Luminosity limitation from a crossing angle

The crossing angle at an IP is usually designed such that more bunches can be accumulated in the collider and higher luminosity can be attained. However, the crossing angle reduces the overlapping region between the two bunches during collision, and the luminosity in each collision decreases, as expressed in the following equation: $L=\frac{n_{\text{b}}{N}_{\text{b}}^{2}f}{4\pi {\sigma }_x^{*}{\sigma }_y^{*}}\frac{1}{\sqrt{1+[(\varphi {\sigma }_z)/(2{\sigma }_x^{*}){]}^2}}.$ (1)

The first term in Eq. (1) is the ideal luminosity, and the second factor is the luminosity reduction owing to the crossing angle. The asterisk indicates the IP value. σ_x* and σ_y* are the RMS horizontal and vertical beam sizes, respectively. L, N_b, n_b, f, φ, and σ_z are the luminosity, bunch population, number of bunches, revolution frequency, crossing angle, and RMS longitudinal beam size, respectively.

The crossing angle at the IP is created using a set of dipoles [15-16]. After adding the crossing angle, the beam orbit is displaced in the IR quadrupoles, leading to unwanted anomalous dispersion at the IP [15, 17]. This can increase the beam size as follows: ${({\sigma }_{x,y}^{*})}^2={\epsilon }_{x,y}{\beta }_{x,y}^{*}+{({\sigma }_{\text{p}}{D}_{x,y}^{*})}^2,$ (2) where ε_x,y, β_x,y and D_x,y are the horizontal and vertical emittance, beta function, and dispersion, respectively, and σ_p is the RMS momentum spread. Eq. (1) shows that the ideal luminosity decreases, and the reduction factor due to the crossing angle increases with a larger horizontal beam size. The derivative of luminosity with respect to σ_x is $\frac{\text{d}L}{\text{d}{\sigma }_x^{*}}=-2f{n}_{\text{b}}{N}_{\text{b}}^2/\left(\pi {\sigma }_y^{*}(4{({\sigma }_x^{*})}^2+{\sigma }_z^2{\varphi }^2)\sqrt{4+\frac{{\sigma }_z^2{\varphi }^2}{{({\sigma }_x^{*})}^2}}\right),$ (3) which shows that the luminosity monotonously decreases with increasing σ_x. In addition, the luminosity is restricted by the synchro-betatron coupling caused by the finite dispersion at the IP [18].

The scheme proposed in Ref. [15] provides a method for correcting anomalous dispersion in both planes. Two pairs of quadrupole correctors are placed in the neighboring arc region (where there was a horizontal dispersion) on both sides of each IR. In each pair, the two quadrupoles are separated by an π in phase advance and have the same strengths and opposite polarities to cancel the changes in the beta function from each quadrupole corrector. For an IR with vertical crossing, a pair of skew quadrupoles is used because these quadrupoles transform the horizontal dispersion into the vertical plane.

The dispersion correction scheme limits the achievable ranges of crossing angle and β*, while maintaining the required level of correction. Moreover, the ranges of β* and crossing angle are determined by the available physical aperture. The minimum aperture occurs in the inner triplet at the location of the maximum β and decreases linearly with increasing crossing angle. Equation (1) shows that the luminosity strongly depends on the transverse beam size and crossing angle. Thus, the targeted level of dispersion correction and the physical aperture limit the attainable luminosity.

Beam behaviors related to the beam-beam interactions

The B-B interactions are nonlinear kicks, so that the particles oscillating with different amplitudes have different tune shifts. The tune distribution of all the particles in the frequency space is called the tune footprint. The theoretical formula to calculate the tune shift in the horizontal plane for particles with different amplitudes is [4] $\begin{array}{l}\Delta {\nu }_x(a_x,a_y,d_x,d_y,r)=\frac{4\pi C}{{\epsilon }_x}\\ {\displaystyle {\int }_0^1\frac{e^{-(p_x+p_y)}}{\upsilon {[\upsilon (r^2-1)+1]}^{1/2}}}{\displaystyle \sum _x{\displaystyle \sum _y\text{d}\upsilon }},\end{array}$ (4) where ${\displaystyle \sum _x={\displaystyle \sum _{k=0}^{\infty }\frac{{\left(\frac{a_x}{d_x}\right)}^k}{k!}}}\Gamma \left(k+\frac{1}{2}\right)\left[{\text{I}}_k(s_x)(\frac{2k}{a_x{}^2}-\upsilon )+{\text{I}}_{k+1}(s_x)\frac{s_x}{a_x{}^2}\right],$ (5) $\begin{array}{l}{\displaystyle \sum _y={\displaystyle \sum _{l=0}^{\infty }\frac{{(\frac{a_y}{d_y})}^l}{l!}}}\Gamma (l+\frac{1}{2}){\text{I}}_l(s_y),\\ C=\frac{N_{\text{b}}{r}_{\text{p}}}{{(2\pi )}^3{\gamma }_{\text{p}}},r=\frac{{\sigma }_y}{{\sigma }_x},p_x=\frac{\upsilon }{2}(a_x^2+d_x^2),p_y\\ =f\frac{\upsilon }{2}(a_y^2+d_y^2)\\ s_x=\upsilon a_x{d}_x,s_y=f\upsilon a_y{d}_y,\text{}f=r^2/(\upsilon (r^2-1)+1),\end{array}$ (6) where υ is the integration variable in Eq. (4); a_x and a_y are the particle amplitudes normalized by the beam transverse size; d_x and d_y are the separations normalized by the beam transverse size; r is the aspect ratio; r_p is the classical proton radius; and γ_p is the relativistic Lorentz factor. I_n(s_n) is the modified Bessel function of the first kind and Γ(n+0.5) is the gamma function. A similar equation exists for Δν_y; more details can be found in Ref. [4]. These expressions show that the tune shifts depend on the normalized separations, amplitudes, and aspect ratio of the strong beam, in addition to the brightness parameter N_b/(γ_pε_x).

