Beam-beam effects in the Tevatron

The Tevatron in Collider Run I1 (2001-present) is opemting with 6 times more bunches many times higher beam intensities and luminosities than in Run I (19%-1995). Electsomagnetic long-range and head-on interactions of high intensity proton and antiproton bearns have been signifimnt sources of beam loss and lifetime limitations. We present observations of the beam-beam phenomena in the Tevatron and results of relevant beam studies. We analyze the data and various methods employed in operations, predict the performance for planned luminosity upgrades, and discuss ways to improve i t

Run II of the Tevatron protm-antiproton mllider began in March 2001. Compard to run I, the beam energy was increased from 900 to 980 GeVand the number of bunches was increased from 6 to 36 in each beam, in order to increase luminosity many times above the run I record peak luminosity of 0.25 X mP2 S K I . Since the start of run II, the Tevatron peak luminosity has steadily improved and reached the level of 1.2 X cmP2 s-I (see Fig. 1)-significantly exceeding the original run IIa peak luminosity goal [I] without using electron cooling of antiprotons in the recycler ring. The progress was a result of more than a dozen improvements in the injectors and the Tevatron itself, each giving a 5%-25% performance increase. The improvements have often been intrduced during regular shutdown periods (8-12 weeks long every autumn). Details of the accelerator complex operations can be found in Ref.
[I] and descriptions of the numerous improvements are given in 121. More than 1 fb-I of integrated luminosity has been delivered to each of the CDF and DO experiments to date. In parallel to the collider operation, we have started a luminosity upgrade project which should lead to peak luminosities of about 2.7 X crrP2 s-l and total integrated luminosity of 4.4-8.5 fbthrough 2009. Table I contains various parameters of the Tevatron beams for present operation and their design values after the planned luminosity upgrades. Figure 2 presents a typical Tevatron operation cycle. A collider fill starts with 150 GeV proton bunches from the main injector injected one bunch at a time onto the central orbit of the Tevatron. The bunches are loaded in three trains of 12 bunches each, with 396 ns bunch separation and 2.6 ps gaps between the trains. Protons and antiprotons circulate in the same beam pipe, so electrostatic separators are used to put the beams onto separate helical orbits. After all 36 proton bunches are loaded the separators are powered ta put the protons on their helical orbit. Antiprotons are loaded onto the antiproton helical orbit, four bunches at a time, into one of the three abort gaps. The antiproton bunches are moved longitudinally relative to the proton bunches ("cogged") to I& room for the next four bunches in the abort gap. In Fig. 2, the coggings are marked by (artificial) spikes in the antiproton bunch intensity caused by instrumentation effects. After injection, the two beams are accelerated to 980 GeV in about 85 sec. A final cogging is done a few seconds after the ramp. Then, the optics are changed in 25 steps to reduce the beta functions at the interaction points (IPS) from 1.6 to 0.35 m. After the final step of this "low-beta squeeze," the beams are brought into collision at the IPS by using separators around the IPS. Next, a dozen collimators are inserted to reduce the beam halo background in the &kctors. A high-energy physics (HEP) store begins shortly thereafter.
It should be noted that because of the way the injection complex operates, the antiprotons bunches vary signifi- cantly in intensity and emittance. As an example, Fig. 3 shows parameters of all antiproton and proton bunches at the start of store #3692 (July 31, 2004). Proton bunch intensities and emittances vary from bunch to bunch by less than 5%, while antiproton bunch intensities vary by a factor of 3, and ernittances by a factor of 1.5. The transverse emittances cited in this paper are 95% norrnalizd ernittances which relate to the rms beam sizes ux,, as ex,^ = 6BrEd,, -~Z,.,(~P/P)~I/IB,,,~ where ~P / P is the rrns momentum spread, and P , , and D,, are the beta and dispersion functions, respectively. Most of the variations have a period of 12 and 4: the intensities and transverse ernittances of bunches at the end of each train are typically smaller than those at the beginning of the trains. Consequently, the instantaneous luminosity per bunch crossing can differ by a factor of 3 or more.
