Beam-beam effects investigation and parameters optimization for a circular e+e- collider TLEP to study the Higgs boson

Several proposals exist for future circular electron-positron colliders designed for precise measurements of the Higgs boson characteristics and electroweak processes. At very high energies, synchrotron radiation of the particles in a strong electromagnetic field of the oncoming bunch (beamstrahlung) becomes extremely important, because of degradation of the beam lifetime and luminosity. We present theoretical calculations of beamstrahlung (including the beam lifetime reduction and the energy spread increase) which are benchmarked against quasi strong-strong computer simulation. Calculation results are used to optimize TLEP project (CERN).


INTRODUCTION
Design study has commenced of high luminosity e + e − collider TLEP for precise measurements of the Higgs boson properties and other experiments at the electroweak scale at CERN.TLEP will be capable to collide beams in wide center-of-mass energy range from 90 to 350 GeV (with an option up to 500 GeV) with luminosity higher than 5 • 10 34 cm −2 s −1 [1].
As mentioned in [2], a key issue that limits luminosity and beam lifetime in circular electron-positron colliders with high energy is beamstrahlung, i.e., synchrotron radiation of a lepton deflected by the collective electromagnetic field of the opposite bunch.Because of this radiation, colliding particles of TLEP at high energy could lose so much energy that they are taken out of the momentum acceptance of accelerator (beam lifetime limitation due to the single beamstrahlung).In the beginning of 2013, V. Telnov estimated lifetime considering single beamstrahlung [3], and set of TLEP parameters using V. Telnov's formula was given in [4].For TLEP at low energies, energy loss because of beamstrahlung is not large enough to kick the particles immediately out of the momentum acceptance; however multiple beamstrahlung increases beam energy spread and bunch length, reducing luminosity owing to the hour glass effect.
We present an analytical approach to calculate the beam lifetime limitation caused by the single beamstrahlung as well as the energy spread and bunch length increase due to the multiple beamstrahlung.Results of the theoretical predictions are compared with weakstrong beam-beam tracking code Lifetrac [5], in which the effect of beamstrahlung was introduced.Set of new parameters of TLEP with higher luminosity or/and better lifetime is presented for further studies.We considered head-on and crab waist [6] collisions schemes.

Beam-beam
The potential of incoming beam is written as where r e -classical electron radius, γ -Lorentz factor, N p -amount of particles, σ x,y,s -horizontal, vertical and longitudinal beam sizes, 2θ -crossing angle,x, y, s -horizontal, vertical and longitudinal coordinates, z = s−ct -particle's position with respect to the center of the bunch describing synchrotron oscillations.For simplicity, we will neglect particle's synchrotron oscillations therefore z = 0.
Equations of motion are written as: In order to calculate effective interaction length L and mean bending radius ρ x,y in hard edge approximation we will neglect σ x,y dependence on s and find expected value of vertical ∆y and horizontal ∆x kicks.After calculations we obtained where φ = σ s θ/σ x is Piwinski parameter, means expected value with respect to the first coordinate in subindex while other one satisfies condition in subindex.Inverse bending radius in corresponding plane is calculated as Finally effective interaction length in each plane is Since log 7)) we will count horizontal and vertical bending radii Beamstrahlung Following approach given in [3] amount of emitted photons and beam lifetime are given in equations ( 10) and (11).The only difference is that we do not make an assumption of 10% of the particles experiencing the maximum field, but use average values calculated in previous paragraph.
where α -fine-structure constant, N ip -number of IPs.
The difference from V. Telnov's calculations is in estimation of interaction length L which is given in equation ( 8) and in expression for the total bending radius ρ given in equation (13) (ρ x and ρ y from (9)) The radiation integrals [7] are modified according to where N ip is a number of interaction points.
Hence, we obtain expression for the beam lifetime where bold symbols are showing the difference from the expression given by V.I.Telnov [3]

