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Luminosity lifetime at an asymmetric e+e− collider
Nuclear Instruments and Methods in Physics Research A302 (1991) 209-216 North-Holland
209
Luminosity lifetime at an asymmetric e + e - collider Frank C. Porter
Charles C. Launtsen Laboratory of High Energy Physics, California Institute of Technology, Pasadena, CA 91125, USA
Received 28 March 1990 and in revised form 21 December 1990
The dependence of the luminosity on time is discussed for an asymmetric e + e - storage ring collider, with emphasis on single-particle scattering mechanisms for beam loss. The "optimal" filling strategy and average luminosity obtainable are also reviewed .
1 . Introduction
2. Beam lifetime - general consideration
Asymmetric e + e - storage ring colliders have become extremely interesting lately, because of their potential role in the study of CP violation in the B meson system . In this application, the term "asymmetry" refers principally to the difference in the beam energies of the two colliding beams . More generally, the beams could be "asynnnetric" for other reasons, such as differences in the currents . In this article, I develop the formulas for the dependence of the luminosity on time, for an asymmetric collider. Specific examples are considered of beam loss from bremsstrahlung and Bhabha scattering interactions between the two beams, and from losses due to the interactions with residual gas in the vacuum chamber . Finally, I investigate the exponential approximation for the luminosity time dependence, and remark on the optimal strategy for fill times, obtaining the maximum average luminosity achievable . While the discussion is in the context of e + e - colliders, much of it may be applied with minor modifications to other storage ring colliders as well . Note that much of the material covered here also exists elsewhere in the form of various internal reports [1] . There exist also a number of relevant published articles, for example : ref. [2] contains a summary of beam-loss and current-limiting mechanisms ; ref . [3] provides a discussion of vacuum considerations, including beam lifetime aspects ; ref. [4] calculates the scattering loss on thermal photons; ref. [5] contains a nice discussion of the operational beam-gas experience in the Tristan accumulation ring.
We denote the luminosity of the accelerator at time t by Y(t) . Each beam may consist of a number of bunches, n b,, with a number of particles per bunch Nb ,(t) . The subscript b is used to indicate that this is a quantity for a single bunch, and the subscript i refers to the beam (i = 1,2) . We assume throughout that the number of particles is the same for all of the bunches in a beam, though the number may be different between the two beams. For some of what we say, this assumption may be relaxed, with the interpretation that Nb, is the average number of particles per bunch. In any event, the total number of particles in a given beam is N, = nb,Nb,, There are two categories of particle loss mechanism which are typically of interest : The first category includes those processes which lead to particle loss because of collisions between the particles in the two beam. If the cross section leading to loss of a particle from beam t is a then the loss rate depends on the luminosity according to :
Work supported in part by the Department of Energy, contract DE-AC03-81ER40050 .
The cross section is assumed to be time-independent. An effective instantaneous beam lifetime is : T, (t) -
~((0 ,
(
The beam-beam interaction may also contribute to the lifetime, by causing particles to eventually encounter the physical aperture . The effective cross section for this loss mechanism may be time-dependent as it depends on time-dependent accelerator parameters . We do not explicitly include this case here, which requires sophisticated methods to obtain lifetime estimates.
0168-9002/91/$03 .50 C 1991 - Elsevier Science Publishers B .V. (North-Holland)
210
F. C Porter / Luminosity lifetime at an asymmetric e + e
The second category includes single-beam particle loss mechanisms, such as intrabeam scattering, quantum fluctuations, and scattering from residual gas in the vacuum chamber. We write the basic equation for this category in the form :
N d (t)= -rr(t)N(t)
N,(t) = N o e - '-, ' .
(4)
We have introduced here the notation N o -- N,(0), and we shall also use ~0 =Y(0), to denote quantities evaluated at t = 0 . In the following discussions, we will assume that r, is independent of time. As long as we evaluate T, at t = 0, this will ordinarily be a conservative assumption, since any time-dependence is likely to be a decrease in the loss rate with time.
3. Luminosity model and luminosity lifetime To go from our beam lifetimes to a luminosity lifetime, we need to know how the luminosity depends on the beam currents . This dependence is determined to some extent by the operation of the storage ring . We consider a simple model for the machine operation which provides a conservative model for the luminosity function . The model is to assume that the bunch sizes do not vary with time, and the luminosity is given by : (t)=
eq. (5) leads to two coupled differential equations in the beam currents : da l = - ka , N, N2 ,
Nb](t)Nb2(tM 2 , r (ai+aX ) ( eyl+ay2)
The vx,y, , in this equation are the transverse rms spot sizes at the interaction point (IP), and should not be confused with the cross sections v, . The bunch crossing frequency is f. . All terms which are time-dependent are so indicated. Note that this is a place where we use the assumption that all bunches in a given beam have the same number of particles. We also assume that any modifications to the above formula from considerations such as finite bunch lengths and nonzero crossing angles are time-independent . If there is more than one interaction region, the appropriate luminosity in this discussion is the sum of the luminosities from each region . For particle loss mechanisms in the first category,
--i-t2
k G2 NI N2 ,
(6)
where k =_
We have kept the possibility here that the loss rate r, depends on time . For example, it may be that the vacuum is a function of the beam current, and hence, the loss rate also depends on beam current . If F, does not depend on time, then the beam current in the second category depends on time according to a simple exponential :
colltder
o NONzo
(7)
The solution is : 1-r N1(t) =Nlo e ct -r >
N2(t)
1-r
-Nzol- re -ct'
where al
G
~°~ N1o
a2 Nzo ~'
(9)
and r = N1°a2 . N2oa1
(10)
The above result yields a time-dependence of the luminosity of : _ )2 . Y ect( Y= Y, l ect rr
(11)
The time, tf , it takes for the luminosity to decrease to a fraction f of its initial value is given by : tf=
G
ln~2f[(1-r)2+2fr
The integrated luminosity up to time t is :
ftY(t') dt'=Y° 1 G r ( ect-r ) .
(13)
A useful limiting case of the above is when r = 1 (and G = 0), such as would be the case in a truly "symmetric" machine. In this limit, we have the simpler result ( ) t ()
1
NO
(14) 1
=YO [ j
(15)
and f = aNe`8o ( VJ
_
1)
.