One can see in Fig. 2 that minor beam losses wcur at every step of the Tevatron cycle. Some losses are intentional, like the few percent loss of protons and antiprotons during the halo removal process. Such scraping greatly reduces background event rates in both detectors and improves their data-taking efficiency. The beam losses during injection, ramp and squeeze phases are mostly caused by beam-beam effects. Since the start of run 11, these losses have k e n reduced greatly, as demonstrated in Fig. 4, such that the total beam intensity loss in the Tevatron prior to initiating collisions is currently = 16%. Details on that subject will be presented below. In "proton-only" or "antiproton-only" stores, the losses do not exceed 2%-3% per specie. So, the remaining 10%-12% loss is caused by beam-beam effects. The proton and antiproton inefficiencies are similar, despite the factor of 6%-10% difference in intensity. Figure 5 shows the decay of instantaneous CDF luminosity over store #3657. The solid line represents the result of a simplified two parameter fit:
In summary, beam-beam effects in the Tevatron account for a 20%-27% loss in the lurninosity integral due to (a) 10%-12% particle loss before the start of collisions and (b) 10%-15% reduction in the luminosity lifetime. This loss is significant now, and it may be even larger after the luminosity upgrades, thus requiring continued systematic attention.
Our operational focus is to maximize integrated lurninosity for the HEP program. Therefore, many studies presented in this paper were conducted parasitically. The luminosity lifetime of 6-8 h), the logarithm changes 01 . (Color) Evolution of instantaneous luminosity at the run 11 stores is presented in Fig. 6, and it shows a significant CDF detector iT1 store #3657 (july 16, 2004). The  machine or beam parameters were rarely set to optimize specific beam-beam effects or to conduct a thorough, dedicated study. Instead, valuable information has been o b tained by studying nonoptimal settings for HEP runs or even unplanned incidents. Nevertheless, this paper presents observations and analysis valuable for future Tevatron operation.

HELICAL ORBITS
Beam-beam interactions differ between the injection and collision stages. The helical orbits should provide sufficient separation between the proton and antiproton beams in order to reduce detrimental beam-beam effects, e.g., tune shifts, coupling, and high-order resonance driving terms. Each bunch experiences 72 long-range interactions per revolution at injection, but at collision there are 70 long-range interactions and two head-on collisions per bunch. In total, there are 138 locations around the ring where beam-beam interactions occur. The sequence of 72 interactions out of the 138 possible ones differs for each bunch, hence the effects vary from bunch to bunch. The locations of these interactions and the beam separations change from injection to collision because of the antiproton cogging.
There are six separator groups (three horizontal and three vertical) in the arcs between the two main interaction points, BO (CDF) and DO. During collisions, these separators form closed 3-bumps in each plane, but the condition of orbit closure prevents running the separators at rnaximum voltages, thus limiting the separation at the nearest parasitic crossings 57 m away from the main IPS. To alleviate this limitation, additional separators can be installed in the arcs such that the separators form 4-bumps.
There is more flexibility in the helix design for the preceding stages: injection, ramp and squeeze. There are still some difficulties at these stages, including the following: (i) irregularities in betatron phase advance over the straight sections, especially AO; (ii) aperture restrictions (physical as well as dynamic) that limit the helix amplitude at injection and at the beginning of the ramp; (iii) the maximum separator gradient of 48 k V l m (limited by separator spark rate) leads to a faster drop in separation d -1 / E than in the beam size (T -1 /~' /~ during the second part of the ramp above E = 500 GeV, (iv) the polarity reversal of the horizontal separation during the squeeze (to satisfy needs of HEP experiments) that leads to a momentary collapse of the helix.
A simple figure of merit is helpful when comparing different helix designs. The conventional choice is the minimum value of the so-called radial separation, S, over all possible parasitic interaction crossing points in units of the rms betatron beam sizes uAyfi: The separation is normalizd to a fixed reference emittance of 15 a rnrn rnrad. Our experience has shown that less than 5 u 4 u separation causes unsatisfactory losses. Figure 7 shows the minimum radial separation S during the ramp and squeeze with the initial helix design (blue, ca. January 2002) and an improved helix (red, ca. August 2004).
Early in 2002, the Tevatron luminosity progress was hampered by a very fast 20%-35% loss of antiprotons occurring at sequence 13 of the low-beta squeeze. Figure 8 demonstrates how the initial luminosity actually decreased when attempting to bring higher intensity proton 0 1 . Any further time reduction is limited by the slew rate of the low-beta superconducting quadrupoles. Since irnplernenting those changes, the antiproton losses in squeeze do not exceed 2%-3%.
Beam separation at the injection energy was the subject of numerous improvements summarized in Table 11. The first two rows present the voltages on each separator plate used for antiproton injection, the corresponding values of minimum radial separation, the maximum absolute values of beam-beam tune shifts and resonance driving terms at the BO location.