The B-B parameter is the largest tune shift owing to the HOIs in a bunch. For these interactions without a crossing angle, it is expressed as ${\xi }_{x,y}=\frac{r_{\text{p}}{N}_{\text{b}}{\beta }_{x,y}^{*}}{2\pi {\gamma }_{\text{p}}{\sigma }_{x,y}({\sigma }_x+{\sigma }_y)},$ (7) where ξ_x,y denote the B-B parameter in the horizontal or vertical plane. In the presence of a crossing angle in the horizontal plane, it is modified as [19] $R=\frac{1}{\sqrt{1+[(\varphi {\sigma }_z)/(2{\sigma }_x^{*}){]}^2}},$ (8) ${\xi }_x=\frac{r_{\text{p}}{N}_{\text{b}}{\beta }_x^{*}{R}^2}{2\pi {\gamma }_p{\sigma }_x({\sigma }_x+R{\sigma }_y)},$ (9) ${\xi }_y=\frac{r_{\text{p}}{N}_{\text{b}}{\beta }_y^{*}R}{2\pi {\gamma }_{\text{p}}{\sigma }_y({\sigma }_x+R{\sigma }_y)}.$ (10)

The B-B parameter is different in both the transverse planes, even for round beams, and it decreases with an increasing crossing angle.

FMA has been widely used as an early indicator of particle instability by calculating the diffusion of tunes. The diffusion parameter is defined as follows [20]: $D={\log }_{10}[\sqrt{{({\nu }_{x2}-{\nu }_{x1})}^2+{({\nu }_{y2}-{\nu }_{y1})}^2}],$ (11) where ν_x1,y1 and ν_x2,y2 are tunes from the first and second halves of the tracking turns using a fast Fourier transform (FFT), respectively. A larger tune diffusion suggests stronger destructive effects of the nonlinearity.

The dynamic aperture is defined as the maximum stable transverse amplitude, which typically requires numerical tracking. We choose the initial transverse distribution to be a 2D Gaussian in the (x, y) space as x = Nσ_xcosθ, y = Nσ_ysinθ, x’ = y’ = 0, and θ ranges from 0 to π/2 in steps of 15°. N is a real parameter whose range varies with the problem under study.

Resonance excitation due to the B-B interactions was the main mechanism affecting the single-particle stability in our study. The HOI excited even resonances, and the LRI activated resonances of any order. In the case of the crossing scheme, the crossing angle can drive additional odd resonances. If the resonances overlap, the particles may be trapped and move outward to the aperture limit defined by the collimation system [21]. Meanwhile, the crossing scheme couples the transverse plane to the longitudinal plane, which activates synchro-betatron resonances, another mechanism that deteriorates beam stability.

Mitigation of long-range interactions with modified beta star and crossing angle

The LRI kick strongly depends on the transverse separation normalized by the RMS transverse size of the strong beam. The transverse separation can be controlled in two ways. One is to increase the crossing angle, which changes the absolute transverse separation; the other is to increase β*, which changes the beam size.

Increasing the crossing angle has many negative effects, such as lower luminosity, enhanced synchro-betatron resonances, lower relative physical aperture, and worse nonlinearities for the inner quadrupole triplet. In this case, the countermeasure is to use crab cavities to recover the luminosity and minimize the synchro-betatron resonances, assuming their successful operation in the LHC. However, the smaller relative physical aperture and worse nonlinearities of the triplet remain important limitations.