THE MODEL USED IN BEAM-BEAM SIMULATIONS
To track a test particle through IP, the opposite (strong) bunch is represented by a number of thin slices.The trajectory's bending radius for each slice can be estimated as where ∆s is effective slice width, ∆p -the transverse component of beam-beam kick.Radiation spectrum corresponds to normal synchrotron radiation from a bending magnet if the following condition is satisfied ∆p Here (∆p/p) total stays for the entire bunch (not a slice!) and can be estimated as 4πξσ ∼ 10 −3÷−4 .The given condition is always satisfied at the large energies (e.g.TLEP, γ ≥ 10 5 ).
The critical energy of radiation u c (in units of mean beam energy where δ E is particle's energy deviation.Hereinafter, the energy of emitted photons is always normalized with respect to critical energy u c .The spectrum density of radiation is Note that at relatively small energies, where (19) becomes invalid, u c drops significantly and we can neglect the whole effect of beamstrahlung, therefore there is no need to be concerned about the spectrum.Taking into account the time of interaction: ∆t = ∆s/c, we obtain the (average) number of emitted photons in a small interval of spectrum: The actual number of emitted photons is given by Poisson distribution.For tracking purposes we replace the continuous spectrum by a sequence of discrete lines, from 0.01 to 20 with a step of 0.01 (all in units of u c ) -2000 in total.The lower and upper limits were chosen from the condition that the radiation power outside the borders is negligible.The step between the lines is small enough to adequately represent the spectrum.Since the critical energy u c also depends on the actual particle's trajectory, the overall spectrum of emitted photons in simulations will be continuous regardless of being discrete in units of u c .Considering randomness (and rather low probability) of photon emission in any given interval of ∆(u/u c ), we conclude that our spectrum simplifications will not affect the final results.
We have ∆(u/u c ) = 0.01 in (22) and our lines correspond to spectrum intervals of 0.005÷ 0.015 (1st), 0.015 ÷ 0.025 (2nd), etc.The integrals of K 5/3 (x) were calculated once and written in a static table for all 2000 points.The sum of all these values is responsible for the total (average) number of emitted photons The overall simulation algorithm is as follows.First, ∆p/p is calculated for each particle after passing a single slice of the opposite bunch.Second, the u c is calculated from ( 18) and ( 20), and n -from (23).Then, the actual number of emitted photons N ph (which can be zero) is obtained from the Poisson distribution with parameter n, using random number uniformly distributed in the interval of [0, 1].The energy of each particular photon is defined according to the relative probabilities (which are proportional to integrals of K 5/3 (x)) for different spectrum lines, using another random number in the interval of [0, 1].In total, the random number generator is called N ph + 1 times for each particle-slice interaction.
It is noteworthy, beamstrahlung simulations are not affected by the number of slices N slif it is large enough to correctly represent the opposite bunch.For example, further increase of N sl leads to proportional decrease of both ∆s and ∆p/p, while ρ and u c remain unchanged.
The total number of emitted photons also does not change: n for each slice decreases with ∆p/p, but it is compensated by N sl increase.
TLEP has 4 interaction points (TABLE I), therefore lattice is assumed to possess 4-fold symmetry and we chose fractional betatron phase advances between IPs (0.53,0.57).
Beamstrahlung influence makes the bunch longer, and also depends on the bunch length.
Therefore simulation was performed by quasi strong-strong method, where in the several repeated iterations the weak and strong bunches exchanged their roles and the length of the weak bunch was assigned geometric mean of strong and weak bunches.The equilibrium of the bunch length was found.Simulation was performed by weak-strong beam-beam tracking code Lifetrac [5].On the contrary to luminosity calculations agreement, the beam lifetime (FIG.3) given by Lifetrac full is consistently smaller than analytical calculations because in analytical calculations particles energy distribution was neglected, however, particle with energy deviation needs to lose different amount of energy in order to be lost.Probability to emit photon with smaller energy is higher, and probability to have a corresponding energy deviation is smaller.The interplay of these probabilities is included in computer simulation, but not in analytical calculations.Additionally, bunch length (Fig. 4) increases, changing deflecting field and so lifetime; beam energy spread (Fig. 5) becomes larger and energy acceptance of the accelerator shrinks to only 7÷10 RMS of energy distribution, thus making particle's loss more probable due to noise excitation.Also, analytical calculations do not include beam sizes dependence on longitudinal position (hour-glass).The Lifetrac threshold simulations of beam lifetime (red crosses on FIG.Performed comparison shows that accurate simulation gives smaller luminosity at TLEPZ, smaller beam life time in all scenarios.At TLEPttH the beam lifetime is so small (2 sec by Lifetrac full and by our analytics) that given scenario is not feasible.Lefitrac results if beamstrahlung is considered for emission of photons with energy higher than acceptance, green dots are our analytical calculations.

NEW SET OF PARAMETERS
Luminosity for flat beams is given by well known expression where I is a full beam current (limited by synchrotron energy loss), e -electron charge, ξ y -vertical beam-beam tune shift parameter, β y -minimum beta function at IP.The given value of β y = 1 mm is already small, further decrease is not reasonable.Hence, luminosity increase is only possible by making ξ y larger.
Analytical calculations and simulation show that beam-beam effects for TLEP are determined by several factors, quantitative relations between which greatly depend on energy.
At high energies (TLEPH and higher) beamstrahlung becomes a main factor which determines beam lifetime.The only way to decrease beamstrahlung influence (11) is to increase ρ.Since, we do not want to make ξ y smaller, the only way is to make interaction length L (8) larger (in head-on collision -by increasing the bunch length).We will assume that bending radius of beamstrahlung is proportional to energy (13) (note that beam sizes and bunch population are changing with energy also).