(RDTs) for 5vx and 7 9 resonances over all antiproton bunches. The tune shifts were calculated for particles with small betatron amplitudes, whereas the RDTs give the increment in the action variable value for a resonant particle with an amplitude of 3 u in the corresponding plane. In the initial helix design, beam-beam effects were dominatd by a single parasitic crossing near A0 (see Fig. 9). By employing two other separators (see Table II Table I I ) .
During acceleration, the separator voltages should increase as ( E = beam energy) in order to maintain Given the 48 k V / m maximum operational gradient, the separators providing the bulk of the separation, B 17H and C17V, reach their maximum voltage at E -500 GeV.
Above this energy, the radial separation drops as 1 /~' f l ( Fig. 7). That leads to enhancement of detrimental beambeam effects and causes particle losses. By employing additional separators (Table II, bottom row), it was possible to increase the separation by more than 50% and to reduce beam losses above 500 GeV significantly. This improvement was achieved mainly by increasing vertical separation which was neither possible at injection, nor early on the ramp due to aperture limitations. The transition between the two types of helix, injection and end-oframp, manifests itself in Fig. 7 as a sharp maximum at 600 GeV. The new helix for the end of ramp and the first part of squeeze was introduced in August 2003. The changes to the helix design, together with the reduction in chromaticity, drastically improved antiproton efficiency through injection, ramp and squeeze to well above 90% (see Fig. 3).

III. BEAM LOSSES DURING INJECTION AND ON RAMP
Although both the proton and antiproton beams stay at 150 GeV for less than an hour, a significant particle loss occurred during that time at the beginning of the run 11. As it will be shown below, the particle losses for both beams were driven by diffusion and exacerbated by small transverse and longitudinal apertures. The problem was alleviated significantly by a comprehensive realignment of many Tevatron elements in 2003-2004, as well as a reduction in the longitudinal ernittances due to improvements in the main injector's bunch coalescing, and an increase of the Tevatron's dynamic aperture. Figure 10 presents the intensity lifetimes of single antiproton bunches after injection for typical stores in 2002 and 2004. It is clearly seen for both stores that the intensity decay is not exponential. Figure 10 shows that the intensities are approximated well by the expression N(r) = iVoe-& that was used for the lifetime fits. Similar & dependence has been observed for the bunch length shaving; Fig. 11 shows an example of such behavior. The transverse ernittances do not exhibit such dependence on 4. Figure 12 shows, for many stores early in 2005, the antiproton bunch loss rate at 150 GeV as a function of the antiproton bunch emittance. One can see that the loss rate scales approximately as the square of the ernittance. The data indicated by blue circles represents the same losses after reducing the chromaticity on the antiproton helix from Q' = d Q / ( d P / P ) = + ( 5 4 ) units to about 3. Although the functional dependence on the emittance is nearly identical, the absolute scale of the losses is reduced by a factor of -5. The example above demonstrates the importance of chromaticity for reducing the losses of both protons and antiprotons. Since the proton and antiproton orbits are separated using the electrostatic separators, their tunes and chromaticities can be controlled independently by using sextupole and octupole circuits, respectively. The major obstacle in attaining the desired chromaticity reduction was a weak head-tail instability in high intensity proton bunches [3]. Early in run II, avoiding this instability required chromaticities as high as 8-12 units at 150 GeV. Reducing the pruton chromaticities down to + (3-4) units became possible after removing unused high-impedance extraction Lambertson magnets, reducing the impedance of the injection Lambertsw magnets by installing condue tive liners, and commissioning active bunch-by-bunch instability dampers for the protons [4]. Decreasing the chromaticities to zero has become possible after recodguring octupole circuits to introduce Landau damping to suppress the head-tail instability. The antiproton bunches do not suffer from that instability since the intensity is much smaller than the protons. Congquently, both Q i and Q , ' are set closer to zero by using differential chromaticity octupole circuits.