Increasing β* reduces β at LRIs, which can improve the beam stability, but causes lower luminosity. However, in future hadron colliders with significant synchrotron radiation, the emittance evolution follows an exponential decay: $\epsilon \left(t\right)=\epsilon \left(0\right)e^{-\left(t/\tau \right)},$ (12) where τ is the transverse emittance damping time (= 2.35 h in the SPPC, see Table 1) The emittance shrinking can help increase the relative transverse separation at LRIs and thus mitigate their strength. This provides the possibility of squeezing β* dynamically to recover luminosity while maintaining the impact of the LRIs under control. However, the emittance reduction will enhance the HOIs and possibly make them become dominant in the later stages of the luminosity run. Dedicated studies of B-B interactions need to be conducted together with the evolution of the beam parameters, which is discussed in Sect. 4.

Compensation of long-range interactions with current carrying wires

The LRI compensation (LRC) with current-carrying wires was first proposed and studied in 2000 [22]. Each wire was placed parallel to the beam trajectory to compensate for the LRI acting on the beam. In general, the wire kick depends on (a) the wire current (I), (b) the wire length (L), (c) β_x,y at the wire, (d) phase advances between the wire and LRI locations, and (e) the transverse distance from the beam to the wire. Parameters (a) and (b) can be determined by assuming the same integrated strength of the wire and the LRI, which can be expressed as [23] $I{L}_{\text{w}}=N_{\text{b}}{n}_{\text{LR}}ec,$ (13) where N_b is the bunch population; n_LR is the number of LRIs that need to be compensated; e is the elementary charge; and c is the speed of light.

Parameters (c), (d), and (e) need to be selected according to the distribution of the LRIs and lattice features with several requirements. First, the ratio β_x/β_y at the wire should match that at the LRI locations [24]. Second, it is preferable to place the wire after the first separation dipole to avoid affecting another beam. Third, the phase advances between the wire and the LRI locations should be an integer multiple of π to better compensate for the LRI kicks [24]. Finally, the transverse normalized separation at the wire must match that at the LRI location. It is impossible to meet all the requirements, and therefore, compromises are necessary in choosing the location of the wires for optimum compensation.

Compensation of head-on interactions with electron lens

HOI compensation (HOC) using electron beams was first proposed and studied in 1993 [25-26]. A low-energy electron beam is introduced to collide with a proton beam, which can compensate for the kick given to this beam delivered by the opposing beam. The device used to provide the electron beam is called an e-lens. Generally, the kinetic energy of electrons is approximately 5-10 keV. To compensate for the effect of HOIs, the following conditions, similar to those for long-range compensation, need to be met [27-28]:

(a) The electron beam should exhibit the same transverse distribution as the proton beam.

(b) The number of electrons (N_e) overlapping with a proton bunch, bunch population of protons (N_p), and relative speed of electrons (β_e) should satisfy $N_{\text{e}}(1+{\beta }_{\text{e}})=N_{\text{p}}.$ (14) Once β_e is selected, the number of electrons can be determined by the bunch population of protons.

(c) The electron beam size should be the same as that of the proton beam at the e-lens.

(d) The ratio of the lattice horizontal and vertical β functions at the e-lens should match the ratio at the IP.

(e) The phase advance between the HOI and e-lens should be an odd multiple of π to compensate for the tune shift and resonance driving terms.

It is also difficult to fulfill all the conditions in practice when compensating for the HOI.

Nominal IR lattice

The IR lattice is a typical left/right antisymmetric optics with the same β* value in the two transverse planes. The free space from the IP to the first triplet quadrupole is 45 m. The inner triplet is responsible for producing a small β* of 0.75 m, which causes a maximum β of 18600 m in the triplet. The two separation dipoles separate the beams into their own pipes. The outer triplet matches the beta and dispersion to the dispersion suppressor. Figure 2 shows the beta functions in the IR with a horizontal crossing angle (IR3). The first-order chromaticity in the ring is corrected by sextupoles, but a second-order chromaticity correction system has not yet been designed and is not considered here.

Fig. 2Full-screen View

(Color online) Beta functions in the interaction region (IR) with a horizontal crossing angle.

In the SPPC, crossing is generated in the horizontal plane at IP3 and vertical plane at IP7 to compensate for the tune shifts from the LRIs. Four dipole correctors per beam in each IR are used to steer the closed orbit, and four quadrupole correctors are placed in each IR to correct the anomalous dispersion owing to the crossing angle. Figure 3 shows the layout of the dipole and quadrupole correctors for the two IRs. The phase advance of each orbit and quadrupole corrector from the IP is marked in the plot. In Table 2, the dispersions at both IPs and the maximum horizontal and vertical dispersions in IR3, IR7, and the entire ring are listed. Comparing the 3rd and 4th columns, the anomalous dispersion is found to be well compensated by the quadrupole correctors. Figure 4 shows the corrected dispersion in the two IRs and their neighboring arc regions. With a corrected dispersion of 0.001 m at the IP, Eq. (2) shows that the luminosity loss with a momentum spread of approximately 0.004% (equal to the nominal RMS value) is negligible.