(16)
Finally, the integrated luminosity in this case is :
f°t~(t') dt'=Y0 1 + (a,~°/No)t
(17)
F C. Porter / Luminosity lifetime at an asymmetric e + e -
For particle losses in category two, say at rate F1 for beam 1 and T2 for beam 2, the time-dependent luminosity in this model is of the simple exponential form : P(t) =Y0 e-(F,+TZ)1
(18)
The model we have just discussed is, of course, not the only possibility . A more optimistic model than the first is to assume that our machine is operated so that the spot sizes are reduced as particles are lost, in such a way that the beam-beam linear tune shifts (Op,,y,,) are constant . For this case, the luminosity equation (again neglecting modifications due to contributions from such things as finite bunch lengths and nonzero crossing angles) may be written as :
f
al
(L1+ a2 X
where a'
_
L
1
1
+
Ovy, ß* '
ay
)1~
_ Y,,Nbl Nbl wy1 ßv2 y2Nb2
wy 2 ßy1
N E N O
4
n 0 0
001
002
003
004
005
Kmin/E
a2 ay (1+a 1 )]
a1(1+a 2 ) + ay a2 (1+a 1
6
(3 .15)
( 19)
If we further suppose that the beta functions are not varied in time, then this equation is expressed in a form such that the only time-dependence appears in the bunch populations, as desired. The differential eq . (1) may then be integrated analytically, although the result is cumbersome, and not particularly illuminating . In any event, we advocate our more conservative model for use as predictor of machine lifetimes . However, before leaving this, we mention an interesting subcase, where we assume that the currents in the two beams have a time-independent ratio. This situation holds for a truly "symmetric" machine, or somewhat more generally, for a machine in which "Garren's rules" [61 (all tune shifts equal to the same value and spot sizes at the IP the same for the two beams) hold independently of time. For this special case, the time-dependence of the luminosity is the same as that for either of the beam currents, and the form is a simple exponential, with decay time constant -r = No/(a, ?0) for category one processes, and T = 1/T, for a category two process. Note that the assumptions of this special situation also constrain the particle loss rates from the two beams to be related.
Fig . 1 . The e + e - -> e - e - y bremsstrahlung beam loss cross section (eq. (20)) as a function of energy aperture for various values of center of mass energy : From bottom to top, Ec ,=1, 10, 100, 1000 GeV
category is loss due to bremsstrahlung (e + e - - e + e - y) of a photon, which can change the energy of a beam particle sufficiently to put it outside the energy acceptance of the accelerator. An excellent approximation for the cross section to lose a particle from beam i due to bremsstrahlung is [71 : Qbrems i
16are 2
+ I (In
In
M2 Ec2m
E' nun r
)2
-
En
(In 2I ) kmin, 3
IT 2
5 8 (20)
In this expression, k,r n, is the minimum energy of the radiated photon which causes loss of a particle from beam t. Thus, kn n,/E, can be taken as the fractional energy aperture x1 of the machine for beam i. In an asymmetric machine, a collision may lead to particle loss from only one beam . Hence, we are interested in the effect on a single beam at a time, and we calculate the desired cross section separately for each beam. The dominant radiation from a given beam is kinematically well separated from the radiation from the other beam ; therefore interference effects may safely be neglected . Fig . 1 shows the bremsstrahlung beam loss cross section calculated according to eq . (20) . The cross section depends slowly on the energy aperture and on Eem . A typical machine might have an energy aperture of order 1% . For Eem = 10 GeV, the cross section is abrems ,
4. Specific particle loss mechanisms We consider here some specific particle loss mechanisms at a level we can treat analytically . All cross sections considered are valid in the high energy limit. Typically the most important mechanism in the first
#1 The energy aperture may be limited by the RF voltage, or by
the transverse aperture as the off-energy particle propagates into dispersive regions of the lattice . We ignore here complications when the dispersion at the interaction point is nonzero .
212
F C. Porter / Luminosity lifetime at an asymmetric e + e - colltder
= 2.6 X 10-25 CM2. Under these conditions, to obtain a two hour luminosity lifetime in a symmetric machine at P= 103acm-2 s-1, assuming model one, we require a total of N, o = nb,Nb,o = 2.9X 10 13 particles in each beam (presumably distributed among a number of bunches) . "Luminosity lifetime", as we use it, means the time it takes the luminosity to decrease to 1/e of its original value, e.g ., as t1le in eq . (12) . Another loss mechanism in the first category, typically not as important as the bremsstrahlung considered above, is the loss due to Bhabha (e +e- -+ e+ e - ) scattering at sufficiently large angles to escape the acceptance of the machine. The lowest order differential Bhabha cross section is : [9J: da d2
a2 2E,m
+ 2 (1
I I
1 + cosy B2
Bcm sina
_2
sine km
2
I,
+ cos20Cm )
(21)
where Bcm is the scattering angle in the center of mass . Note that cross sections in units of GeV -2 may be converted to cm2 by multiplying by 3.89 X 10 -28 GeV 2 cm2 (or by the replacement a2 _ rné e2) . We suppose that a particle will be lost from beam i if it is scattered by an angle larger than some minimum Bcm min 1 , Thus, we need the integrated cross section from this minimum to 180' : oBhabha(xmax) =2 -
~I
xmax 1
8Tra 2 2
Ecm
+
2
3
-
dv ~jdcos 0cm
1 ( 1 - xmax ) In 2 + 1 - xmax 3
4
B > Omm = min
) . a(s ß()ß(s)
(1 - xmax )
+ ô \1 - xmax )2 - 48 l1 - xmax )3 J (22) where xm_ = cos Bcm mm, for whichever beam is under consideration . In the following, it will be sufficient to neglect all but the leading order for xmax near one. At this point, let us develop a simple ansatz for the minimum scattering angle which leads to particle loss, which may provide a reasonable estimate in the absence of more detailed tracking calculations . Suppose that an elastic scatter at longitudinal coordinate so in the ring causes a change in angle of the trajectory B. Propagating around the ring, the particle tends to sample a displacement due to this kick of roughly d(s) = B ,ß (so )ß (s) .
(23)
We have suppressed any labelling for the transverse dimension - this minimum angle may be smaller in one transverse dimension than the other. To keep the analysis simple, we conservatively take the minimum angle considering all transverse directions #2. It is often convenient to express the limiting aperture in units of the rms beam spot size, that is, in terms of n(s), the number of standard deviations of clearance : Bmm = E/ß(so) minn (s) .