Time after injection [min]
During the roughly 20 rnin needed to load antiprotons into the Tevatron, the proton lifetime degrades as more antiproton bunches are injected (see Fig. 2). Figure 13 shows an approximately linear dependence of the proton loss rate at 150 GeV on the number of antipratons in the Tevatron. The proton loss rate without antiprotons is about 4% per hour (25 h lifetime), whereas it grows to about 16% per hour (6 h lifetime) when all antiproton bunches are loaded. A similar linear dependence of the antiproton loss rate on proton intensity can be seen in the store-to-store antiproton inefficiency variations, but it cannot be demonstrated as clearly as in Fig. 13, since the proton intensity at injection remains fairIy constant over many months of operation.  where the index a or p stands for antiprotons or protons, s is transverse ernittance, N is total number of particles in the opposite beam, Qt is the chromati~ity on the corresponding helix, and the factor F emphasizes the fa~A that Iosses also depend on the longitudinal ernittance s~ , separation S (size of the helix and cogging stage) and b n e Q. Over years of operation, the betatron tunes on both helices at injection were optimized to be close to QJQ, = 20.584/20.576, i.e., above 7th order resonances at 4/7 = 0.5714, but close to the 12th order resonance 7/12 = 05833. Significant variations of the tune (in excess of ?(X002) often led to lifetime reduction, especially if the (vertical) tune approached the 4/7 resonance. Detailed work on optinlization of beam-beam separation Sa+ was presented in Sec. 11.
We believe that the observed 4 dependence of beam intensity decay and bunch length is driven by particle diffusion leading to partide loss at physical or dynamic apertures. The major diffusion mechanisms are intrabeam scattering (IBS), scattering on the residual gas, and diffusion caused by rf phase noise.
For example, if the available machine aperture is smaller than the beam size of the injected beam, the beam is clipped on the first turn with an instantaneous particle loss. Such a clipping creates a steplike discontinuity at the boundary of the beam distribution that causes very fast particle loss due to diffusion. The diffusion wave propagates inward, so that the effective distance is proportional to &. Consequently, the particle loss is also proportional to d. To estimate such a "worst-case loss," consider an initially uniform beam distribution: f (I) = fo = l/Io, where I. is the action at the boundary. For sufficiently small time t Io/D, where D is diffusion coefficient, the diffusion can be considered one-dimensional in the vicinity of the beam boundary. Solving the diffusion equation gives the result: By integrating it over I, one obtains the dependence of particle population on time: 2 rad. The above numbers are not well known, but we believe they are in the indicatd ranges.
In reality, the machine acceptance is determined by the interplay between the physical and dynamic apertures. The latter is a strong function of the synchrotron action, and beam-beam interactions drastically reduce the dynamic aperture for synchrotron oscillation amplitudes close to the bucket size. Naturally, such an aperture reduction is stronger for larger values of chromaticity.
Several phenomena contribute to the losses observed during acceleration in the Tevatron. These include losses caused by shaving on a physical aperture, the limited dynamic aperture (DA) due to machine nonlinearities, the reduction of rf bucket area during the initial stages of the ramp, and beam-beam effects. Figure 14 shows the relative change of intensity during acceleration in store #3717 Dedicated studies were done in 2002-2003 to identify loss mechanisms that are unrelated to beam-beam effects. In several proton-only studies, protons with different intensities, transverse and longitudinal emittances were injected into the Tevatron and accelerated. These studies showed very clearly that the proton losses were determind by the longitudinal emittance and the longitudinal bunch profile. Short Gaussian bunches with bunch lengths <2 ns at 150 GeV suffered the least losses -2%, while long and non-Gaussian bunches suffered losses close to 10%. There was almost no dependence on the bunch intensity. Improvements in bunch coalescing in the main injector have improved the beam quality significantly. In recent fit 0.97+1.78 Na I cantly lower transverse antiproton ernittances [2]. Changes were also made to the helix during the second half of the ramp, as describd in a previous section. All of these changes have lowered the antiproton losses during acceleration to around 4%-5% in recent stores (ca. April 2005). Figure 16 shows the dependence of antiproton losses during acceleration on the vertical emittance for two different stores. Store #3711 was a "mixed-source" store which included antiprotons from both the accumulator and the recycler, while store #3717 had only accumulator antiprotons. Figure  4-5 arnrnrnrad smaller than those from the accumulator, (ii) there exists a clear correlation between the losses and the vertical, and (iii) the antiproton losses are close to zero for vertical ernittances below 6 mrnrn rnrad. The losses do not correlate as strongly with the horizontal emittance, suggesting that the physical or dynamic aperture limitation on the antiproton helix is in the vertical plane. The antiproton losses up the ramp in the same two stores have almost no dependence on longitudinal emittance. In summary, combining observations presented in Figs. 15 and 16, beam losses on the ramp scale similarly to Eq. (3) as Losses during the ramp are tolerable at present, but there is room for further improvements. Lowering chromaticities during the ramp with the help of octupoles will reduce proton losses that are mainly in the longitudinal plane. As antiproton intensities increase, beam-beam induced losses of protons during the acceleration may also increase. Smaller transverse proton emittances would help. Additional reductions in antiproton losses are possible with smaller antiproton transverse emittances, and that requires more bunches injected from the recycler.

