The Lie-algebraic method is used to develop generalized Courant-Snyder invariant in the presence of an arbitrary number of beam-beam collisions, head-on or long-range, in a storage ring collider. The invariant is obtained by concatenating nonlinear beam-beam maps in the horizontal plane and to first order in the beam-beam parameter. Tracking evidence is presented to illustrate that with LHC parameters the invariant is indeed preserved and can be used to predict the smear of horizontal emittance observed in tracking simulations. We discuss the limits of applicability of this model for realistic LHC collision schemes

Herr, Werner

Kaltchev, D I

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ANALYTICAL CALCULATION OF THE SMEAR FOR LONG-RANGEBEAM-BEAM INTERACTIONSDobrin Kaltchev, TRIUMF, Vancouver, CanadaWerner Herr, CERN, Geneva, SwitzerlandAbstractThe Lie-algebraic method is used to develop generalizedCourant-Snyder invariant in the presence of an arbitrarynumber of beam-beam collisions, head-on or long-range,in a storage ring collider. The invariant is obtained byconcatenating nonlinear beam-beam maps in the horizontalplane and to first order in the beam-beam parameter. Track-ing evidence is presented to illustrate that with LHC pa-rameters the invariant is indeed preserved and can be usedto predict the smear of horizontal emittance observed intracking simulations. We discuss the limits of applicabilityof this model for realistic LHC collision schemes.INTRODUCTIONThe beam-beam interaction, in the weak-strong model,can be studied with transfer maps. Assuming the weak-beam particle motion to consist of passages through thestrong beam (collisions) alternating with linear-motion sec-tions, the ring transfer map is given by the product:M =NIP∏k=1e:F(k):e:F(k)2 : = e:h:,where NIP is the number of collision points, head-on orlong-range, e:F (k): and e:F(k)2 : are Lie operators associatedwith a linear matrix and a beam-beam kick respectivelyand h is a generalized Courant-Snyder invariant. By usingthe properties of Lie operators for F-factors in action-anglerepresentation it is not difficult to derive an expression for hto first order of the beam-beam parameter in the horizontalplane. For multiple head-on interaction points in the LHCsuch an expression was reported previously [1],[2], wherealso a condition was derived for suppression of resonances.In this paper we present an invariant h valid for both head-on and long-range collisions and verify with tracking thatin absence of other lattice perturbations h is indeed pre-served. We compare the distortion of h with the smear ob-served in Sixtrack [5] simulations and discuss the limits ofagreement of such a first-order theory with tracking.For a ring with a single (NIP = 1) head-on collisionpoint, both the nonresonant and resonant h have been ana-lyzed by Dragt [3] (see also Chao’s lectures [4]).INVARIANT FOR MULTIPLE COLLISIONPOINTSFourier Expanded PotentialIn the LHC, a particle moving in the horizontal planesees both horizontal (near IP5) and vertical (near IP1) head-on and long-range collisions. Assuming round Gaussianbeam profile at all collision points, the kick is: Δx′ = λf ,wheref(x) =2(x + dx)(x + dx)2 + d2yexp[− (x + dx)2 + d2y2σ2], (1)λ ≡ Nbr0γ , Nb is the number of particles per bunch, r0 –the classical particle radius, γ - the relativistic parameterand β, σ =√β and dx,y are the beta function, transversebeam size and the transverse separations at the location ofthe kick;  is the emittance.The beam-beam potential is λF (x) = λ∫ x0 f(x′)d x′.We first transform F (x) to action-angle coordinates (A, φ)by substituting in it x =√2Aσ sinφ and then expand it inFourier series. The result is:F (A, φ) =∫ 10dtt(1− e−t[(√A sin φ+ d˜x√2)2− d˜2y2])==∑∞n=−∞ cn(A) einφ , (2)where d˜x,y ≡ dx,y/σ. The complex coefficients satisfyc∗n = c−n and are here found numerically 1. If d˜x,y = 0(head-on collision) then cn are expressed via Bessel func-tions [4]. In what follows we will need the linear (F1 ∼ x)and