B
cosy
Thus, if the transverse half-aperture of the ring is a(s), the particle will be lost if :
(24)
Of course, for Bhabha scattering losses, ß(so) = ß *, for whichever transverse dimension is used . Note that Bm,n is measured in the frame of the machine, while the Bhabha cross section is evaluated in the center of mass frame. However, in the limit that the angle is small, we may use eq . (22) with the substitution (the j subscript equals 2 if i = 1, and is 1 if t = 2) #3 : E Bcm nun , = .Bxnm
(25)
Then to a good approximation we may use the following estimate for the cross section to lose a particle from beam i due to Bhabha scattering (keeping only the leading term in the small angle limit) : OBhabha i -
16-na 2 EJ E.2
cm
1
E B2 ' mm
(26)
For example, if E1 = 3 GeV, E2 = 9 GeV (Ecm = 10 GeV), and Omm 1 = 0.01 radians, then aBhabha t = 3 X 10 -28 cm2. This is considerably smaller than the cross section in our earlier bremsstrahlung loss example. Another important lifetime consideration is the loss of beam due to scattering on residual gas in the vacuum
#2On the other hand, we are not being conservative by neglecting the natural emittance (e) and dispersion (il) of the beam. A better approximation is (r 2 t~a I -e2 min Omm ß(S.) s (La(S)2p-r'(S)2s2~ N(S)
where 8 is the relative beam energy spread, and the formula is evaluated in any desired transverse coordinate. Eq. (23) is justified as long as the aperture is large compared with the beam size . #3 Note that this is equivalent to the expression km mini _ (2E, /Ecm) Bm,n since E = 4E,E_, in the high energy limit.
m
213
FC. Porter / Luminosity lifetime at an asymmetric e + e - collider
chamber. Such losses are in the second category, and eq . (3) applies, with F,(t)-
v,CP,(t) TT(t) ,
(27)
where a, is the cross section to lose a particle due to a beam gas collision, c is the speed of light, P, is the residual gas pressure (including the contribution from beam-induced desorption), k is Boltzmann's constant, and T, is the temperature (c/k = 2 .89 x 10 33 s -1 Km -2 Torr -1 ) . We may make a conservative simplification by assuming that P,/T, is constant at the peak value (i.e ., equal to the value at peak current, rather than using the base no-beam pressure), in which case the beam loss from this source has simple exponential behavior . If, in some situation this approach is overly conservative, a more accurate model may be employed, for example by assuming that the running pressure depends on the beam current according to a constant base pressure term plus a term proportional to the current . Refs . [1,3] contain further discussion . Similarly to the beam-beam bremsstrahlung, we may write the cross section formula to lose a particle due to beam-gas bremsstrahlung. However, in this case, we scatter from an atom, so the form is complicated. We use the "complete screening" result of Tsai, plus the Thomas- Fermi-Molière model accurate for Z >- 5 [8] : z dae t Z brems , _ 4are - 3 dy y {3y+y j z z] x [ Z (Lrad - f[( aZ ) )+ZLrad] +(1-y)(Z2+Z)/9 },
(28)
with y --- k,/E and 184 .15 Lrad = In ZF3 , 1194 ( 29) , Lrad =1n Z2/3 f(x) = x
~t
n(nz+x)
1 .202x - 1 .0369x2
+ 1 .008x 3 /(1 +x) . Integrating this expression gives, in the limit kn n , << E, : 16aZre2 3 ~[Z(Lrad - f [(a2) 2 ]) +Lrad, ae t Z brems i E, 5 - 8) rmn ,
x(ln k
+(Z+1)ln (
E, -1)/121 . knun i
100
EU N N
b X T
0 U
e0
60
b 40 0
001
002
003 Kmin/E
004
005
Fig. 2. The cross section to lose a beam particle due to bremsstrahlung on a molecule of carbon monoxide as a func tion of the energy aperture of the machine, according to eq . (30) .
where A is the molecular weight, NA is Avogadro's number, and X0 is the radiation length . The cross section to lose a beam particle due to bremsstrahlung on a carbon monoxide gas molecule is shown as a function of the energy aperture in fig . 2 . We assume that the cross section to scatter on a molecule is the sum of the cross sections on the constituent atoms . As an example, suppose that our vacuum pressure is 10 -8 Torr at a temperature of 40 ° C, and that the energy aperture is 1 % . In this case, ae t Z brems , = 6.7 x 10 -24 cm2 , and 1/T, = 4 .5 h, or the luminosity lifetime (1/e point) is approximately 2 .2 h if the loss rates are the same in both beams . Another potentially important beam-gas loss mechanism is Coulomb scattering resulting in orbits outside the machine aperture . For this case, we must evaluate the minimum scattering angle by making a suitable pressure-weighted average of eq. (23) or (24) around the ring. However, a reasonable first estimate may be obtained by inserting the average ft for ß(s 0 ) . The Coulomb differential cross section for a beam particle of energy E to scatter by an angle 0 is given approximately by the Mott cross section (for a = 1) : 2 (32) Z2a 9 (1-sin2 0) . da 4E 2 sin4
2
(30)
We may typically neglect the term proportional to (Z + 1) . It is then convenient to write this expression in terms of the radiation length of the material : E, - 5 ), 4 A 1 (31) ae t brems = 3 NA X0 (In kmtn , g
We have neglected a number of effects in this simple formula : 1) We have neglected scattering off of the atomic electrons, reasonable when Z >> 1 (i .e ., correction terms are of order 1/Z times the above) . 2) We have neglected screening by the atomic electrons . This is justified in the limit that the wavelength of the virtual photon transferred in the collision from
214
F. C. Porter / Luminosity lifetime at an asymmetric e + e
the beam particle is small compared with the atomic size, i.e., as long as Bm,n >> am,/E, neglecting a weak dependence on Z. 3) We have neglected nuclear form factor effects, and treated it as a static, spinless target . The form factor may be neglected as long as the wavelength of the virtual photon is large compared with the nuclear size . Since the cross section grows rapidly at small angles, this will be a good approximation as long as Bm,n < - 0.1 GeV/E. Likewise, the static approximation is reasonable for (E/M)Bn2 n << 1, where M is the nuclear mass . This condition is also sufficient to justify neglect of the magnetic moment effects. The Coulomb cross section to scatter by an angle greater than Bm,n, is thus approximately given by : 4. Te2Z2 4.rrZ2a2 (33) oe±Z CO-1 1 = = Y,202 E202 e nun ~ nun ~ As an example, on carbon monoxide, the cross section to lose a 5 GeV beam particle, if the minimum aperture is 15a, the emittance is 10 -s mrad, and the average beta function is 10 m, is 4.6 X 10 -24 cm2. At a pressure of 10 -s Torr and 40'C, this yields a single beam lifetime of 1/I; = 6.5 h (realistic cases are usually better than this). At an asymmetric collider, the strong energy dependence of the cross section makes it unlikely that the loss rates due to this mechanism will be matched between the two beams. We remark that the damping times in an e + estorage ring are sufficiently short so that multiple collisions according to the above mechanisms may be neglected.
collider
10
08
04
02
00 L 0
"
7
2
3
r
t/tye Fig. 3. The luminosity (m units of peak luminosity) as a function of time (in units of the actual time it takes the luminosity to fall by a factor of 1/e) for category one processes. Various cases and approximations are shown: i) The solid curve is a simple exponential, with time constant equal to one in these units. ii) The long-dash curve is the actual time dependence for r=1 (eq. (15) . iii) The dot-dash curve is the actual time dependence for r =1/4 and r = 4 (eq. (11)). iv) The dotted curve shows the time dependence for a simple exponential with time constant given by the combinations of the actual 1/e times of the individual beams, for r =1 (i .e ., this would be the appropriate curve for r=1 if the beam currents decayed according to exponentials) . v) The short dash curve is similar to iv), but for r =1/4 and r = 4.
