Amplitude dependent closest tune approach

Recent observations in the LHC point to the existence of an amplitude dependent closest tune approach. However this dynamical behavior and its underlying mechanism remain unknown. This effect is highly relevant for the LHC as an unexpected closest tune approach varying with amplitude modifies the frequency content of the beam and, hence, the Landau damping. Furthermore the single particle stability would also be affected by this effect as it would modify how particles with varying amplitudes approach and cross resonances. We present analytic derivations that lead to a mechanism generating an amplitude dependent closest tune approach.


I. INTRODUCTION
The closest tune approach, represented as ΔQ min , is the minimum distance between the fractional parts of the transverse tunes that can be achieved when trying to bring them together in presence of linear transverse coupling [1].Colliders traditionally operate very close to the diagonal (Q x ≈ Q y ) where large resonance-free regions can accommodate the tune spread generated by beam-beam interactions.Figure 1 shows an illustration for the LHC.A hypothetical ΔQ min ¼ 8 × 10 −3 would drastically reduce the available resonance-free space.Significant effort is invested in controlling the linear and the chromatic coupling in the LHC [2,3].
Compelling experimental observations in the LHC [5] suggested the existence of an amplitude dependent closest tune approach.These experimental observations have been further complemented with computer simulations of the LHC in [6] identifying the key ingredients to reproduce the observations, namely linear coupling and normal octupole fields.Furthermore, [6] also shows the possibility to penetrate the linear coupling stopband for increasing amplitudes thanks to octupoles.This implies a reduction of the closest tune approach with amplitude.The underlying mechanisms generating amplitude dependent closest tune approach are explored in this paper for the first time.
In [7] emittance exchange during resonance crossing is studied in the presence of space charge.This phenomena has great similarities with emittance exchange driven by linear coupling [8].An intensity dependent closest tune approach is proposed and evaluated through multiparticle simulations in [7].This could be interpreted as the result of the amplitude dependent closest tune approach generated by the nonlinear space charge fields acting on the ensemble of particles.
The influence of linear difference coupling resonance in the long-term particle stability has been thoroughly studied [9][10][11][12].The mechanisms are twofold, linear coupling directly modifies the excitation of lattice resonances but it also affects how resonances are approached and crossed via the transverse emittance exchange and the closest tune approach.For example, in [13,14] a skew octupolar Hamiltonian term (h 1012 ), which can be generated via linear coupling and octupoles, is identified as particularly relevant for the long-term particle stability.An amplitude dependent closest tune approach would further contribute to the previous mechanisms.
Landau damping is generated in the LHC via strong octupoles [15] and it is fundamental to suppress instabilities from collective effects.It has been observed that linear coupling can destabilize the beam in HERA [16] and possibly also in LHC [17,18].An amplitude dependent closest tune approach would modify the frequency content of the beam altering the effective Landau damping.The impact of this mechanism is being investigated in [18].
The structure of the paper is as follows.Section II introduces the linear coupling theory and the nomenclature.Section III illustrates that an amplitude detuning closest tune approach is not generated in a trivial manner.Section IV identifies a mechanism for the appearance of an amplitude dependent closest tune approach based on the interplay between linear coupling and cross amplitude detuning (h 1111 ).Other mechanisms contributing to the amplitude dependent closest tune approach are not discarded.This is illustrated in Sec.V by exploring the interplay between linear coupling and the Hamiltonian term h 2002 .The complexity of the equations avoids the explicit identification or refutation of any amplitude dependent closest tune approach.In the last Sec.VI LHC simulation results under realistic conditions with linear coupling and octupoles show qualitative agreement with the predictions from Sec. IV, which assumed the interplay between linear coupling and the h 1111 amplitude detuning term as the main mechanism for amplitude dependent closest tune approach.