I 1 DIFFERENCES IN BUNCH-BY-BUNCH DYNAMICS
Remarkably, beam-beam effects in the Tevatron cause nearly every measurable indicator of beam dynamics to vary as a function of position within a bunch train. As mentioned above in Sec. I, the 36 bunches for each beam are arranged in three trains of 12 bunches each, and the spread of intensities and ernittances among the proton bunches is small. Consequently, a threefold symmetry is expectd [6] in the antiproton bunch dynamics. We have observed such behavior, so most of the plots below refer only to a single train of 12 bunches. For example, Fig. 17 shows that the helical orbits of antiproton bunches at 150 GeV and at low-beta differ by some 40 to 50 p m in a systematic, ladderlike fashion. Such variation in the closed orbits was predicted long ago [7], and agrees well with analytical calculations (see the comparisons in Figs. 17(b) and 17(c)).
Two (vertical and horizontal) 1.7 GHz Schottky detectors [8] allow continuous, nondestructive measurements of betatron tunes and chromaticities for each proton and antiproton bunch during HEP stores. The tunes measured by the detectors represent an average over all particles in a bunch. The tune and chromaticity accuracies for single bunch measurements are better than 0.001 and 1 unit, respectively. A single measurement can be ma& in approximately 20 sec. Figure 18 presents the distribution of antiproton vertical and horizontal tunes along a bunch train. It is remarkable that bunches #1 and #12 have vertical and horizontal tunes, respectively, much lower (by more than 0.003) than the other ten bunches. Long-range beam-beam interactions at the parasitic IPS produce such significant bunch-by-bunch tune differences. The variation was expected before the start of run 11 [6] and was studied experimentally in 1996 [9] using helical orbits somewhat different from what has been used in run 11. More detailed theoretically analyses are presented in [10,11]. The data shown in Fig. 18 agree with analytic calculations if one takes into account that the measured tune is averagd over a weighted particle distribution, and, thus, the effective head-on tune shift is approximately half of the maximum beam-beam incoherent tune shift: where r, denotes the classical proton radius, Np is the bunch intensity, E , is the emittance, and the factor of 2 accounts for the two head-on interaction points. For nominal bunch parameters at the beginning of an HEP store (see Table I), the head-on tune shift for antiprotons is 6 -0.020, while 8 a 0.004 for protons. Using the Schottky tune measurements, and taking (8) into account, the tune footprint of all proton and antiproton bunches at the beginning of a Tevatron store is plotted in Fig. 19.
The antiproton tunes decrease over the course of a store with characteristic decay times of 1 1-15 h, caused by the reduction of the head-on tune shift, which itself is mostly due to the increase of proton ernittances (by more than factor of 2) and the decrease of proton bunch intensities (by more than 25%). The time evolution of the measured antiproton tunes for two selected bunches in store #3678 is shown on Fig. 20.
Within the accuracy of the detectors, the proton tunes are identical for all bunches, and are usually stable over duration of HEP stores (16-30 h). Small, but noticeable, decreases of both vertical and horizontal tunes by a (0.00054.001) over the first few hours agree with the expected decrease of the head-on tune shifts for protons. The chromaticity measured by the same system is remarkably stable within 1 unit during the store. Since no time dependence is observed, averaging the data over the entire store seems fair. Even so, the chromaticity does depend on the bunch number within a train, as shown in Fig. 21. Chromaticity varies by about 6 units in both planes along a bunch train, and that is in acceptable agreement with theory that considers both parasitic beam-beam interactions, as in [ l l ] (which predicts the variation to be significant only in horizontal plane), and the energydependence of the beta functions at the main IPS.