020
---------------
015
5. Approximation by exponential form It would be convenient to be able to approximate the luminosity dependence on time by an exponential, for example in order to combine lifetime expectations from various loss mechanisms. In this section, we show how this can be done with small error for our preferred (for conservative estimates) model of machine operation, for losses from processes in the first category. We are generally interested in times short, or comparable with, the 1/e decay time of the luminosity. Thus, an approximation which works well in this time range will be acceptable. The natural thing to try is to approximate the actual luminosity by an exponential with the 1/e time adjusted to be the same as the true 1/e point of eq . (12) or eq. (16) . This in fact works quite well, as may be seen in fig. 3, which compares an exponential with the dependence of eq. (11) (and eq. (15)) for various values of r, as a function of time in units of ttle . Curves for r and 1/r are identical, since the luminosity formula (see eq .
010
005
0 00
0 05
t/tl ie
3
Fig. 4. The relative error made in the integrated luminosity, for category one processes, as a function of time (in units of the actual time it takes for the luminosity to decrease by 1/e from the peak value). Both curves are for the worst case : r =1 . The solid curve shows the error made by using an exponential form with time constant equal to one in these units. The dashed curve shows the error made by using a simple exponential with time constant given by the appropriate combination of the actual 1/e times of the individual beams, in the approximation that the beam current time dependence is exponential.
RC. Porter / Luminosity lifetime at an asymmetric e + e (11)) is unchanged under the transformation N10/at .For t < the the exponential N20/a2, N20/a2 -> N10/al approximation slightly overestimates the luminosity, and for times t > t1/, it underestimates the luminosity . The approximation is poorest for the case r = 1, but even here the largest error made in the integrated luminosity is only 4.6%, for t < t,/e, as shown in fig . 4 . For times larger than - 2 .7t m/e the error made in the integrated luminosity becomes larger, approaching an asymptotic value of approximately 50% for r = 1, but we are ordinarily not interested in discussing the luminosity after such long times . The integrated luminosity is overestimated until t - 1 .9tm/e , beyond which it is underestimated . Thus, we find that a useful approximation for most practical purposes is to assume an exponential form for the luminosity for category one processes, with time constant given by the according to eq . (12) (or eq . (16)) . Note that a substantially poorer approximation is achieved if we first approximate the individual beam currents by exponentials (with given ttle) and then combine these exponentials to get an exponential for the time-dependence of the luminosity (see figs. 3 and 4) .
6. Optimizing the average luminosity If we accept that the luminosity time-dependence may be approximated by an exponential, then the optimization of the average luminosity under specified in jection conditions is readily obtained . Thus, we start with a luminosity which depends on time according to : P(t) =Yo e - `iT .
(34)
We suppose that a data taking period, T, for the experiment is long compared with the injection time and the beam coast time (i.e ., that there are many such fills in a data run) . In this case, it is sufficient to replace the actual distribution of injection times with a single average injection time, which we call t I . We further assume that we take data for a fixed time interval, t c , following injection, prior to beginning the next injection ; and that each fill begins with the same initial luminosity (Yo) . Finally, we assume that no useful data is accumulated during injection. Our discussion does not depend on whether the injections are "top-off' or start from zero beam current. It is our present interest to find the optimal value for tc . The total integrated luminosity which we accumulate during our data run is given by :
Y dt = n f1'Y,
fo
0
e`/' di,
(35)
where n = T/(t, + t t ) is the number of injection/coast cycles in the run. The ratio of the actual integrated
215
collider
t,it
if .
Fig. 5 . The solid curve shows the optimal coast time between injections, in units of the luminosity lifetime (1/e decay time), as a function of the injection time (also in units of the luminosity lifetime) . The dashed curve gives the average luminosity, in units of the peak luminosity, if the optimal coast time is used .
luminosity to that obtained were the machine capable of running the entire time at its peak luminosity is then : . 0 7 f0
Y dt = t~+ tt ( 1 -
(36)
This quantity (and hence the actual integrated luminosity) is maximized when t. is chosen to satisfy the condition : (tc+ t,)/,r = e 'c /' - 1 .
(37)
(Y) =Yo e `~~T .
(38)
Thus, the maximum average luminosity possible is : Fig . 5 may be used to evaluate t c and the average luminosity given t t and T. For example, if T is 1 h, and t t is 10 min, then the optimal coast time is 32 min, and the average luminosity is 59% of the peak luminosity .
7 . Summary We have investigated the dependence of the luminosity on time in an asymmetric e + e - storage ring coltider . A conservative model for making design estimates is advocated where the bunch sizes are taken to be constant in time . The dependence of the luminosity on time may be reasonably approximated by an exponential law, with time constant given by the 1/e time of the actual time function. The optimal coast time and maximum average luminosity are obtained in this approximation, as a function of the injection time. In situations of current interest, the loss of particles due to e + e - e + e - y bremsstrahlung collisions dominates over loss
F.C. Porter / Luminosity lifetime at an asymmetric e +e - collider
216
due to Bhabha scattering . Depending on details, both
bremsstrahlung
and
Coulomb
losses are potentially significant.
beam-gas
scattering
Acknowledgements I
would
like
to thank Swapan Chattopadhyay Claudio Pellegrini, John Rees, and Michael Zisman for useful discussions . I am grateful to D. Mohl and K. Hiibner for catching an error in the original manuscript.
[2] [3] [4] [51 [6] [7]
References [11 I am indebted to the reviewers for bringing to my attention a recent report which also discusses the asymmetric case : H. Braun, K. Hiibner and W. Joho, PS/LP note 90-11, PSI-TM-12-90-05 (March 1990). That report contains references to several earlier unpublished notes concerning beam
[8] [91
hfetimes . Another recent report of interest is the Feasibility Study for a B-Meson Factory in the CERN ISR Tunnel, ed . T. Nakada, CERN 90-92, PSI PR-90-08 (30 March 1990). J. Le Duff, Nucl . Instr. and Meth . A239 (1985) 83 . C. Benvenuti, R. Calder and O. Gr6bner, Vacuum 37 (1987) 699. V.I . Telnov, Nucl . Instr. and Meth . A260 (1987) 304. T. Momose et al ., IEEE Trans. Nucl . Sci. 32 (1985) 3809 . A. Garren et al ., LBL-26982, Proc . IEEE Particle Accelerator Conf . 1989, Chicago. G. Altarelli and F. Buccella, Nuovo Cimento 34 (1964) 6385 . Note that eq. (20) can also be obtained by integrating eq . (3 .87) of ref. [81 below, paying attention to the definitions of the kinematic quantities, and recognizing that Tsai includes radiation from both particles. Y.S. Tsai, Rev. Mod. Phys . 46 (1974) 815. See, for example, J.D . Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill Book Company, New York, 1964) pp . 135-140 . We disagree with the cross section in M. Conte, Nucl. Instr. and Meth . 228 (1985) 236.
