a large rf voltage (100 MV) when op- in the design of newly proposed linear colliders and B-factory erating at a frequency of 379 MHz. The problem can be solved projects. For a hadron collider, this scheme can also be em-

by crab crossing the two counter-circulating proton beams to ployed to lower at the interaction point for a higher luminos- make them collide head-on. ity. In Section II of this paper, we first summarize the princi- In this paper, we first review the principle and operational re- ple and operational requirements of three crabbing schemes for quirements of various crab crossing schemes for storage ring storage ring colliders. A Hamiltonian formalism is developed in colliders. A Hamiltonian formalism is developed to study the Section III to calculate the emittance growth and crabbing qual- dynamics of crab crossing and the related synchro-betatron cou- ity degradation produced by the errors in voltage and betatron- pling. Requirements are obtained for the operational voltage phase matching of the crab cavities. The results are applied to a and frequency of the crab cavities, and for the accuracy of volt- conceptual design of angle crossing in the proposed high-field age matching and phase matching of the cavities. hadron collider. Conclusions and a discussion are given in Sec- For the recently proposed high-field hadron collider,[2, 3] a

tion IV.

0:1 deflection crabbing scheme can be used to reduce from m to 0.05 m and below, without a loss of luminosity due to angle II. CRAB CROSSING SCHEMES crossing. The required voltage of the storage rf system is re- The goal of crab crossing in a storage ring collider is to make duced from 100 MV to below 10 MV. With the same frequency the two counter-circulating angle-crossing beams collide head- of 379 MHz operating in a transverse mode, the required voltage on at the IP without sacrificing beam quality and luminosity of the crab cavities is about 3.24.4 MV. The required accuracy lifetime. In this section, we present three schemes: deflection

of voltage and betatron-phase matching is about 1%.

crabbing,[1] dispersive crabbing,[4] and crabbing.[5] The z I. INTRODUCTION angle crossing is assumed to occur in the horizontal (x– )plane. The subject of angle crossing at the interaction point (IP) of A. Deflection Crabbing a storage ring collider has been studied for many years with the With the deflection crabbing scheme, as shown in Fig. 1, two realization that the synchro-betatron resonance induced by the transverse deflectors (rf cavities operating at their transverse crossing angle severely limits the luminosity. In 1988, a beam- beam collision scheme was invented by Palmer[1]to allow a large crossing angle for a linear collider without a loss of lumi- p p nosity. The Palmer scheme (or deflection crabbing scheme) em-

ploys transverse rf deﬂectors placed at locations where the beta-

90 tron phase advance is from the IP. Both colliding bunches are tilted by the cavities by half the crossing angle at the IP α so that they collide head-on. Subsequently, several alternative schemes[4, 5, 6, 7] were also introduced to apply crab crossing to storage ring colliders. Recently, various crab crossing sce- IP narios have been incorporated in the design of newly proposed linear colliders[8] and B-factory projects.[9] The design goal of a high-ﬁeld hadron collider[2, 3, 10] is

a 50 TeV storage ring that can achieve a peak luminosity of

34 2 1 10 cm s with a small number of interactions per bunch- Figure 1: Schematic view of deﬂection crab crossing of two

bunch collision. With an experimental drift space of 25 m and counter-circulating beams. The crossing angle between the two a focusing strength of 360 T/m at the triplet quadrupoles, a of beam trajectories is . 0.1 m can be achieved at the energy of 50 TeV. To enjoy the ben-

346 modes) are positioned on each side of the IP, preferably at high-

Table I: Comparison between nominal and crabbing operations.

locations with betatron phase advances of 90 from the IP.

s 90

At an azimuthal location 0 with a betatron phase of from

the IP, the particle receives a kick in the horizontal direction x, Quantity Unit Nominal Crab I Crab II

p=p

along with a change in momentum ( ), m 0.1 0.1 0.05

r707097

K R h

0 c

V 0

rf MV 100 10 10

x = z ; sin

h R

V c

cr ab MV 0 3.2 4.4

(1)

cr ab

MHz – 379 379

z mm 22 41 41

= K x cos z ;