It is not surprising that with such significant differences in orbits, tunes and chrornaticities, the antiproton bunch intensity lifetime and emittance growth rates vary considerably from bunch to bunch. For example, Fig. 22 presents the beam-beam induced intensity loss rates for antiproton bunches observed in the first two hours of 20 HEP stores during summer 2004. To calculate such a loss rate, called the nonlurninous (NL) loss rate, one subtracts the particle losses due to collisions at the main IPS dlnN/dt = Lu,/N (luminosity L is measured bunch-by-bunch by both detectors, cr,, a 70 mb at the Tevatron center-ofmass collision energy [12]) from the total measured bunch intensity loss rate d lnN/dt. The error bars represent the rms store-to-store fluctuations in the loss rates. One can see that bunch #1 systematically loses less intensity than the others (of about 0.3% per hour or 300 h of lifetime), while bunches #4 and 812 lose more than 1% per hour (i.e., their NL lifetime is 70-90 h). In comparison, the average Bunch Number  luminous antiproton loss rate d lnN/dt = Lsm,/N is about 3%/h or 30 h of lifetime for a typical high luminosity store. Thus, beam-beam effects account for, on average, up to = 15 8 of the antiproton loss rate (and a 30% for bunches #4 and #12). Later in stores, the luminous losses decrease faster than NL losses, and the two often become comparable or the NL losses can even dominate. Other mechanisms of NL beam loss, like collisions with residual gas and losses from the rf buckets, are much weaker than beam-beam effects, and they account for less than 0. l%/h of the intensity loss. Experiencing the largest beam-beam tune shift in any hadron collider, antiproton bunches in the Tevatron may suffer ernittance growth as a result of strong higher-order resonances if the working tune point is not optimized. As with the significant bunch-by-bunch tune variations, this growth can be quite different for different bunches in the bunch train. As an illustration, Fig. 23(a) presents the time evolution of the vertical emittance of bunches #1, #6, #11, #12 after collisions began in store #3554 (June 2, 2004). One can see that within 10-15 min, some bunches experience 10% -20% transverse ernittance blowup that reduces collider luminosity. Figure 23(b) summarizes the total ernittance blowup in that store for one train of antiproton bunches. One can see a remarkable distribution along the bunch train which gave rise to term "scallops" (three scallops in three trains of 12 bunches) for this phenomenon-the end bunches of each train have lower emittance growth than the bunches in the middle of the train. The scallops depend strongly on the machine working point (vertical and horizontal tunes) since the tune shift for a given bunch depends on its position within a train. Figure 24 shows a two-dimensional contour plot of the maximum (over all antiproton bunches) ernittance blowup Eq. (7), one can conclude that the vertical emittance blowup is strongest when the core particle vertical tune approaches either the 5th order resonance or the 12th order resonance Q, = 7/12 = 0.583. In the horizontal plane, the scallops are small if tune is set away from Q, = 3/5 = 0.600. Scallops were first observd in 2003, when the headon tune shift parameter increased to 0.02. Various methods have been employed to minimize the development of scallops (including a successful attempt to compensate one bunch emittance growth with a Tevatron electron lens [2,13]), but carefully optimizing the machine tunes was found to be the most effective. As one can understand from Fig. 19, one should balance between the desire to lower the core antiproton tunes away from 3/5th resonance store demonstrate the volatility of the losses. It is note-Much smaller scallops of -0.5 anlmmrad were ob-wotzhy that proton N L loss rates are often much higher served infrequently in proton bunches when their tunes than the intensity decay rate due to luminosity, which is of were set near 1 2th order resonances. but they were cm-the order of Q2%-0.3%lh for typical initial luminosities, rected easily by tune adjustments and have not been as and higher than antiproton N L loss rates-compare the serious an issue as the antiproton emittance growth.
vertical scales in Figs. 22.26. and 27. T h e NLproton losses Another interesting beam-beam related phenomenon happens to the proton beam. It was originally observed in the Fall of 2003 that the proton halo count rates in the C D F detector follow the proton intensity loss rates that vary significantly by a factor of 4-6. in a systematic fashion. along a bunch train as seen in Fig. 25: the losses were lower are also much higher than losses due to collisions with residual gas and losses out of the rf buckets, both of which are less than O.l%/h. The volatility and scale of the NL proton losses are of concern for the detectors since high halo rates deteriorate their data-taking efficiency and, in general, they reduce the luminosity lifetime and the inte grated luminosity per store. Again, the most effective way to control the losses has been to adjust the working point. In particular, it was found that the losses were much higher when the proton tunes lay over the 12th order resonance lines, and Fig. 25 shows that proton lifetime there was only 25-30 h on average. After the proton tunes were moved below the 12th order resonances (as shown in Fig. 19), the lifetime improved.