209
Luminosity lifetime at an asymmetric e + e - collider Frank C. Porter
Charles C. Launtsen Laboratory of High Energy Physics, California Institute of Technology, Pasadena, CA 91125, USA
Received 28 March 1990 and in revised form 21 December 1990
The dependence of the luminosity on time is discussed for an asymmetric e + e - storage ring collider, with emphasis on single-particle scattering mechanisms for beam loss. The "optimal" filling strategy and average luminosity obtainable are also reviewed .
1 . Introduction
2. Beam lifetime - general consideration
Asymmetric e + e - storage ring colliders have become extremely interesting lately, because of their potential role in the study of CP violation in the B meson system . In this application, the term "asymmetry" refers principally to the difference in the beam energies of the two colliding beams . More generally, the beams could be "asynnnetric" for other reasons, such as differences in the currents . In this article, I develop the formulas for the dependence of the luminosity on time, for an asymmetric collider. Specific examples are considered of beam loss from bremsstrahlung and Bhabha scattering interactions between the two beams, and from losses due to the interactions with residual gas in the vacuum chamber . Finally, I investigate the exponential approximation for the luminosity time dependence, and remark on the optimal strategy for fill times, obtaining the maximum average luminosity achievable . While the discussion is in the context of e + e - colliders, much of it may be applied with minor modifications to other storage ring colliders as well . Note that much of the material covered here also exists elsewhere in the form of various internal reports [1] . There exist also a number of relevant published articles, for example : ref. [2] contains a summary of beam-loss and current-limiting mechanisms ; ref . [3] provides a discussion of vacuum considerations, including beam lifetime aspects ; ref. [4] calculates the scattering loss on thermal photons; ref. [5] contains a nice discussion of the operational beam-gas experience in the Tristan accumulation ring.
We denote the luminosity of the accelerator at time t by Y(t) . Each beam may consist of a number of bunches, n b,, with a number of particles per bunch Nb ,(t) . The subscript b is used to indicate that this is a quantity for a single bunch, and the subscript i refers to the beam (i = 1,2) . We assume throughout that the number of particles is the same for all of the bunches in a beam, though the number may be different between the two beams. For some of what we say, this assumption may be relaxed, with the interpretation that Nb, is the average number of particles per bunch. In any event, the total number of particles in a given beam is N, = nb,Nb,, There are two categories of particle loss mechanism which are typically of interest : The first category includes those processes which lead to particle loss because of collisions between the particles in the two beam. If the cross section leading to loss of a particle from beam t is a then the loss rate depends on the luminosity according to :
Work supported in part by the Department of Energy, contract DE-AC03-81ER40050 .
The cross section is assumed to be time-independent. An effective instantaneous beam lifetime is : T, (t) -
~((0 ,
(
The beam-beam interaction may also contribute to the lifetime, by causing particles to eventually encounter the physical aperture . The effective cross section for this loss mechanism may be time-dependent as it depends on time-dependent accelerator parameters . We do not explicitly include this case here, which requires sophisticated methods to obtain lifetime estimates.
0168-9002/91/$03 .50 C 1991 - Elsevier Science Publishers B .V. (North-Holland)
210
F. C Porter / Luminosity lifetime at an asymmetric e + e
The second category includes single-beam particle loss mechanisms, such as intrabeam scattering, quantum fluctuations, and scattering from residual gas in the vacuum chamber. We write the basic equation for this category in the form :
N d (t)= -rr(t)N(t)
N,(t) = N o e - '-, ' .
(4)
We have introduced here the notation N o -- N,(0), and we shall also use ~0 =Y(0), to denote quantities evaluated at t = 0 . In the following discussions, we will assume that r, is independent of time. As long as we evaluate T, at t = 0, this will ordinarily be a conservative assumption, since any time-dependence is likely to be a decrease in the loss rate with time.
3. Luminosity model and luminosity lifetime To go from our beam lifetimes to a luminosity lifetime, we need to know how the luminosity depends on the beam currents . This dependence is determined to some extent by the operation of the storage ring . We consider a simple model for the machine operation which provides a conservative model for the luminosity function . The model is to assume that the bunch sizes do not vary with time, and the luminosity is given by : (t)=
eq. (5) leads to two coupled differential equations in the beam currents : da l = - ka , N, N2 ,
Nb](t)Nb2(tM 2 , r (ai+aX ) ( eyl+ay2)
The vx,y, , in this equation are the transverse rms spot sizes at the interaction point (IP), and should not be confused with the cross sections v, . The bunch crossing frequency is f. . All terms which are time-dependent are so indicated. Note that this is a place where we use the assumption that all bunches in a given beam have the same number of particles. We also assume that any modifications to the above formula from considerations such as finite bunch lengths and nonzero crossing angles are time-independent . If there is more than one interaction region, the appropriate luminosity in this discussion is the sum of the luminosities from each region . For particle loss mechanisms in the first category,
--i-t2
k G2 NI N2 ,
(6)
where k =_
We have kept the possibility here that the loss rate r, depends on time . For example, it may be that the vacuum is a function of the beam current, and hence, the loss rate also depends on beam current . If F, does not depend on time, then the beam current in the second category depends on time according to a simple exponential :
colltder
o NONzo
(7)
The solution is : 1-r N1(t) =Nlo e ct -r >
N2(t)
1-r
-Nzol- re -ct'
where al
G
~°~ N1o
a2 Nzo ~'
(9)
and r = N1°a2 . N2oa1
(10)
The above result yields a time-dependence of the luminosity of : _ )2 . Y ect( Y= Y, l ect rr
(11)
The time, tf , it takes for the luminosity to decrease to a fraction f of its initial value is given by : tf=
G
ln~2f[(1-r)2+2fr
The integrated luminosity up to time t is :
ftY(t') dt'=Y° 1 G r ( ect-r ) .
(13)
A useful limiting case of the above is when r = 1 (and G = 0), such as would be the case in a truly "symmetric" machine. In this limit, we have the simpler result ( ) t ()
1
NO
(14) 1
=YO [ j
(15)
and f = aNe`8o ( VJ
_
1)
.