34 2 1

R L

ini 10 cm s 1.1 1.1 2.2

dx=ds z

where x , is the longitudinal displacement from the

= E=c C =2R rf bucket center, p is the momentum, is the

which results in an offset in the betatron closed orbit, K

circumference, and the strength 0 of crabbing is related to the

V 0

peak voltage c of the cavity by

x = ;

0 0

(7)

qeV h

c0 c

V h

= ; 0 0 K where and are the peak voltage and harmonic number of

0 (2)

RE the rf cavity. At the IP where the dispersion is zero, the bunch

qe h

with the electric charge of the particle, and c ,aninteger,the is tilted by an angle , with

harmonic number of the cavity. If the crabbing wavelength is s

x 0

much larger than the bunch length, i.e.,

tan (8)

2 z

;

z (3)

c V

The voltage 0 required is thus

0 s

the kick in x is approximately linearly proportional to the dis-

0 z

placement z . At the IP, this kick results in a -dependence of

qeV = tan :

0 (9)

h 2

0 0

the horizontal displacement x. Thus, the bunch can be tilted by

=2 x z tan( =2) K 0 an angle in the – plane with 0 ,

A second cavity located at a place with a betatron phase of

where and 0 are the functions at the IP and the cavity

+180 from the IP operates at the same voltage as the ﬁrst one s

( 0 ), respectively. The voltage required is thus

to restore the particle motion. The dispersion and function at

RE the second cavity needs to be the same as at the ﬁrst one.

qeV = tan : 0

c (4)

h The dispersive scheme usually requires a large dispersion at

c 0

the cavity locations along with a large operating rf voltage. For

Obviously, for given and , a high- location and a high the high-ﬁeld collider, the rf cavities are assumed to be posi- h

operational frequency (or harmonic number c ) is preferred for

200

tioned at places with 0 m. Even with a high disper-

the cavity, provided that the condition Eq. 3 is satisﬁed. The

=10

sion of 0 m, an impractically high voltage of more than

s +90

second cavity located at 1 with a betatron phase of from

=70 1 GV is required for a r crossing. Furthermore, since the IP needs to operate at a voltage the required dispersions at the two cavities are the same, addi-

s tional dipoles are needed to make the dispersion at the IP zero.

= V

V The longitudinal slippage between the two cavities produced by

1 c0

c (5)

1 these dipoles will inevitably degrade the crabbing accuracy.

to restore the particle motion to its unperturbed state. C. Crabbing

For the high-ﬁeld hadron collider, the crab cavities can be Another scheme for crab crossing is not to employ any ded-

50 positioned at places with 0 km. With a frequency of icated cavities. Instead, the storage rf cavity is placed near the

379 MHz, the required voltage for the transverse cavities is be- IP,and the dispersion at the IP is made non-zero. With a peak

tween 3.2 and 4.4 MV for a crossing angle between 70 and

V h

voltage rf and harmonic number , the rf cavity changes the

97 r,asshowninTableI. momentum of the particle,

B. Dispersive Crabbing qeV h

= sin z : (10)

An alternative scheme for crab crossing is to employ two reg- 2

E R

ular (instead of transverse deﬂecting) cavities located at disper-

sive regions, where the betatron phase advances are 180 from Due to the dispersion , the horizontal displacement at the

IP resulting from this momentum change produces a tilt in the

the IP. At the ﬁrst cavity where the dispersion is 0 , the particle

tan( =2) =z

receives a kick in momentum, bunch with . The voltage required is thus

qeV h

0 0

= sin z ;

qeV = tan :

(6) rf (11)

E R

h 2

347 V

For a moderate voltage rf , this scheme often requires a large average the contribution of the storage rf cavities over the cir-

dispersion at the IP. Such a dispersion effectively enlarges cumference, and deﬁne

the horizontal beam size at the IP, which inevitably causes a 2

h ! qeV

; and C ; C W

degradation in luminosity. Furthermore, the dispersion mod- (15)

ulates the beam-beam interaction to give nonlinear synchro- h

= h =i1= betatron coupling. where x is the slippage factor. The Hamil-

For the high-ﬁeld collider, we assume that an rf cavity of tonian (Eq. 13) can be expressed in terms of the action-angle

J J J s

x y y z z

50 MV is located near each one of the two IPs. To achieve variables ( x , , , , , ; ) using a canonical transfor-

=70

an angle of r, the needed is more than 4 m. With a mation. The new variables are related to the old ones as