Fig. 3Full-screen View

(Color online) Sketch of dipole and quadrupole correctors in IR3 with horizontal crossing (a) and in IR7 with vertical crossing (b). D1 is the first separation dipole and the star symbols show the interaction points (IPs). The blue and red rectangles indicate dipole correctors for Beam 1 and Beam 2, respectively. The overlapping rectangles show the correctors for both beams in different beam pipes. The numbers in brackets are the phase advances from the element to the IP in the horizontal plane (a) or in the vertical plane (b). (HX, VX) are the dipole correctors generating crossing angles in IR3 and IR7, respectively. (HQ, VQ) are the quadrupole correctors in IR3 and IR7, respectively. (F, D) represents (focusing, defocusing) quadrupoles, respectively.

Fig. 4Full-screen View

Dispersion in two IRs and its neighboring arc regions with quadrupole correctors in IR3 (a) and IR7 (b). β* is 0.75 m and crossing angle is 110 μrad at both IPs.

As mentioned in Sect. 2.1, the values of the crossing angle and β* affect the validity of the dispersion-correction scheme. Three β* values and different initial parasitic separations were considered to study the dispersion correction scheme. The initial parasitic separation in the drift space before the triplet quadrupoles is normalized by the RMS beam size σ and can be approximated by the following formula for a small β*: $d=\varphi s/\sqrt{\epsilon ({\beta }^{*}+\frac{s^2}{{\beta }^{*}})}\approx \frac{\varphi \sqrt{{\beta }^{*}}}{\sqrt{\epsilon }},$ (15) where s is the distance of the parasitic interaction location from the IP.

First, the initial separation was maintained constant at 12σ; β* was varied; and the crossing angle scale was $1/\sqrt{{\beta }^{*}}$. Table 2 indicates that the maximum (horizontal, vertical) dispersions (D_x^max, D_y^max) are (2.8 m, 0 m), respectively, without a crossing angle. With the crossing angles, the quadrupole correctors can correct D_x^max to approximately 2.8 m when β* is 0.5, 0.75 and 1.00 m. The dispersion at the IP was corrected to less than 0.005 m, decreasing the luminosity by less than 0.2%. Figure 4 shows that the correction of the vertical dispersion in IR7 is slightly worse than that of the horizontal dispersion in IR3. This occurs because the skew quadrupole correctors are not at the ideal locations, where the phase advances between the quadrupole correctors and the IP should be half-integer multiples of π, but the errors are still within the acceptable level.

Next, we increased the initial separation from 12σ to 20σ for the same three values of β* indicated above. Similarly, the horizontal dispersion was well corrected in each case, but the vertical dispersion, D_y, was not. Figure 5 shows the maximum vertical dispersion, D_y^max, with and without the quadrupole correctors. In the entire range of separations, D_y^max can be corrected to less than 0.5 m when β* is 0.75 and 1.00 m, but D_y^max exceeds 1 m after the correction when β* is 0.5 m. The dispersion correction worsens with a smaller β* and larger separation. Table 3 lists some key parameters for an initial parasitic separation of 20σ. It was assumed that the luminosity reduction was recovered with the use of crab cavities. When β* is 0.5 m, the smallest physical aperture in the inner triplet drops to an unacceptably low value of 9σ and the beta-beating is intolerably large. This study indicates that large separations require a β* greater than 0.5 m.

Fig. 5Full-screen View

(Color online) Maximum vertical dispersion D_y^max around the ring as a function of the initial parasitic separations with (solid) and without (dashed) quadrupole correctors.

Beam-beam effects with constant emittance

3.2.1

Tune footprint and frequency map analysis

In the SPPC, the nominal tunes were (120.31, 117.32), and the fractional parts were the same as that in the LHC. The B-B parameter was 0.0075 for each IP. The tune footprints of a nominal bunch from both the theory mentioned in Sect. 2.2 and the simulations with all the B-B interactions for two different initial tunes are shown in Fig. 6 (previously used in a review article [29]). In the simulations, the tune footprint was obtained by tracking the particles with amplitudes ranging from 0 to 10σ_x,y for 2048 turns. Chromaticity correction and momentum deviation were not included to consider the tune spread from only B-B interactions. All the sum and difference resonances up to the fourth order are shown. In both plots, the theoretical calculation predicts the main part of the tune footprint quite well, particularly in the area far from the resonances that are not included in the theory. Both plots show the effects of the difference coupling resonance driven by the skewed quadrupole components of the LRI. The figure on the left shows that a few particles are captured by the 3rd order resonance 2ν_x + ν_y = 1, which is also driven by the LRI. In the plot on the right, the harmful low-order sum resonances are far away, and there is no evidence of particle trapping.