II. LINEAR COUPLING THEORY
In [1] the Hamiltonian of the linear coupled motion is given as including only skew quadrupoles as perturbing Hamiltonian, H 1 ¼ Kxz.x, z, p x , p z are the canonical variables and K 1;2 represent the linear uncoupled gradients.The general solution of the unperturbed Hamiltonian is given by where u, v and ū, v are the Floquet functions and their complex conjugates, respectively.a 1 , a 2 , ā1 , ā2 are the constants of motion.Always following [1] the equations of motion in presence of the perturbing Hamiltonian are derived using the former constants of motion as new variables, where U is the perturbing Hamiltonian as function of the new variables and admits the following Fourier expansion, where and h jklm are the Hamiltonian terms.
Considering only the slowly varying U term close to the difference coupling resonance (Q H − Q V ¼ p) the equations of motion are approximated by The general solution of these coupled differential equations follows, where A AE are complex constants of motion and w AE are the frequencies given by 2jκj is therefore the minimum separation between the two frequencies, i.e. the closest tune approach: ΔQ min ¼ 2jκj.Some relevant quantities follow, The reader can check that the quantities are invariants of the motion.Furthermore, we define F as which, for Δ ¼ 0, follows a second order differential equation given by This tight connection between F and the coupling coefficient κ or, equivalently, the closest tune approach ΔQ min , is exploited in the following.

III. A FIRST EXPLORATION
Finding mechanisms that truly generate an amplitude dependent closest tune approach is not trivial.In this section we try to find a Hamiltonian that would lead to the following equations of motion which are identical to Eqs. ( 6) but with an amplitude dependent κ via the only action-like invariant of the motion C.This change is transparent to the differential equations and yields the same solutions as in Eqs. ( 7) and ( 8) but keeping the C dependency in κ, implying a pure amplitude dependent closest tune approach.Now it only remains to identify which kind of Hamiltonian terms could produce such differential equations with κðCÞ.From Eqs. (3), and using Eqs.( 14) yields, This condition imposes severe constraints in the possible κðCÞ.From Eq. ( 10) ∂C ∂a j ¼ a j ; with j ∈ f1; 2g; ð17Þ which turns Eq. ( 16) into taking the absolute value yields, and as a 1 and a 2 are independent in general, this leaves as the only possible solution which implies the exact opposite of our initial quest.Therefore there is no Hamiltonian that leads to Eqs. ( 14) with κ being a function of C, which would have automatically led to an amplitude dependent closest tune approach.Of course, this does not exclude other mechanisms for the appearance of an amplitude dependent closest tune approach, such as the one identified in the following section.

IV. AMPLITUDE DEPENDENT CLOSEST TUNE APPROACH VIA LINEAR COUPLING AND h 1111
The Hamiltonian term h 1111 generates cross amplitude detuning.The differential equations in presence of linear coupling and this Hamiltonian term follow (h We assume κ to be a real positive number without any loss of generality as its phase can be evenly split between a 2 and a 1 in Eqs.(21).In the absence of coupling a 1 a 1 corresponds to the action invariant of the motion J x .According to Eqs. (21) its derivative versus θ is expressed as from these equations it can be seen that we obtain Inspecting the real and imaginary parts of Eq. ( 26) the following expression is obtained, This equation can actually be integrated, resulting in where Q is another constant of the motion yielding to the invariant Note that these equations require jκj > 0. For jκj ¼ 0 the motion is simply an amplitude dependent betatron oscillation.Computing the second derivative of a 1 a 1 , and therefore using F ¼ C − 2a 1 a 1 and Eq. ( 28) For h 1 ¼ 0 the linear motion is retrieved.To find the closest tune approach we are interested in Δ ¼ 0, This equation can be transformed into the cnðx; kÞ Jacobi elliptic differential equation, however a perturbative approach is enough to illustrate the appearance of the amplitude dependent closest tune approach.By assuming with jBj ≪ jAj and neglecting terms of order above h 2 1 we obtain a new amplitude dependent κ and B given by Note that the assumption jBj ≪ jAj implies 16κ ≫ h 1 A.
The choice of F implies that the initial a 1 ā2 is real and This equation shows the appearance of a nonlinear closest tune approach when both κ and h 1 are different than zero.An important feature of this equation is that the closest tune approach can increase or decrease with amplitude depending on the phase space initial conditions and the sign of the Hamiltonian terms.Idealized simulations containing only κ and h 1 for LHC parameters have been used to verify the above findings and equations.However, realistic simulations of the LHC, with all nonlinear components, are used in Sec.VI to qualitatively benchmark the predictions of the theory in a realistic configuration.A more quantitative comparison would require extending the theory to more octupolar Hamiltonian terms, which is considerably harder as shown in the next section.