(16)
Finally, the integrated luminosity in this case is :
f°t~(t') dt'=Y0 1 + (a,~°/No)t
(17)
F C. Porter / Luminosity lifetime at an asymmetric e + e -
For particle losses in category two, say at rate F1 for beam 1 and T2 for beam 2, the time-dependent luminosity in this model is of the simple exponential form : P(t) =Y0 e-(F,+TZ)1
(18)
The model we have just discussed is, of course, not the only possibility . A more optimistic model than the first is to assume that our machine is operated so that the spot sizes are reduced as particles are lost, in such a way that the beam-beam linear tune shifts (Op,,y,,) are constant . For this case, the luminosity equation (again neglecting modifications due to contributions from such things as finite bunch lengths and nonzero crossing angles) may be written as :
f
al
(L1+ a2 X
where a'
_
L
1
1
+
Ovy, ß* '
ay
)1~
_ Y,,Nbl Nbl wy1 ßv2 y2Nb2
wy 2 ßy1
N E N O
4
n 0 0
001
002
003
004
005
Kmin/E
a2 ay (1+a 1 )]
a1(1+a 2 ) + ay a2 (1+a 1
6
(3 .15)
( 19)
If we further suppose that the beta functions are not varied in time, then this equation is expressed in a form such that the only time-dependence appears in the bunch populations, as desired. The differential eq . (1) may then be integrated analytically, although the result is cumbersome, and not particularly illuminating . In any event, we advocate our more conservative model for use as predictor of machine lifetimes . However, before leaving this, we mention an interesting subcase, where we assume that the currents in the two beams have a time-independent ratio. This situation holds for a truly "symmetric" machine, or somewhat more generally, for a machine in which "Garren's rules" [61 (all tune shifts equal to the same value and spot sizes at the IP the same for the two beams) hold independently of time. For this special case, the time-dependence of the luminosity is the same as that for either of the beam currents, and the form is a simple exponential, with decay time constant -r = No/(a, ?0) for category one processes, and T = 1/T, for a category two process. Note that the assumptions of this special situation also constrain the particle loss rates from the two beams to be related.
Fig . 1 . The e + e - -> e - e - y bremsstrahlung beam loss cross section (eq. (20)) as a function of energy aperture for various values of center of mass energy : From bottom to top, Ec ,=1, 10, 100, 1000 GeV
category is loss due to bremsstrahlung (e + e - - e + e - y) of a photon, which can change the energy of a beam particle sufficiently to put it outside the energy acceptance of the accelerator. An excellent approximation for the cross section to lose a particle from beam i due to bremsstrahlung is [71 : Qbrems i
16are 2
+ I (In
In
M2 Ec2m
E' nun r
)2
-
En
(In 2I ) kmin, 3
IT 2
5 8 (20)
In this expression, k,r n, is the minimum energy of the radiated photon which causes loss of a particle from beam t. Thus, kn n,/E, can be taken as the fractional energy aperture x1 of the machine for beam i. In an asymmetric machine, a collision may lead to particle loss from only one beam . Hence, we are interested in the effect on a single beam at a time, and we calculate the desired cross section separately for each beam. The dominant radiation from a given beam is kinematically well separated from the radiation from the other beam ; therefore interference effects may safely be neglected . Fig . 1 shows the bremsstrahlung beam loss cross section calculated according to eq . (20) . The cross section depends slowly on the energy aperture and on Eem . A typical machine might have an energy aperture of order 1% . For Eem = 10 GeV, the cross section is abrems ,
4. Specific particle loss mechanisms We consider here some specific particle loss mechanisms at a level we can treat analytically . All cross sections considered are valid in the high energy limit. Typically the most important mechanism in the first
#1 The energy aperture may be limited by the RF voltage, or by
the transverse aperture as the off-energy particle propagates into dispersive regions of the lattice . We ignore here complications when the dispersion at the interaction point is nonzero .
212
F C. Porter / Luminosity lifetime at an asymmetric e + e - colltder
= 2.6 X 10-25 CM2. Under these conditions, to obtain a two hour luminosity lifetime in a symmetric machine at P= 103acm-2 s-1, assuming model one, we require a total of N, o = nb,Nb,o = 2.9X 10 13 particles in each beam (presumably distributed among a number of bunches) . "Luminosity lifetime", as we use it, means the time it takes the luminosity to decrease to 1/e of its original value, e.g ., as t1le in eq . (12) . Another loss mechanism in the first category, typically not as important as the bremsstrahlung considered above, is the loss due to Bhabha (e +e- -+ e+ e - ) scattering at sufficiently large angles to escape the acceptance of the machine. The lowest order differential Bhabha cross section is : [9J: da d2
a2 2E,m
+ 2 (1
I I
1 + cosy B2
Bcm sina
_2
sine km
2
I,
+ cos20Cm )
(21)
where Bcm is the scattering angle in the center of mass . Note that cross sections in units of GeV -2 may be converted to cm2 by multiplying by 3.89 X 10 -28 GeV 2 cm2 (or by the replacement a2 _ rné e2) . We suppose that a particle will be lost from beam i if it is scattered by an angle larger than some minimum Bcm min 1 , Thus, we need the integrated cross section from this minimum to 180' : oBhabha(xmax) =2 -
~I
xmax 1
8Tra 2 2
Ecm
+
2
3
-
dv ~jdcos 0cm
1 ( 1 - xmax ) In 2 + 1 - xmax 3
4
B > Omm = min
) . a(s ß()ß(s)
(1 - xmax )
+ ô \1 - xmax )2 - 48 l1 - xmax )3 J (22) where xm_ = cos Bcm mm, for whichever beam is under consideration . In the following, it will be sufficient to neglect all but the leading order for xmax near one. At this point, let us develop a simple ansatz for the minimum scattering angle which leads to particle loss, which may provide a reasonable estimate in the absence of more detailed tracking calculations . Suppose that an elastic scatter at longitudinal coordinate so in the ring causes a change in angle of the trajectory B. Propagating around the ring, the particle tends to sample a displacement due to this kick of roughly d(s) = B ,ß (so )ß (s) .
(23)
We have suppressed any labelling for the transverse dimension - this minimum angle may be smaller in one transverse dimension than the other. To keep the analysis simple, we conservatively take the minimum angle considering all transverse directions #2. It is often convenient to express the limiting aperture in units of the rms beam spot size, that is, in terms of n(s), the number of standard deviations of clearance : Bmm = E/ß(so) minn (s) .
B
cosy
Thus, if the transverse half-aperture of the ring is a(s), the particle will be lost if :
(24)
Of course, for Bhabha scattering losses, ß(so) = ß *, for whichever transverse dimension is used . Note that Bm,n is measured in the frame of the machine, while the Bhabha cross section is evaluated in the center of mass frame. However, in the limit that the angle is small, we may use eq . (22) with the substitution (the j subscript equals 2 if i = 1, and is 1 if t = 2) #3 : E Bcm nun , = .Bxnm
(25)
Then to a good approximation we may use the following estimate for the cross section to lose a particle from beam i due to Bhabha scattering (keeping only the leading term in the small angle limit) : OBhabha i -
16-na 2 EJ E.2
cm
1
E B2 ' mm
(26)
For example, if E1 = 3 GeV, E2 = 9 GeV (Ecm = 10 GeV), and Omm 1 = 0.01 radians, then aBhabha t = 3 X 10 -28 cm2. This is considerably smaller than the cross section in our earlier bremsstrahlung loss example. Another important lifetime consideration is the loss of beam due to scattering on residual gas in the vacuum
#2On the other hand, we are not being conservative by neglecting the natural emittance (e) and dispersion (il) of the beam. A better approximation is (r 2 t~a I -e2 min Omm ß(S.) s (La(S)2p-r'(S)2s2~ N(S)
where 8 is the relative beam energy spread, and the formula is evaluated in any desired transverse coordinate. Eq. (23) is justified as long as the aperture is large compared with the beam size . #3 Note that this is equivalent to the expression km mini _ (2E, /Ecm) Bm,n since E = 4E,E_, in the high energy limit.