Fig. 6Full-screen View

(Color online) Footprint caused by beam-beam (B-B) interactions for different initial nominal tunes: (a) (0.31, 0.32) and (b) (0.17, 0.19). The red and green points represent the theoretical calculation and simulations, respectively. The lines show all the nearby resonances, up to the fourth order. The numbers (m, n) in brackets define the resonances mν_x + nν_y = p, where p is an integer.

To determine the contribution of different nonlinear kicks, Fig. 7 presents FMA plots for three cases comparing the effects of the HOIs and LRIs. The FMA plots were obtained by tracking the particles for 4096 turns. The amplitudes extend to a physical aperture of 23σ in the first plot compared to 10σ in the other two plots. This clearly shows that the tune diffusion increases significantly with the LRI compared to the HOI. The tune variation of particles with amplitudes between 6σ and 10σ increases by more than four orders of magnitude in the last two plots, which indicates that the LRI is the main source of particle instability. This conclusion is the same as that for LHC-like configurations, as expected [3, 13-14]. Although the LRI mainly impacts particles at large amplitudes, small-amplitude particles are also affected.

Fig. 7Full-screen View

(Color online) Frequency map analysis (FMA) plots with different B-B interactions: (a) Head-on interaction (HOI); (b) Long-range interaction (LRI); (c) HOI + LRI. Sextupole kicks are included in all cases.

3.2.2

Dynamic aperture with nominal tunes

Particle instability based on FMA calculations must be checked using longer-term tracking. Thus, the DA was calculated by tracking the particles over 10⁶ turns. The initial longitudinal distribution in z is Gaussian-truncated to 2σ_z and has a uniform momentum spread of 1σ_p. The chromaticity was corrected to +1 in both planes. The average DA was obtained by averaging over 15 azimuthal angles.

Four nonlinear cases are studied. The average DA values for both the nominal and Pacman bunches are listed in Table 4. Pacman bunches denote those bunches with fewer LRI in an IR because of the time gap between the bunch trains. In Table 4, the Pacman bunch has 41 LRIs in each IR, which is the least number of LRIs among all such bunches. The physical aperture (23σ) is taken as the average DA when the simulated average DA exceeds the physical aperture. We found that the DA for the nominal bunch is larger than the physical aperture without the LRI. These interactions reduce the DA to approximately 6σ The HOI has a relatively small impact on the DA. Therefore, this demonstrates again that the LRI is the main source of particle loss among the included interactions. The same conclusion is true for the Pacman bunches. Figure 8 shows the distribution of the LRI separations in IR3. The first 12 LRIs before the first quadrupole occurred at a constant separation of 12σ. The minimum separation was 9σ-10σ. There are also six LRIs with a constant separation of 17σ on the right-hand side. Figure 9 shows the DA for different numbers of LRIs with different separations.

Table 4Full-screen View

Average dynamic aperture (DA) for four different cases. The physical aperture is 23σ

	DA-nominal (σ)	DA-Pacman (σ)
Sextupole	23	23
Sextupole + HOI	23	23
Sextupole + LRI	6.2	9.8
Sextupole + HOI + LRI	5.5	7.8

Show more

Fig. 8Full-screen View

B-B separations in IR3 are normalized by their horizontal beam sizes. The blue dots denote the separations of LRIs and the red dot represents the HOI at the IP.

Fig. 9Full-screen View

DA in amplitude space for different numbers of LRIs. The HOIs are included in all settings. The numbers in brackets show the number of LRIs and average DA. The blue/red/cyan lines show the average DAs with all LRIs, 48 LRIs at separations of 12σ, and LRIs at separations of (9-10)σ.

3.2.3

Dynamic aperture with tune scan

To operate accelerators effectively, the working points must be carefully selected; otherwise, the beam may be sensitive to errors and easily lost. Initially, the fractional tunes of the SPPC are the same as those in the LHC design, but a tune scan is performed to find better working points [30]. The DA is calculated with different fractional tunes, and the integer part remains unchanged. In the tune space, the resonance-free spaces are wider along the diagonal; thus, the initial tunes are scanned along the diagonal from (0.10, 0.10) to (0.46, 0.46) with a step size of 0.01. The tunes are split by keeping |ν_x-ν_y| constant to avoid the strong influence of difference resonance. The tune separations are set to the values of (± 0.01, ± 0.02) to study the coupling effects due to the LRI. LHC studies also revealed that a better DA was observed with tunes close to the diagonal [31].

For a tune separation of 0.01, Fig. 10 shows the average DA versus the horizontal fractional tune, with and without a nominal crossing angle of 110 μrad. Independent of the crossing angle and momentum deviation, the DA declines rapidly when the fractional tunes approach the lower-order resonances in the 5th order (0.2), 4th order (0.25), and 3rd order (0.33), while the drop at 0.4 only appears in the case with a crossing angle. Both the crossing angle and momentum spread cause some reduction in DA. The synchro-betatron resonances driven by the crossing angle are believed to cause this reduction [32].