V. DOES h 2002 GENERATE AMPLITUDE DEPENDENT CLOSEST TUNE APPROACH?
The Hamiltonian term h 2002 is investigated in this section.The differential equations in presence of linear coupling and this Hamiltonian term follow (h After some algebra similar to the previous section we find For simplicity we assume that both h 2 and κ are real numbers.Separating S into its real and imaginary parts S ¼ R S þ iI S Eq. ( 38) is rewritten as The first and the third equations can be combined to reach the following integrable equation, resulting, after integration, in where X is a constant of the motion.This equation allows us to express F as a function of R S .Taking the second derivative of R S in Eqs.(38) and operating an equation including only R S is obtained, This equation is highly complicated and no amplitude dependent closest tune approach can be easily identified, contrary to the previous case with h 1111 .Nevertheless it cannot be discarded that h 2002 generates amplitude dependent closest tune approach.

VI. OBSERVATIONS FROM SIMULATIONS
Simulations are presented in this section supporting the existence of an amplitude dependent closest tune approach and in qualitative agreement with predictions from Sec. IV.A model of the LHC beam 2 at injection is tracked using MADX and PTC.The LHC is equipped with 2 families of Landau damping octupoles (MOF and MOD) and their nominal settings corresponding to the first part of 2012 is −3 m −4 .The vertical tune is matched to values ranging from 59.28 to 59.30 and for each of these settings kicks between 0.1 mm to 4.5 mm were performed (at β x ¼ 175 m and β y ¼ 179 m).The actions are reconstructed using the amplitude and the beta functions for each beam position monitor (BPM), as described in [5].
Using the nominal model without any skew quadrupolar components the detuning behaves linearly, as seen in Fig. 2. We observe that none of the points are on the diagonal which could indicate a small closest tune approach without a clear trend with amplitude.
Coupling is introduced using the skew quadrupoles placed in the LHC arcs to generate a closest tune approach of 0.015.Figure 3 shows the tunes for different initial tune splits demonstrating that there is a mechanism pushing the tunes away from each other already far away from the linear closest tune approach (light red area).This is accompanied with an enhanced exchange of transverse emittances compared to the expectation from linear coupling, as shown in [5].
Figure 4 shows the situation where the focusing octupoles (MOF) are powered with the opposite strength compared to the defocusing octupoles (MOD) but with same absolute value as in the previous case.This configuration causes the h 1111 Hamiltonian term to be very close to zero.In this case we observe that tunes reach the linear coupling stopband independently of the amplitude.This is consistent with the fact that the amplitude dependent closest tune approach requires both κ and h 1111 to be different than zero.This configuration with opposite MOF and MOD polarities might be interesting for the LHC operation to feature Landau damping but with fully suppressed amplitude dependent closest tune approach even in the presence of linear coupling.
The same tracking procedure is repeated for kicks in the horizontal plane and nominal octupole powering.The horizontal tunes are changed but the linear coupling and FIG. 2. Tunes from particle tracking for different initial tunes and for increasing amplitudes of the vertical kicks.The color code represents the sum of the horizontal and vertical actions.The black diagonal line indicates the resonance Q x ¼ Q y .FIG. 3. Tunes from tracking with linear closest tune approach of 0.015 and nominal octupoles.The vertical kicks ranged from 0.5 mm to 4.5 mm while the horizontal were kept at 0.5 mm.The light red area delimits the linear closest tune approach of 0.015.vertical tune are kept the same and the magnitude of the horizontal kicks were increased, see Fig. 5.It is a remarkable observation that for some of the kicks, starting close to the linear closest tune approach, the particles penetrate the stopband.This means that the tunes can approach each other closer than what is possible in linear coupling theory at larger amplitudes.
To assess the relevance of the h 2002 Hamiltonian term, which as described in Sec.V did not feature an easily identified amplitude dependence of the closest tune approach, tracking simulations of the nominal LHC optics (Q x ¼ 64.28 and Q y ¼ 59.31) were compared with results for an optical configuration with the same integer tunes in the horizontal and vertical planes (Q x ¼ 62.28 and Q y ¼ 62.31).The linear ΔQ min was held constant for the two configurations with jC − j ¼ 0.01, and octupole settings in both cases were set to −3 m −4 .In this way the average value of h 2002 was suppressed by a factor ∼2 while retaining the amplitude detuning coefficients (and hence h 1111 ) close to their nominal values.Table I shows a comparison of h 1111 and h 2002 for the two optics configurations.Results from tracking simulations of the two configurations are shown in Fig. 6, where as in the previous figures the applied vertical kick has been increased incrementally.It is seen that in spite of a substantial reduction of hjh 2002 ji relative to h 1111 the closest tune approach has not been substantially affected, implying that the h 2002 plays a minimal role in the observed behavior.
Similar tracking simulations, performed with a variety of different nonlinear elements, were used to exclude alternative sources of amplitude dependent closest tune approach.In particular skew octupoles on their own did not generate any similar behavior.An optics with nominal FIG. 4. Tunes from tracking with linear closest tune approach of 0.015 and h 1111 ¼ 0 by powering MOF with opposite polarity than MOD.The vertical kicks range from 0.5 mm to 4.5 mm while the horizontal were kept at 0.5 mm.The light red area delimits the linear coupling stopband of 0.015.FIG. 5. Tunes from tracking with linear closest tune approach of 0.015 and nominal octupoles.The kicks in the horizontal plane ranged from 0.5 mm to 4.5 mm while the kicks in the vertical plane were kept at 0.5 mm.The light red area indicates the linear stopband of 0.015 for all cases.The two black lines show the resonances Q x ¼ Q y and Q y ¼ 1=3 respectively.octupoles, zero linear coupling, and unsplit tunes also failed to produce an amplitude dependent closest tune approach.