m
213
FC. Porter / Luminosity lifetime at an asymmetric e + e - collider
chamber. Such losses are in the second category, and eq . (3) applies, with F,(t)-
v,CP,(t) TT(t) ,
(27)
where a, is the cross section to lose a particle due to a beam gas collision, c is the speed of light, P, is the residual gas pressure (including the contribution from beam-induced desorption), k is Boltzmann's constant, and T, is the temperature (c/k = 2 .89 x 10 33 s -1 Km -2 Torr -1 ) . We may make a conservative simplification by assuming that P,/T, is constant at the peak value (i.e ., equal to the value at peak current, rather than using the base no-beam pressure), in which case the beam loss from this source has simple exponential behavior . If, in some situation this approach is overly conservative, a more accurate model may be employed, for example by assuming that the running pressure depends on the beam current according to a constant base pressure term plus a term proportional to the current . Refs . [1,3] contain further discussion . Similarly to the beam-beam bremsstrahlung, we may write the cross section formula to lose a particle due to beam-gas bremsstrahlung. However, in this case, we scatter from an atom, so the form is complicated. We use the "complete screening" result of Tsai, plus the Thomas- Fermi-Molière model accurate for Z >- 5 [8] : z dae t Z brems , _ 4are - 3 dy y {3y+y j z z] x [ Z (Lrad - f[( aZ ) )+ZLrad] +(1-y)(Z2+Z)/9 },
(28)
with y --- k,/E and 184 .15 Lrad = In ZF3 , 1194 ( 29) , Lrad =1n Z2/3 f(x) = x
~t
n(nz+x)
1 .202x - 1 .0369x2
+ 1 .008x 3 /(1 +x) . Integrating this expression gives, in the limit kn n , << E, : 16aZre2 3 ~[Z(Lrad - f [(a2) 2 ]) +Lrad, ae t Z brems i E, 5 - 8) rmn ,
x(ln k
+(Z+1)ln (
E, -1)/121 . knun i
100
EU N N
b X T
0 U
e0
60
b 40 0
001
002
003 Kmin/E
004
005
Fig. 2. The cross section to lose a beam particle due to bremsstrahlung on a molecule of carbon monoxide as a func tion of the energy aperture of the machine, according to eq . (30) .
where A is the molecular weight, NA is Avogadro's number, and X0 is the radiation length . The cross section to lose a beam particle due to bremsstrahlung on a carbon monoxide gas molecule is shown as a function of the energy aperture in fig . 2 . We assume that the cross section to scatter on a molecule is the sum of the cross sections on the constituent atoms . As an example, suppose that our vacuum pressure is 10 -8 Torr at a temperature of 40 ° C, and that the energy aperture is 1 % . In this case, ae t Z brems , = 6.7 x 10 -24 cm2 , and 1/T, = 4 .5 h, or the luminosity lifetime (1/e point) is approximately 2 .2 h if the loss rates are the same in both beams . Another potentially important beam-gas loss mechanism is Coulomb scattering resulting in orbits outside the machine aperture . For this case, we must evaluate the minimum scattering angle by making a suitable pressure-weighted average of eq. (23) or (24) around the ring. However, a reasonable first estimate may be obtained by inserting the average ft for ß(s 0 ) . The Coulomb differential cross section for a beam particle of energy E to scatter by an angle 0 is given approximately by the Mott cross section (for a = 1) : 2 (32) Z2a 9 (1-sin2 0) . da 4E 2 sin4
2
(30)
We may typically neglect the term proportional to (Z + 1) . It is then convenient to write this expression in terms of the radiation length of the material : E, - 5 ), 4 A 1 (31) ae t brems = 3 NA X0 (In kmtn , g
We have neglected a number of effects in this simple formula : 1) We have neglected scattering off of the atomic electrons, reasonable when Z >> 1 (i .e ., correction terms are of order 1/Z times the above) . 2) We have neglected screening by the atomic electrons . This is justified in the limit that the wavelength of the virtual photon transferred in the collision from
214
F. C. Porter / Luminosity lifetime at an asymmetric e + e
the beam particle is small compared with the atomic size, i.e., as long as Bm,n >> am,/E, neglecting a weak dependence on Z. 3) We have neglected nuclear form factor effects, and treated it as a static, spinless target . The form factor may be neglected as long as the wavelength of the virtual photon is large compared with the nuclear size . Since the cross section grows rapidly at small angles, this will be a good approximation as long as Bm,n < - 0.1 GeV/E. Likewise, the static approximation is reasonable for (E/M)Bn2 n << 1, where M is the nuclear mass . This condition is also sufficient to justify neglect of the magnetic moment effects. The Coulomb cross section to scatter by an angle greater than Bm,n, is thus approximately given by : 4. Te2Z2 4.rrZ2a2 (33) oe±Z CO-1 1 = = Y,202 E202 e nun ~ nun ~ As an example, on carbon monoxide, the cross section to lose a 5 GeV beam particle, if the minimum aperture is 15a, the emittance is 10 -s mrad, and the average beta function is 10 m, is 4.6 X 10 -24 cm2. At a pressure of 10 -s Torr and 40'C, this yields a single beam lifetime of 1/I; = 6.5 h (realistic cases are usually better than this). At an asymmetric collider, the strong energy dependence of the cross section makes it unlikely that the loss rates due to this mechanism will be matched between the two beams. We remark that the damping times in an e + estorage ring are sufficiently short so that multiple collisions according to the above mechanisms may be neglected.
collider
10
08
04
02
00 L 0
"
7
2
3
r
t/tye Fig. 3. The luminosity (m units of peak luminosity) as a function of time (in units of the actual time it takes the luminosity to fall by a factor of 1/e) for category one processes. Various cases and approximations are shown: i) The solid curve is a simple exponential, with time constant equal to one in these units. ii) The long-dash curve is the actual time dependence for r=1 (eq. (15) . iii) The dot-dash curve is the actual time dependence for r =1/4 and r = 4 (eq. (11)). iv) The dotted curve shows the time dependence for a simple exponential with time constant given by the combinations of the actual 1/e times of the individual beams, for r =1 (i .e ., this would be the appropriate curve for r=1 if the beam currents decayed according to exponentials) . v) The short dash curve is similar to iv), but for r =1/4 and r = 4.