Fig. 10Full-screen View

Average DA versus different horizontal tunes (ν_y = ν_x + 0.01). Blue: without a crossing angle and dp/p = σ_p; Red: with a crossing angle and no momentum deviation; Green: with a crossing angle and dp/p = σ_p.

A larger tune separation of 0.02 is also tested. The 3rd order resonances still reduce the DA but the 4th and 5th order resonances do not affect it. This implies that the tune separation of |ν_x - ν_y| > 0.01 seems to be safer. Table 5 summarizes the DA with the six best-found tunes and nominal tune. We find that the smallest DA is always approximately 1σ less than the average DA. Figure 11 shows the FMA plots in both the amplitude and frequency spaces for the nominal and two best tunes. We observe from the amplitude space plots that the tune diffusion is smaller at the best tunes for particle amplitudes ranging from 6σ to 8σ. The plots in the tune space show that the footprint for the nominal tune is crossed by the 3rd order resonances, and the footprints for the other two tunes are crossed by 4th and 6th order sum resonances. As expected, the lower the order of the resonance, the smaller is the dynamic aperture. The tunes identified here could serve as good initial candidates for further detailed studies.

Table 5Full-screen View

DA for the six best tunes and nominal tune.

Tunes	Average DA (σ)	Smallest DA (σ)
(0.12, 0.13)	7.13	6.25
(0.17, 0.19)	7.12	6.25
(0.27, 0.26)	7.02	6.00
(0.37, 0.35)	6.70	5.75
(0.19, 0.17)	6.57	5.75
(0.38, 0.37)	6.50	6.25
(0.31, 0.32)	5.50	4.75

Show more

Fig. 11Full-screen View

(Color online) FMA plots in the amplitude (top) and frequency (bottom) spaces for the nominal tune (a,d), (0.27, 0.26) tune (b,e), and (0.17, 0.19) tune (c,f). The numbers (m, n) in brackets define the resonances mν_x + nν_y = p, where p is an integer.

Mitigation of the beam-beam effects

3.3.1

Long range interactions with different parasitic separations

In the baseline design, the B-B transverse separation was set to 12σ at the first parasitic interaction with a full crossing angle of 110 μrad and β* of 0.75 m. As discussed in Sect. 2.3, the LRI can be mitigated by increasing the parasitic separation. The separation at the first parasitic interaction was increased from 12σ to 20σ by adjusting the crossing angle and β*. The separations at the other LRIs increase simultaneously, but not necessarily by the same amount. As stated in Sect. 3.1, β* = 0.5 m and 1.0 m are also considered, and the same settings as those in Sect. 3.2.2 are used for the chromaticity correction and momentum spread.

Figure 12 shows the average DA versus the separation at the first parasitic interaction for the three values of β*. As expected, increasing the separation is useful for the average DA, which is almost independent of β* because the crossing angle is increased by scaling as $1/\sqrt{{\beta }^{*}}$. However, increasing the crossing angle causes luminosity loss, although the crab cavities can recover the loss; however, other negative effects still exist, as discussed in Sect. 2.3. Thus, active LRC is considered to improve beam stability or even allow operation at smaller crossing angles.

Fig. 12Full-screen View

(Color online) Average DA versus initial parasitic separation for different β* values.

3.3.2

Compensation of long-range interactions

Figure 13 shows the ratio β_x/β_y at different parasitic interaction locations in IR3 at nominal β* = 0.75 m. The ratio β_x/β_y is not constant, but varies from 0.2 to 4.6. Figure 14 shows the locations for installing the current wires for long-range compensation and e-lens for head-on compensation in IR3. The phase advances from the IP to the locations of the LRI are all close to π/2 in the two transverse phase planes. As mentioned in Section 2.4, it is impossible to fully compensate for the LRI with the current wires. The ideal location is approximately 497 m away from the IP with a ratio β_x/β_y = 1, and the phase advance between the IP and the wire is approximately 3π/2 in both planes. There are four current wires in total, with one wire on each side of the IP. Each wire is responsible for compensating for 41 LRIs on its side in the IR. The phase advance between the wire and the LRI is nearly π in both planes, and the transverse separation from the weak beam at the wire is 12σ. According to Eq. (13), the current for a wire 2.5 m long is 118.1 A. The tune footprint after LRC is shown in Fig. 15. As expected, the tune footprint becomes much smaller with the use of wires and is close to that with only the HOI.

Fig. 13Full-screen View

Ratio β_x/β_y at all parasitic interaction locations in IR3.

Fig. 14Full-screen View

(Color online) Phase advance and beta functions in IR3. The locations with β_x/β_y = 1 for the current wires and e-lens are marked with arrows.