TL OTHER EXPERIMENTAL OBSERVATIONS
The two types of the beam-beam effects in the Tevatron, long-range and head-on, have quite different rnanifestations. In general, the long-range effects should depend on (a) beam separation; bunches were loaded instead of the usual 36 due to problems in the injector chain. As a result, proton bunches #9-#12 did not collide "Read-on" with any antiproton bunches at the IPS, but experienced most of the possible long-range interactions (except for some of the parasitic collision points nearest to the IPS). The measured intensity loss for those particular proton bunches were extremely small: 0.03%4.06%/h. That rate is consistent with the 1000-2000 h lifetime expected solely from beam-gas interactions. One can conclude that the long-range beam-beam interactions with antiprotons do not affect pmton bunch lifetime. The other bunches shown in Fig. 28 were colliding head-on with various antiproton bunches of various emittances, and their pattern of the rates follows Eq. (9). Bunch antiprotons protons expected if antiprotons were affected mainly by parasitic collision points near the main IPS. The proton losses went up in a much more uniform manner, with a small variation due to differences in the opposing antiproton bunch emittances (similar to Fig. 25). Figure 30 shows the antiproton nonluminous loss rate dependence on the Relix size in about 35 HEP stores in March-April 2005. In each of these stores, voltages of all 24 separators were scaled from their nominal values either by + 10% (1 1 stores) or by -10% (6 stores) or set nominal (18 stores). The voltages stayed the same for the entire length of store, and the voltages were changed only on a store-to-store basis. The typical initial luminosities were similar for all three sets (from SO X lQ30 cmP2 s-' to 1 15 X 1030 cmP2 s-I). There was no systematic variation in proton nonluminous lifetime for these stores. In contrast, the nonluminous antiproton loss rates decreased as the helix size S increased approximately as l/s3: they varied by &30% for ? 10% variation of the helix size.
All three facts presented above point to two conclusions:  The proton and antiproton dynamics also differ in the evolution of their longitudinal distribution functions during HEP stores. Figure 3 1 shows that at the beginning of the store #3678 (July 2004), both proton and antiproton distributions are contained within 5.0 eV sec. For protons, diffusion due to IBS and rf phase noise over 34 h led to an increase of both the average action and tails beyond 5.0 eV sec. For antiprotons, there is no tail seen in the final distribution, although the average action clearly increased. The antiprotons with large synchrotron amplitudes have a higher transverse diffusion rate due to multiple crossings of higher-order beam-beam resonances, consequently they have shorter lifetime.
One can summarize all data on antiproton intensity lifetime in collisions presented in this and previous chapters as following: where M stands for bunch position in bunch train, E L is the longitudinal ernittance.
All beam-beam effects observed in the Tevatron depend strongly on particle tunes or working points (WPs). Dedicated experiments to explore these effects have not been conducted because that would require wasting antiprotons needed for WEP-the scans can be quite detrimental and lifetime can deteriorate significantly. Instead, proton and antiproton tunes at injection energy and in collisions have been changed only slightly and not very often over periods of weeks or months. Most operational efforts were focused on keeping machine WPs as close as start of store #3678 -.-34 hours later possible to the "golden ones" (those where machine performance is the best or most reliable). As mentioned previously, deviation of the beam tunes from those optimal values by few 0.001 usually resulted in significant changes (typically deterioration) of Tevatron efficiencies andlor lifetimes. Nevertheless, at the end of HEP stores, when luminosity is many times smaller than the peak, the experiments are more willing to sacrifice a few hours of integrated luminosity and to accept higher than usual background radiation rates. They usually turn off power to the most critical systems, like silicon vertex detectors, and only leave on the instrumentation needed for the accelerator physics experiments, such as luminosity counters and halo monitors. During studies in which beam position is changed, these counters correctly reflect variations of corresponding beam lifetimes. Since these counters are very sensitive to losses and have large bandwidth (report data at least once a second), they can be used for fast WP scans near the optimal working points. Existing beam diagnostics provide bunched beam intensity measurements with a precision of 0.2%-0.6%, so significant time would be needed to determine beam lifetime if it exceeds 10 h. The use of the detector halo rate counters is limited by their maximum counting rates-counters usually saturate if the lifetime drops below 1-2 h. The contour plots presented in Fig. 32

VL DISCUSSION AND CONCLUSIONS
As mentioned in the introduction, the luminosity integral I = J Ldt-the sole critical parameter for HEP experiments-depends on the product of peak luminosity and the luminosity lifetime, e.g., for a single store with initial luminosity Lo and duration T, the integral is I a

L O~t ln(1 + T / T~) .