2 10

momentum spread of p in the beam, such a large

2 J

x;y x;y

x; y = cos( + );

x;y x;y

dispersion makes the spot size at the IP intolerably large. x;y p

III. THEORETICAL ANALYSIS s

2pJ

In this Section, we develop a Hamiltonian formalism to study x;y

p = [ cos ( + )+

x;y x;y x;y x;y x;y

the dynamics of deﬂection crab crossing, which so far is the x;y

only practical scheme for the high-ﬁeld hadron collider. Using

+ sin( + )] ;

x;y x;y

this formalism, we evaluate the sensitivity of beam quality and x;y 1

machine performance to errors in voltage and phase matching X

cos [m ( + )] ; = 8 z

of the crab cavities. A matrix formalism is also introduced for z

=1;odd

the linearized system to describe the coupling. m

2 C

A. Hamiltonian Formalism

W = c sin [m ( + )] ;

m z z

K(k ) C

The single-particle motion can be described by a Hamiltonian W

=1;odd

m (16)

x p y p W s y

expressed in terms of the variables ( , x , , , , ; )as

where the betatron phase x;y and the azimuthal angle are

H = H + H ; 1 0 (12)

deﬁned as Z

s ds

; :

x;y (17)

= hz =R W E =h! ! s

where is the rf phase, s , is

x;y x;y

E = cp H

the angular revolution frequency, and . Here, 0 W The nonlinear synchrotron motion of and is reﬂected by governs the unperturbed particle motion,[11, 12]

the series expansion with the coefﬁcients

1 p

2 2 2 2

H = p + p K x + K y + p +

K( 1 k )

0 x y

x y

2p 2

c ; exp ;

1 (18)

K(k )

2 2

1 h !

k = H =C 1 J

0 is related to the action variable by the

3 2

2 relation

qeV

(s nC )

J = Wd =8 (k 1)K(k )+E(k) ;

z (19)

n=1

k ) E(k )

where K( and are the complete elliptical integrals[13]

h hs

( p p cos + + x) +

x x s of ﬁrst and second kind. The unperturbed synchrotron tune

R Rp

can be expressed in terms of the linear tune s as

= ; ! = C C :

; + sin ( p p x)

z s0 s0 s W

x x s (20)

2K(k )

Rp (13)

2 2

sin = d =d

= d =ds H

Using the relation[14, 15] , the new

x 1 s0

with x the dispersion, ,and represents the

H =0

K K

Hamiltonian can be obtained as 0 ,and 1

contribution from the two crab cavities of strength 0 and

s s

X 1

located at 0 and , respectively,

[K (s nC s )+ K (s nC s )] H =

0 0 1 1 1

pR h

n=1

H = x sin

h h

2 R

1=2

(2p J ) cos ( + )

x x x x x

hK (k )

[K (s nC s )+K (snC s )] :

0 0 1 1

n=1

(14) X

mc cos [m ( + )] :

z z

For simplicity without losing generality, we assume that the m

=1;odd

crab cavities operate at the same frequency as the storage rf m (21)

h = h J z

system ( c ), and that the machine operates above transi- Note that the dependence on the longitudinal action is con-

= c k m tion in storage mode with the synchronous phase s .We tained in and .

348 B. Error-Induced Coupling Resonance For a difference (or sum) resonance above (or below) transition,

Eq. 21 indicates that if the crabbing is not completely com- the growth rate of the bunch area S is

pensated, synchro-betatron couplings can be excited. Suppose

2 2

1 2m hc iK E dS

m 0

1=2

that is the deviation of the betatron phase advance from

= (2 )

0 x

S dt hS K 180 between the two crab cavities, and that is the devi-

(31)

ation of the crabbing strength from the matching value, "

1=2

; +()

= +; and K = (K +K):

1 0 1

0 (22)

hc i

where the quantity m can be approximated by[14]

Keeping only slowly varying resonance terms satisfying

h ! S h i

s max

hc i ; h i ; max

m (32)

m = l + ; 1; m =odd;

2 z

x (23)

8 E z

and performing another canonical transformation with

with max the maximum phase of the particle synchrotron os- ~

~ cillation.