Fig. 15Full-screen View

(Color online) Tune footprints with amplitude from 0 to 15σ_x,y including all the interactions with and without LRC, and tune footprint including only HOI.

Although the tune spread shrinks using wires, the resonance strength is more important for beam stability. Thus, DA simulations were performed to verify the effectiveness of compensation. The simulations for the HL-LHC found that the wires positioned at β_x/β_y = 0.5 or 2 were the most favorable, because the resonance driving term was found to be minimum [33]. Similar simulations of the DA were carried out for the SPPC, but the results showed that the compensation at β_x/β_y = 1 was still the best [34].

3.3.3

Compensation of head-on interactions

Although the above study shows that the LRI is dominant in determining the DA, the large tune spread caused by the HOI will have greater importance when all nonlinear components in the ring are included. Head-on compensation is considered an option, or even necessary, for future hadron colliders. In this subsection, the results of the compensation with an e-lens are presented.

Considering the conditions defined in Section 2.5, the electron beam was assumed to have a Gaussian distribution, similar to a strong beam. The kinetic energy of electrons is 10.53 keV, corresponding to a β_e of 0.2. With Eq. (14), N_e was calculated to be 1.25 × 10¹¹. Two e-lenses were distributed along the SPPC ring, one for each IR, as shown in Fig. 14. The e-lenses are located at the location with the same horizontal and vertical β but the phase advances between the e-lens and the IP are (1.42π, 9,54π instead of the odd multiples of π The effective length of the e-lens is 2 m.

Figures 16 and 17 show the tune footprints and tune diffusion with and without HOC. The tune spread was dramatically reduced with the compensation, but the tune diffusion rate increased at all particle amplitudes. Thus, we find that the HOC does not necessarily improve beam stability.

Fig. 16Full-screen View

(Color online) Tune footprints with amplitude from 0 to 8σ_x,y with (green) and without (red) HOC. The HOI denotes the interaction without a crossing angle.

Fig. 17Full-screen View

(Color online) FMA plots in the amplitude space without (a) and with (b) HOC.

Beam-beam effects with different emittances

This subsection presents studies on the effects of both HOIs and LRIs with varying emittance values. The simulations were carried out with different normalized emittances ranging from 2.4 μm (nominal value) to 0.2 μm, and the natural damping is estimated to require 6 h. The emittance damping is given by Eq. (12). The emittance values in the simulations were far greater than the calculated equilibrium emittance of 0.02 μm. Other parameters, such as the bunch intensity, were unchanged in these simulations. The results with a crossing angle of 110 μrad and longitudinal motion are shown in Fig. 18. The physical aperture and DA values in this figure were scaled by the RMS beam size at the normalized emittance to show the relative beam stability.

Fig. 18Full-screen View

(Color online) (a) Average DA and B-B parameter versus emittance with a crossing angle of 110 μrad and longitudinal motion (HOI alone); (b) average DA versus emittance with different B-B effects and correction schemes (LRI alone, HOI-LRI, HOI-LRI-LRC, HOI-LRI-LRC-HOC).

Figure 18a shows that as the emittance decreases from 2.4 μm to 0.2 μm, the physical aperture normalized by RMS beam size and B-B parameter increase as expected. The HOI alone does not cause beam loss at the initial stages of shrinking emittance but from 1.2 μm downward, the relative DA decreases. This verifies that the HOI gradually becomes considerably more important as the emittance decreases. As expected, the DA with the LRI alone improves with a smaller emittance, as shown in Fig. 18b. With combined HOI and LRI, the beam quality is dominated by the LRI from 2.4 μm to 1.2 μm and long-range compensation is effective in increasing the DA. However, the HOI dominates at smaller emittances, and both the LRC and HOC gradually lose their effectiveness. Thus, to maintain beam stability, for example, with a relative DA larger than 8σ, it will be necessary to slow down the fall in emittance by an emittance heating mechanism.

Beam-beam effects with emittance damping and particle burn-off

In addition to emittance damping, beam loss from luminosity also reduces the strength of the B-B interactions. Thus, we now consider both emittance shrinkage and beam loss in the DA simulations. The emittance evolution is assumed to follow Eq. (12). The loss rate of the bunch population owing to beam collisions follows from $\frac{\text{d}{N}_{\text{b}}}{\text{d}t}=-{\sigma }_{\text{tot}}{n}_{\text{IP}}\frac{L}{n_{\text{b}}},$ (16) where n_IP is the number of interaction points; n_b is the number of bunches; and σ_tot is the total p-p cross section. Then, the bunch population evolution can be calculated by the following expression, which is derived by combining Eqs. (1), (12), and (16): $N_{\text{b}}\left(t\right)=\frac{1}{\frac{1}{N_{\text{b}}\left(0\right)}-\frac{2K\tau {(H{e}^{t/\tau }+1)}^{1/2}}{H{e}^{t/\tau }\epsilon (0)}+\frac{2K\tau {(H{e}^{2t/\tau }+1)}^{1/2}}{H{e}^{t/\tau }\epsilon (0)}},$ (17) $K=\frac{{\sigma }_{\text{tot}}f{n}_{\text{IP}}}{4\pi {\beta }^{*}}$, $H=\frac{{({\sigma }_z)}^2{\varphi }^2}{4\epsilon (0){\beta }^{*}}$,