The initial luminosity can be obtained from a well-known formula for luminosity in head-on which depends on the ratio of the rms bunch length us and beta function at IPS p , y is the relativistic factor, and fB is the frequency of bunch collisions. Beam losses at 150 GeV and up the energy ramp are mostly due to beam-beam interactions. They are accompanied by small longitudinal emittance reduction, but they do not result in significant changes of transverse emittance. Presently, these losses account for a total of 3%-9% at 150 GeV and 6%-10% on the ramp. What is remarkable is that the fractional losses of the "strong" (higher intensity) proton beam are of the same order, or sometimes even exceed, the losses from the weak antiproton beam. Equation (3) explains that phenomenon: indeed, the proton intensity is 6-9 times higher and the transverse ernittance of protons is some 50% larger, but the chromaticity w the proton helix has to be held two or more times higher than on the antiproton helix in order to control the head-tail instability. In any event, the root cause for both proton and antiproton losses are parasitic long-range beam-beam interactions. Rapid antiproton emittance growth after initiating headon collisions ( scallops) of the order of 2 a rnrn rnrad led to a peak luminosity reduction d L / L --~E , / ( E , + s p ) of about 6% until a better working point was implemented. Beam-beam effects, if noticeable, usually manifest themselves in reduction of the beam emittances or their growth rates rather than in increases. The antiproton bunch intensity lifetime T, -20-25 h is dominated by the luminosity bum rate which accounts for 80%-90% of the lifetime, while the remaining 10%-20% comes from parasitic beam-beam interactions with protons. Proton intensity loss is driven mostly by head-on beam-beam interactions with smaller size antiprotons at the main IPS, and varies in a wide range t, -35-200 h. The proton lifetime caused by inelastic interactions with antiprotons in collisions and with residual gas molecules varies from 200 to 400 h.
The hourglass factor decays with TH -7&80 h due to the IBS, again, mostly in proton bunches. Beam-beam effects may lead to reduction of the proton bunch length growth (longitudinal "shaving") in a poorly tuned machine. Antiproton bunch lengthening slows down later in the store when approaching a dynamic aperture due to V. SRLTSEV er al.
beam-beam effects, as was shown in Sec. V. Combining all of these loss rates together as in Eq. (12), one gets the observed initial luminosity lifetime (averaged over the first two hours of store) of about tL -7.5-9 h, as shown in

IS%.
The goal of the run 11 luminosity upgrade project is to attain 3 times more antiprotons delivered to colljsims in the Tevatron by improving the antiproton production rate in the source [2]. The parameters of proton bunches are not expected to differ much from present values, while antiproton transverse emittances may be up to SO% larger than the present (see Table I). By applying the scaling laws from Eqs. (3), (7), (9), and (lo), one expects the total beam losses preceding collisions (at injection and on the ramp) will increase from 19% now to about 42%, while the luminosity lifetime will be reduced a similar 10%-15% (though the lifetime itself will be significantly smaller). Even if the emittances of antiprotons cooled in the recycler ring will be the same as for present operations, the inefficiency before the collisions still will be about 30%. Note, that according to the same scaling laws, increasing the proton bunch intensity by 25% should not change the beam-beam inefficiencies drastically (increase the antiproton losses 2%-3%) if the proton emittances would remain the same.
The numbers for the upgrade parameters do not look very optimistic, so we plan to continue to counteract the adverse beam-beam effects. The planned measures include: (a) increasing beam separation on the ramp and in collisions by using additional separators or higher voltage separators; (b) reducing chromaticity on the ramp and in collisions by the pssible use of octupoles or by employing transverse instability dampers; (c) moving the proton WP above the 7/12 resonance; (d) stabilizing the antiproton and proton tunes during HEP stores; (e) reduce antiproton and proton emittances; (f) compensating beam-beam tune shifts with electron lenses; (g) betatron phase adjustment between two IPS. This article is the most systematic presentation to date of beam-beam phenomena in the Tevatron and the results of relevant beam studies. We have shown that beam-beam effects dominate beam losses at the 150 GeV injection energy and on the ramp, and significantly reduce beam lifetime during collisions. Antiproton losses at all stages of the Tevatron stores are caused by long-range interactions with protons. Proton losses before colljsions are also due to long-range effects, while the proton lifetime reduction in collisions is mostly due to Read-on interactions with smaller size antiproton bunches. Currently, various bearnbeam effects reduce the integrated luminosity by 20%-25%. Several scaling laws were derived to summarize beam-beam observations in the Tevatron. They predict that after anticipated upgrades of the antiproton production complex and a threefold increase of antiproton intensity, the beam-beam effects can reduce the luminosity integral by as much as 4%-50% if not counteracted. Therefore, the work on understanding and mitigation of the beam-beam effects will continue.