J = J ; = ;

x x x x

(24) For a hadron storage ring, the synchrotron tune is usually

small. The number m satisfying the resonance condition Eq. 23

~ ~

J =J ; = ;

z z z z

0:003

is very large. In the case of the high-ﬁeld collider, z ,

=0:18 hc i < 10 m> 10 m the ﬁnal Hamiltonian becomes max ,and for . The emittance

growth caused by the synchro-betatron coupling, which is ex-

H = J ; z

0 (25) cited by the crabbing errors, is negligible. mR

On the other hand, for an electron-electron or electron- 1

X positron storage ring collider, the synchrotron tune is often

H = [K (s nC s )+K (snC s )]

0 0 1 1

1 large. The mismatch in crabbing voltage and phase can ex-

n=1

cite strong synchro-betatron couplings of low order m,which

2 results in emittance growth in both horizontal and longitudinal

m Rc

1=2

(2p J ) cos ( m ) ;

x x x z

2 directions.

K (k )

h (26) where the tildes in J and are omitted for brevity. From Eq. 26 C. Off-Resonance Crabbing Degradation

and the canonical equations of motion, the relation Even though the beam is off resonance, the perturbation in

action can still make the crabbing process less accurate. If x

mJ J = constant z

x (27) is the average horizontal tune spread in the bunch, the charac-

teristic decoherence time for the betatron motion is turns.

m = l z holds near a resonance x . Hence, the growth in

action is limited for a sum (or difference) resonance above (or The off-resonance condition for an accurate crabbing is thus J

below) transition. However, since x is usually much smaller

1: J (33)

than z , the growth in the betatron amplitude is important even

for a difference resonance above transition.[12] The change of x

m =1 J

J Using Eqs. 28, 29, and 32 with , this condition can be z action x and in one revolution can be evaluated as

sufﬁciently met if the accuracy of voltage and betatron-phase

m Rc

1=2

J = (2p J ) [K sin ( m )+

matching between the two cavities satisﬁes

x x x 0 x z

hK (k )

1=2

2 1=2

K 2h!

s x x

+(K +K) sin ( m +)] ;

+() ;

x z

0 (34)

K h i cK 2

max 0 0

J = mJ : x

z (28)

h i

where max is given by Eq. 32. Deﬁned the rms horizontal emittance x and longitudinal bunch For the high-ﬁeld hadron collider, the tune spread produced

area S , by beam-beam interactions, magnetic multipoles, etc. is of the

= hJ i and S =2hJ i;

x z x (29) order 10 . If the tolerable deviation in x is 1%, the matching p crabbing voltage needs to be accurate to about 1%, and the beta-

where hidenotes the average over all the particles. The growth tron phase advance between the two cavities needs to be within

rate of the emittance x can be obtained as about 2 of 180 (Eq. 34).

1=2

1 d 2mhc i cK 2

x 0 0

m D. Matrix Formalism

dt h x x The coupling between the horizontal and longitudinal motion

(30) caused by imperfect crabbing can also be illustrated by a matrix

1=2

K formalism. Consider the one-turn matrix at the location of the

+() : 90

deﬂection crabbing cavity, betatron phase from the IP, for K

349

x z variables ( x, , , ). The matrix can be derived by linearizing below. At the same time, luminosity degradation caused by

the kicks by the crabbing cavities, the angle crossing is eliminated, as shown in Table I. Since 3

2 a longer bunch length can be tolerated when crab crossing is n

M employed, the required voltage of the 379 MHz storage rf sys- 5

4 (35) tem is greatly reduced from 100 MV to about 10 MV. With the N m same frequency of 379 MHz operating in a transverse mode,

the required voltage for the crab cavities is about 3.2 MV for

where, to the ﬁrst order of error in strength and phase ,

=0:1 =0:05 m, and 4.4 MV for m. The needed relative

accuracy of voltage and betatron-phase matching is about 1%.

2 3

+ sin sin cos The beam emittance growth caused by error-induced synchro-

x 0 x 0 x

6 7

cos betatron coupling is negligible.