where N_b(0) and ε(0) are the initial bunch population and emittance, respectively. The constant K is related to the ideal luminosity, and H is related to the luminosity reduction factor owing to the crossing angle. The remaining parameters are defined in Section 2. With a zero-crossing angle, the bunch population evolution is [35] $N_{\text{b}}\left(t\right)=\frac{1}{\frac{1}{N_{\text{b}}\left(0\right)}-\frac{K\tau }{\epsilon (0)}+\frac{K\tau e^{t/\tau }}{\epsilon (0)}}.$ (18) The emittance and bunch population evolutions obtained using Eqs. (12) and (17) at a few discrete moments are shown in Fig. 19. The DA simulations in this section were performed with the inclusion of the crossing angle and synchro-betatron motion; all other conditions remained unchanged. In addition, a simulation with a zero-crossing angle for the HOI or without longitudinal motion was also carried out to understand the role of synchro-betatron coupling.

Fig. 19Full-screen View

(Color online) Evolutions of emittance and bunch population at different times. The initial and final values are marked.

The simulation results are presented in Fig. 20 (previously used in a review article [29]). The B-B parameter increases monotonically with time. The DA with the combined HOI and LRI increases at first and starts decreasing at approximately 3 h. After compensating for the LRI, the DA improves by at least 2σ in the first 2 h and gradually declines until approximately 4 h when the LRC is no longer effective. This demonstrates that LRI is still the main limitation of beam stability in the initial stages, and HOI together with synchro-betatron coupling become more important in the later stages.

Fig. 20Full-screen View

(Color online) Evolutions of the average DA (circles) and B-B parameter (squares) with time including particle burn-off.

When the HOI and LRI are both compensated, the DA is nearly the same as that with LRC alone in the first 4 h, but HOC increases the DA from 4 h to 5.5 h, as shown in Fig. 20. This demonstrates that HOC is effective when the HOI is dominant. The simulations also show that after 5.5 h, both HOC and LRC have no beneficial effects in improving the DA, but the DA without longitudinal motion or with zero crossing angle still increases in this period. This can be explained by the synchro-betatron coupling at a smaller emittance, which is the driving mechanism of DA reduction. As mentioned in Sect. 2.3, crab cavities at IPs can be used to suppress the synchro-betatron coupling.

The DA in the LHC [36] and HL-LHC [14] must be at least 6σ. The simulations for HL-LHC found that the multipole magnetic field errors did not have a significant impact on the beam dynamics in the case of strong B-B interactions [14]. In the SPPC, the average DA goal is set to 8σ considering that the smallest DA is approximately 1σ smaller than the average DA, and the contribution of high-order magnetic field errors is not included. However, as shown in Fig. 20, the average DA is below 8σ after 4.5 h, even with both HOC and LRC. To restore the beam stability without using an emittance heating mechanism that sacrifices luminosity, a DA goal of 8σ can be achieved by jointly applying crab cavities and LRC.

In the SPPC, luminosity optimization and leveling have been studied for different possible operation scenarios by considering limitations, such as those from the B-B parameter and event pileup per crossing [37]. When the B-B parameter sets the limit of luminosity optimization in the SPPC, a comparatively realistic operation scheme is shown in Fig. 21. The evolutions of some important parameters (luminosity for each IP, bunch population, emittance, and B-B parameter for two IPs) with the run time in two cycles are presented. We assume that with crab cavities and LRC present, the DA is no longer the limiting factor; thus, the emittance is naturally damped before reaching the B-B parameter limit of 0.06 with two IPs when a new collision cycle is started. A peak luminosity of 1.885 × 10³⁵ cm^-2s^-1 occurs in the middle of a physics run. Finally, for each IP, an optimum average luminosity of 1.19 × 10³⁵ cm^-2s^-1 and annual integrated luminosity of 1.15 ab^-1 are obtained, assuming an operation time of 160 days and a machine availability of 70%. Accordingly, if the maximum B-B parameter should be set to a lower value (e.g., 0.03 with two IPs), emittance heating must be applied earlier and the integrated luminosity will be lower.

Fig. 21Full-screen View

(Color online) Evolution of the collision parameters over two cycles. The red, blue, magenta, and green colors denote the luminosity (in 10³⁵ cm^-2s^-1), emittance (in μm), bunch population (in 10¹¹), and B-B parameter.