020
---------------
015
5. Approximation by exponential form It would be convenient to be able to approximate the luminosity dependence on time by an exponential, for example in order to combine lifetime expectations from various loss mechanisms. In this section, we show how this can be done with small error for our preferred (for conservative estimates) model of machine operation, for losses from processes in the first category. We are generally interested in times short, or comparable with, the 1/e decay time of the luminosity. Thus, an approximation which works well in this time range will be acceptable. The natural thing to try is to approximate the actual luminosity by an exponential with the 1/e time adjusted to be the same as the true 1/e point of eq . (12) or eq. (16) . This in fact works quite well, as may be seen in fig. 3, which compares an exponential with the dependence of eq. (11) (and eq. (15)) for various values of r, as a function of time in units of ttle . Curves for r and 1/r are identical, since the luminosity formula (see eq .
010
005
0 00
0 05
t/tl ie
3
Fig. 4. The relative error made in the integrated luminosity, for category one processes, as a function of time (in units of the actual time it takes for the luminosity to decrease by 1/e from the peak value). Both curves are for the worst case : r =1 . The solid curve shows the error made by using an exponential form with time constant equal to one in these units. The dashed curve shows the error made by using a simple exponential with time constant given by the appropriate combination of the actual 1/e times of the individual beams, in the approximation that the beam current time dependence is exponential.
RC. Porter / Luminosity lifetime at an asymmetric e + e (11)) is unchanged under the transformation N10/at .For t < the the exponential N20/a2, N20/a2 -> N10/al approximation slightly overestimates the luminosity, and for times t > t1/, it underestimates the luminosity . The approximation is poorest for the case r = 1, but even here the largest error made in the integrated luminosity is only 4.6%, for t < t,/e, as shown in fig . 4 . For times larger than - 2 .7t m/e the error made in the integrated luminosity becomes larger, approaching an asymptotic value of approximately 50% for r = 1, but we are ordinarily not interested in discussing the luminosity after such long times . The integrated luminosity is overestimated until t - 1 .9tm/e , beyond which it is underestimated . Thus, we find that a useful approximation for most practical purposes is to assume an exponential form for the luminosity for category one processes, with time constant given by the according to eq . (12) (or eq . (16)) . Note that a substantially poorer approximation is achieved if we first approximate the individual beam currents by exponentials (with given ttle) and then combine these exponentials to get an exponential for the time-dependence of the luminosity (see figs. 3 and 4) .
6. Optimizing the average luminosity If we accept that the luminosity time-dependence may be approximated by an exponential, then the optimization of the average luminosity under specified in jection conditions is readily obtained . Thus, we start with a luminosity which depends on time according to : P(t) =Yo e - `iT .
(34)
We suppose that a data taking period, T, for the experiment is long compared with the injection time and the beam coast time (i.e ., that there are many such fills in a data run) . In this case, it is sufficient to replace the actual distribution of injection times with a single average injection time, which we call t I . We further assume that we take data for a fixed time interval, t c , following injection, prior to beginning the next injection ; and that each fill begins with the same initial luminosity (Yo) . Finally, we assume that no useful data is accumulated during injection. Our discussion does not depend on whether the injections are "top-off' or start from zero beam current. It is our present interest to find the optimal value for tc . The total integrated luminosity which we accumulate during our data run is given by :
Y dt = n f1'Y,
fo
0
e`/' di,
(35)
where n = T/(t, + t t ) is the number of injection/coast cycles in the run. The ratio of the actual integrated
215
collider
t,it
if .
Fig. 5 . The solid curve shows the optimal coast time between injections, in units of the luminosity lifetime (1/e decay time), as a function of the injection time (also in units of the luminosity lifetime) . The dashed curve gives the average luminosity, in units of the peak luminosity, if the optimal coast time is used .
luminosity to that obtained were the machine capable of running the entire time at its peak luminosity is then : . 0 7 f0
Y dt = t~+ tt ( 1 -
(36)
This quantity (and hence the actual integrated luminosity) is maximized when t. is chosen to satisfy the condition : (tc+ t,)/,r = e 'c /' - 1 .
(37)
(Y) =Yo e `~~T .
(38)
Thus, the maximum average luminosity possible is : Fig . 5 may be used to evaluate t c and the average luminosity given t t and T. For example, if T is 1 h, and t t is 10 min, then the optimal coast time is 32 min, and the average luminosity is 59% of the peak luminosity .
7 . Summary We have investigated the dependence of the luminosity on time in an asymmetric e + e - storage ring coltider . A conservative model for making design estimates is advocated where the bunch sizes are taken to be constant in time . The dependence of the luminosity on time may be reasonably approximated by an exponential law, with time constant given by the 1/e time of the actual time function. The optimal coast time and maximum average luminosity are obtained in this approximation, as a function of the injection time. In situations of current interest, the loss of particles due to e + e - e + e - y bremsstrahlung collisions dominates over loss
F.C. Porter / Luminosity lifetime at an asymmetric e +e - collider
216
due to Bhabha scattering . Depending on details, both
bremsstrahlung
and
Coulomb
losses are potentially significant.
beam-gas
scattering
Acknowledgements I
would
like
to thank Swapan Chattopadhyay Claudio Pellegrini, John Rees, and Michael Zisman for useful discussions . I am grateful to D. Mohl and K. Hiibner for catching an error in the original manuscript.
[2] [3] [4] [51 [6] [7]
References [11 I am indebted to the reviewers for bringing to my attention a recent report which also discusses the asymmetric case : H. Braun, K. Hiibner and W. Joho, PS/LP note 90-11, PSI-TM-12-90-05 (March 1990). That report contains references to several earlier unpublished notes concerning beam
[8] [91
hfetimes . Another recent report of interest is the Feasibility Study for a B-Meson Factory in the CERN ISR Tunnel, ed . T. Nakada, CERN 90-92, PSI PR-90-08 (30 March 1990). J. Le Duff, Nucl . Instr. and Meth . A239 (1985) 83 . C. Benvenuti, R. Calder and O. Gr6bner, Vacuum 37 (1987) 699. V.I . Telnov, Nucl . Instr. and Meth . A260 (1987) 304. T. Momose et al ., IEEE Trans. Nucl . Sci. 32 (1985) 3809 . A. Garren et al ., LBL-26982, Proc . IEEE Particle Accelerator Conf . 1989, Chicago. G. Altarelli and F. Buccella, Nuovo Cimento 34 (1964) 6385 . Note that eq. (20) can also be obtained by integrating eq . (3 .87) of ref. [81 below, paying attention to the definitions of the kinematic quantities, and recognizing that Tsai includes radiation from both particles. Y.S. Tsai, Rev. Mod. Phys . 46 (1974) 815. See, for example, J.D . Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill Book Company, New York, 1964) pp . 135-140 . We disagree with the cross section in M. Conte, Nucl. Instr. and Meth . 228 (1985) 236.