0 x

6 7

M = 6

7 V. ACKNOWLEDGMENTS

4 5

cos sin

sin The author would like to thank G. Dugan, M. Harrison, and

x x 0 x

0 S. Peggs for many helpful discussions.

2 3

1 CU + C K C 0

0 VI. REFERENCES 4 5

N =

U K 1 0

0 [1] Energy Scaling, Crab Crossing and the Pair Problem,R.B.

Palmer, Invited talk at the DPF Summer Study Snowmass 88,

2 3

( K +K) C K

C SLAC-PUB-4707, Stanford 1988.

0 0 0 0

4 5 =

m [2] Beyond the LHC: A Conceptual Approach to a Future High En-

K K K

0 0 0 0 ergy Hadron Collider, M. J. Syphers, M. A. Harrison, S. Peggs,

Dallas, PAC Proc., 1995, p. 431.

2 3

(K cos sin K ) 0

0 x x

0 [3] Really Large Hadron Colliders, G. Dugan, Invited talk at 1996

4 5

= ;

n DPF/DPB Summer Study on New Directions for High-Energy

K (sin + cos ) 0

x 0 x 0 Physics, Snowmass, 1996. (36)

[4] Dispersive Crab Crossing: An Alternative Crossing Angle

U = qeV h=RE =2

x x 0 0

where rf , ,and and are the Scheme, G. Jackson, Fermilab Note, FN-542, 1990.

s x z

lattice functions at 0 . The amount of global – coupling is [5] often characterized[7] by the quantity Single Beam Crab Dynamics, T. Chen and D. Rubin, Conf.

Record of 1991 IEEE Oart. Accel. Conf., Vol. 3 LBL, SLAC,

2 2

y LANL, San Francisco (1991), p. 1642.

det(m + n ) = K C () sin +

0 x 0

[6] Beam-Beam Collision Scheme for Storage-Ring Colliders,K. K

Oide, K. Yokoya,Physical Review A, 40, 315 (1989).

+ (cos sin )

x 0 x K 0 (37) [7] Angle Crossing of Bunched Beams in Electron Storage Rings,T.

Chen, Ph. D. Dissertation, Cornell University, 1993.

2 K

[8] International Linear Collider Technical Review Committee Re-

sin : x

K port, 1995. 0

[9] CESR-B Conceptual Design for a B Factory Based on CESR, 2

For the high-ﬁeld collider with a matching error of 10 , CESR Group, CLNS 91-1050, 1991.

y 10

m + n ) x z det( is of the order of 10 , i.e., the global – cou- [10] Interaction Region Analysis for a High-Field Hadron Collider,J. pling is very small. Wei, et. al., these proceedings (1996). IV. CONCLUSIONS AND DISCUSSION [11] An Introduction to the Physics of Particle Accelerators, M. Conte and W. W. MacKay, World Scientiﬁc Pub. Co., Singapore, 1991. In this paper, we have reviewed the principle and operational [12] Synchro-betatron Resonance Driven by Dispersion in RF Cavi- requirements of various crab crossing schemes for storage ring ties, T. Suzuki, Particle Accelerators, 18, 115 (1985). colliders. A Hamiltonian formalism has been developed to study the crabbing dynamics. Using this formalism, we eval- [13] Beam Life-Time with Intra-beam Scattering and Stochastic Cool- uate the sensitivity of crabbing performance to errors in voltage ing, J. Wei and A. G. Ruggiero, Proc. 1991 Particle Accelerator Conference, San Francisco, 1869-1871 (1991). and phase matching of the crab cavities. The emittance growth caused by matching error induced synchro-betatron coupling is [14] Synchro-betatron Resonance Driven by Dispersion in RF Cavi- also estimated. Requirements are obtained for the voltage and ties: A Revised Theory, R. Baartman, TRIUMPH Design Note, TRI-DN-89-K40 (1989). frequency of the crab cavities to achieve a head-on collision dur- ing angle crossing, and for the accuracy of voltage and phase [15] Hamiltonian Treatment of Synchro-betatron Resonances,G. matching in a deﬂection crabbing scheme. Guignard, CERN Fifth Advanced Accelerator Physics Course,ed.

For the high-ﬁeld hadron collider, a deﬂection crabbing S. Turner, 1995, p. 187.

0:1 scheme can be used to reduce from mto0.05mand

350