CONCLUSION
We have considered different aspects of the beamstrahlung influence on the parameters of the high-energy high-luminosity e+e-storage ring collider TLEP operating in the energy range from Z-pole up to the t t threshold.Consideration only of the single beamstrahlung is not sufficient to optimize the machine specifications in the entire energy range.Energy loss due to the multiple beamstrahlung increases bunch length and energy spread, modifies probability to emit photons.Particle with energy deviation might emit photon with smaller energy but higher probability to be outside of energy acceptance.Thus, the beam lifetime could be several times smaller than that predicted by the single beamstrahlung formalism.
Accurate consideration of beamstrahlung influence requires quasi strong-strong or strongstrong simulation with damping and noise excitation.Analytical approach does not consider all the effects, however gives sufficient estimation.The new set of parameters enhances performance of TLEP.
We would like to thank V.Telnov, M. Koratzinos and F. Zimmermann for fruitful discussions.The work is supported by Russian Ministry of Education and Science.

FIG. 1 .
FIG. 1. Luminosity for different scenarios of TLEP operation.Blue squares are taken from TABLE

FIG. 2 .
FIG. 2. Transverse beam distribution in normalized betatron amplitudes for TLEPZ.Left is without beamstrahlung, right is with beamstrahlung.The counter lines are equidistant.

3 )
correspond well to initial CERN results (blue squares), because simulation used assumption made by V. Telnov.Our analytical calculations(green dots on FIG. 3) are closer to Lifetrac full, especially at TLEPttH.Bunch length (FIG.4) and energy spread (FIG.5) for Lifetrac threshold (red crosses) do not change in calculations because of made assumptions.The discrepancy of bunch length and energy spread between scenarios (red crosses) corresponds to different optics.

FIG. 3 .
FIG. 3. Beam lifetime for different scenarios of TLEP operation.Blue squares are taken from

FIG. 4 .
FIG. 4. Bunch length for different scenarios of TLEP operation.Blue squares are taken from

FIG. 5 .
FIG. 5. Energy spread for different scenarios of TLEP operation.Blue squares are taken from

FIG. 7 .
FIG. 7. Beam lifetime for different scenarios of TLEP operation.Blue squares are Lifetrac calculations with full spectrum of beamstrahlung for TABLE I, red diamonds are Lifetrac results with full spectrum of beamstrahlung for the new set of parameters, green dots are analytical calculations for the new set of parameters.
FIG. 9. Energy spread for different scenarios of TLEP operation.Blue squares are Lifetrac calculations with full spectrum of beamstrahlung for TABLE I, red diamonds are Lifetrac results with full spectrum of beamstrahlung for the new set of parameters, green dots are analytical calculations for the new set of parameters.

TABLE I
[4]ed diamonds are Lifetrac results with full spectrum of beamstrahlung, red crosses are Lefitrac results if beamstrahlung is considered for emission of photons with energy higher than acceptance, green dots are our analytical calculations.COMPARISON OF OUR RESULTS WITH PREVIOUSInitially, we compared our simulation and analytical formula (16) with the calculations made in CERN.We used a table of parameters for TLEP given on 24.09.2013workshop[4], which are summarized in TABLE I. Analytical calculations and simulation by Lifetrac and given parameters from TABLE I of luminosity, beam lifetime, bunch length, energy spread are plotted on Figures 1, 3, 4, 5 respectively.In all figures CERN stands for CERN calculations from the base table (

TABLE I
we consider simulation as the most accurate calculation and compare everything against it.Luminosity calculations by different approaches are consistent except TLEPZ scenario.The difference in the calculated luminosities for TLEPZ scenario is because analytical and prob-

TABLE I
, red diamonds are Lifetrac results with full spectrum of beamstrahlung, red crosses are Lefitrac results if beamstrahlung is considered for emission of photons with energy higher than acceptance, green dots are our analytical calculations.Lifetimes for TLEPZ and TLEPW are so large, therefore not plotted.

TABLE I
, red diamonds are Lifetrac results with full spectrum of beamstrahlung, red crosses are Lefitrac results if beamstrahlung is considered for emission of photons with energy higher than acceptance, green dots are our analytical calculations.

TABLE I ,
red diamonds are Lifetrac results with full spectrum of beamstrahlung, red crosses are

TABLE II
FIG. 6. Luminosity for different scenarios of TLEP operation.Blue squares are Lifetrac calculations with full spectrum of beamstrahlung for TABLE I, red diamonds are Lifetrac results with full spectrum of beamstrahlung for the new set of parameters, green dots are analytical calculations for the new set of parameters.