quadratic (F2 ∼ x2) parts of the potential:F1 =2√2A sinφd˜2d˜x(1− exp −d˜22), (3)F2 =2A sin2 φd˜4××[−d˜2x + d˜2y + (d˜2x + d˜4x − d˜2y + d˜2xd˜2y) exp−d˜22],where d˜2 ≡ d˜2x + d˜2y .Nonresonant Invariant hUsing the Campbell-Baker-Hausdorff formula it can beshown [1] that the quantity h is given by:1μh(A, φ) = −A +NIP∑k=1λ(k)h˜(k), (4)h˜(k) = c(k)0 +m∑n=1(−1)n n2 sin nμ2[c(k)n ein( 12μ−μ(k)−φ) + c.c.],c(k)n (A) =12π∫ 2π0e−inφ F (k)(A, φ) dφ (n = 0, . . . ,m).1The integral in (2) can be expressed through the incomplete gammafunction: F (A, φ) = γ + Γ [0, P ] + log P , where P = 12(d˜2x + d˜2y) +√2A d˜x sinφ + A sin2 φ (γ = 0.577216 is the Euler’s constant).TH6PFP096 Proceedings of PAC09, Vancouver, BC, Canada3934Beam Dynamics and Electromagnetic FieldsD02 - Non-Linear Dynamics - Resonances, Tracking, Higher OrderHere h˜(k) is the contribution to the invariant of the k-th collision point located at betatron phase μ(k), whileμ = 2πQx (Qx is the ring tune). The Lie-factor F (k) isgiven by (2) with d˜x,y replaced with d˜(k)x,y – normalized sep-arations at the k-th collision point, The number of Fourierharmonics m taken is adjusted to correspond to nine sig-nificant figure accuracy. The factors λ(k) allow to modellumped collisions as in [6], [7] (not used here).Since the coefficients c(k)n (A) only weekly depend on A,they can be computed only once for, say, the initial (firstturn) value of A. This claim is verified in the next section.We see then that the contribution of each collision point tothe distortion of phase space at some amplitude is definedby an unique set of coefficients. This set depends on thenormalized separations at this collision point.VERIFICATION OF THE INVARIANTThis section presents numerical evidence that h definedwith (4) is indeed preserved for some sample (minimal) setof long-range collisions in the LHC: 12 long-range inter-actions at locations with separations 9.4, 8.5, and 5 sigmapositioned symmetrically on both sides of IP5 and IP1.Comparison with a Simple Tracking ModelThe model tracking is based on kicks f (k)(x)− f (k)(0),where f (k)(x) is as in (1), alternating with unperturbed lin-ear matrices. The procedure is to iterate an initial condition(A0, φ0) = (n2σ2 , π/2) thus getting a sequence of pointsAi, φi, where i is the turn number, and then compare thequantities Ai and h(Ai, φi). Figure 1, analogous to one in[3], shows that h is indeed more constant than the ordinaryCourant-Snyder invariant (−A). The coefficients c (k)n (A0)in (4) are computed from potentials with subtracted linearterm: F (k) − F (k)1 .3 2 1 0 1 2 3Φ504540353025h/μ− AFigure 1: Plots of the quantities h/μ (red) and (−A)(blue) for increasing starting amplitude x = nσσ; nσ =7, 8, 9, 10. Large variations of (−A) occur for larger initialbetatron amplitude.To compute smear from the invariant we use the relationh(A, φ) = h(A0, φ0) to express A as a function of φ. Fig-ure 2 illustrates that Ai, φi lay on the resultant theoreticalcurve A(φ). By covering the curve A(φ) with some denseequidistant set of points, smear is defined as the r.m.s. de-viation of the A values from this set divided by the mean.3 2 1 0 1 2 3Φ40.240.440.640.84141.241.441.6AnΣ9 xsmear  0.76 %Figure 2: Same as case nσ = 9 on Fig. 1, but a differ-ent way to show the agreement (as in [1]). Red: theoreti-cal A(φ) dependence used to define smear. Black: modeltracking.Comparison with SixtrackFor comparison with Sixtrack the action is defined vialinearly-perturbed β function and correspondingly c (k)nare computed from potentials with subtracted linear andquadratic parts: F (k) − F (k)1 − F (k)2 . See Figure 3. Theagreement is shown on Figure 4.3 2 1 0 1 2 3Φ4241.54140.54039.53938.53 2 1 0 1 2 3Φ4241.54140.54039.53938.5Figure 3: Same as case nσ = 9 on Fig. 1, but invariantcomputed from linearly perturbed β at the initial point fortracking. Left: potential used is F − F1. Right: the poten-tial is F − F1 − F2 (in this case h is more constant).2 4 6 8 10 12nΣ0.250.50.7511.251.51.75xsmear %SixtrackmodelLie