VII. CONCLUSION
Motivated by LHC experimental observations and simulations we have found analytically a mechanism to generate an amplitude dependent closest tune approach based on the interplay between linear coupling and the cross term amplitude detuning h 1111 given by with jκj > 0. This equation predicts that the amplitude dependence of the closest tune approach can be positive or negative depending on the phase space coordinates a 1;2 , the linear coupling κ and the octupolar term h 1111 .Simulations have confirmed that both linear coupling and h 1111 are fundamental to generate a sizable amplitude dependent closest tune approach, and that this can be both positive and negative depending on the excitation plane.The linear stopband has been penetrated in simulations with octupoles at large oscillation amplitudes.A reduction of the Hamiltonian term h 2002 by about a factor 2 while keeping constant all detuning terms has not shown any significant change in the amplitude dependent closest tune approach.A particularly interesting case for the LHC consists in canceling the h 1111 term while keeping a large Landau damping by using opposite strength in one of the LHC octupole families.In this configuration, as expected, there is no amplitude dependent closest tune approach.
The new concept of amplitude dependent closest tune approach can be described by the mechanism outlined in this paper.A complete description of amplitude dependent behavior in the presence of linear coupling will require an extension of the theory to include additional octupolar Hamiltonian terms.

FIG. 1 .
FIG.1.LHC beam-beam tune footprint up to 6 beam sigma (in blue).Resonance lines up to 10th order are shown[4].A hypothetical ΔQ min ¼ 8 × 10 −3 stopband is drawn in light red to illustrate the reduction of the available resonance-free area.

TABLE I .
Detuning coefficients, jh 1111 j and hjh 2002 ji for the optics configurations studied, where hi represents the average value around the ring.Detuning terms and Hamiltonian coefficients were determined using PTC.
6IG.6.Comparison of tracking simulations with incremental vertical kicks, for optics configurations with Q x ¼ 64.28, Q y ¼ 59.31 and Q x ¼ 62.28, Q y ¼ 62.31.In both cases jC − j ¼ 0.01 and Landau octupoles are applied with a strength of −3 m −4 .