algebraFigure 4: Agreement between Lie algebra, tracking modeland Sixtrack for a sample set of long-range collisions inthe LHC. Shown is smear of the horizontal emittance com-puted over 210 turns as a function of initial x = nσσ.Proceedings of PAC09, Vancouver, BC, Canada TH6PFP096Beam Dynamics and Electromagnetic FieldsD02 - Non-Linear Dynamics - Resonances, Tracking, Higher Order 3935REALISTIC LHC COLLISION SCHEMELimits of ApplicabilityFigure 5 allows to estimate the limits of our first orderLie-algebra model. Two scenarios are considered: the caseof 25 ns bunch separation, corresponding to one head-onand 30 long-range collisions near each of the two main in-teraction points IP5 and IP1, and the 50-ns case when thenumber of long-range collisions is about twice smaller. In-spection of Figure 5 shows that the model is able to predictthe smear to amplitude 12σ for intensities up to 1/2 of nom-inal for the 50 ns scenario, and to about 1/4 of the nominalfor the 25 ns case. If the bunch population is larger thanthese limits, then for amplitudes above 6 σ maps of higherorder are needed.0 2 4 6 8 10 12nΣ00.511.52xsmear%50 nsNb012 Nb015 Nb00 2 4 6 8 10 12nΣ0.511.522.533.54xsmear%25 nsNb012 Nb015 Nb0Figure 5: Agreement between Lie-algebra model (red) andSixtrack (blue) for increasing bunch population N b and twoLHC collision schemes “50 ns” and “25 ns” (head-on isincluded). The curves correspond to 1, 1/2 and 1/5 timesthe nominal bunch population N 0b = 1.15 1011.Individual Contributions of Long-Range CollisionIt is of interest to compare the individual termsh˜(k)(A, φ) in Eqn. 4. Each such term describes the con-tribution of a single collision point to distortion of the ordi-nary Courant-Snyder invariant and hence to smear. One ex-ample is shown on Figure 6 for the “25-ns” case of Figure 5(right), initial amplitude 6 σ and Nb = 1/2N 0b . Only long-range collisions near IP5 (30 in total) are included sinceboth the contribution from IP1 and the total contributionfrom head-on events are small. The top plot correspondsto nominal values of the separations while the bottom one,to the early separation scheme [7]. In the latter case, hor-izontal smear is dominated by contributions (light blue) ofcollisions in immediate vicinity of IP5, which change theird˜x value from±9.4 to ±5σ.3 2 1 0 1 2 3Φ0.020.0100.010.020.030.04AnΣ610.210.49.68.27.26.97.18.9.9.49.49.49.49.49.39.39.49.49.49.49.49.911.513.313.512.19.88.7.47.3color codedx~3 2 1 0 1 2 3Φ0.080.060.040.0200.020.04AnΣ610.210.49.68.27.26.97.18.9.9.49.48.36.64.94.94.94.96.68.39.49.49.911.513.313.512.19.88.7.47.3color codedx~Figure 6: Individual contributions to h of the 30 long-rangecollisions near IP5 (horizontal crossing) for two scenarios:nominal (top) and early separation [7] (bottom).REFERENCES[1] W. Herr, D. Kaltchev, Effect of phase advance between in-teraction points in the LHC on the beam-beam interactions,LHC Project Report 1082, Apr 1 2008[2] D. Kaltchev , On beam-beam resonances observed in LHCtracking, TRI-DN-07-9.[3] Alex J. Dragt, Oleg G. Jakubowicz, Analysis of the Beam-Beam Interaction Using Transfer maps, Proceedings of theBeam Beam Interaction Seminar, Stanford Ca, May 22-23,1980, SLAC-R-541[4] A. Chao, Lie Algebra Techniques for Nonlinear Dynamics,http:www.slac.stanford.edu/˜achao[5] F. Schmidt, SixTrack, V.4.2 Single Particle Tracking CodeTreating Transverse Motion with Synchrotron Oscillationsin a Symplectic Manner, CERN/SL/94-56, Jul 2008.[6] Y. Papaphilippou, F. Zimmermann, Weak-strong beam-beam simulations for the LHC, PRST Vol. 2, 104001 (1999).[7] Sterbini, G Koutchouk, Jean-Pierre, An Early Beam Separa-tion Scheme for the LHC Luminosity Upgrade EPAC 2006,Edinburgh and LHC-PROJECT-Report-972.PointsTH6PFP096 Proceedings of PAC09, Vancouver, BC, Canada3936Beam Dynamics and Electromagnetic FieldsD02 - Non-Linear Dynamics - Resonances, Tracking, Higher Order

Analytical calculation of the smear for long-range beam-beam interactions

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