Beam Blow Up due to Beamstrahlung in Circular $e^+e^-$ Colliders

After the discovery of the Higgs boson at the Large Hadron Collider in 2012, several possible future circular colliders -- Higgs factories are proposed, such as FCC-ee and CEPC. At these highest-energy $e^+e^-$ colliders, beamstrahlung, namely the synchrotron radiation emitted in the field of the opposing beam, can greatly affect the equilibrium bunch length and energy spread. If the dispersion function at the collision point is not zero, beamstrahlung will also increase the transverse emittances. In this letter, we first show that, for circular Higgs factories, a classical description of the beamstrahlung is adequate. We then derive analytical formulae describing the equilibrium beam parameters, taking into account the variation of the electromagnetic field during the collision. We illustrate the importance of beamstrahlung, including the increase of bunch length and the implied tolerance on the spurious dispersion function at the collision point, by considering a few examples.

In most electron storage rings operated so far, the equilibrium transverse emittances, energy spread and bunch length were, or are, determined by a balance of quantum excitation and radiation damping, both arising from the synchrotron radiation emitted when the charged ultrarelativistic beam particles pass through the accelerator magnets, in particular through the bending magnets [1]. Future high-energy circular colliders, like FCC-ee [2] or CEPC [3], are proposed as high-precision Higgs factories, to study the Higgs boson discovered at the Large Hadron Collider [4], or, more generally, as "electroweak factories". In these future circular colliders, for the first time, also the synchrotron radiation emitted during the collision in the electromagnetic field of the opposing beam becomes important. This particular type of synchrotron radiation is called "beamstrahlung" [5][6][7][8][9][10][11]. A beam particle is lost whenever, during the collision, it radiates a photon of an energy high enough that the emittance particle falls outside the momentum acceptance. Through this process, the high-energy tail of the can severely limit the beam lifetime [12,13]. Design parameters for FCC-ee and CEPC are taking into account this lifetime limitation along with additional constraints imposed by a coherent beam-beam instability [14].
There is yet another novel effect of beamstrahlung in circular Higgs factories. Namely, at the aforementioned colliders the beamstrahlung significantly increases the equilibrium bunch length and energy spread of the colliding beams [15][16][17]. Furthermore, with a non-zero dispersion at the IP, beamstrahlung can also affect the transverse beam emittance [17,18]. Such nonzero dispersion can either be due to incompletely corrected optics errors ("spurious dispersion") or be intentionally introduced for the purpose of reducing the centre-of-mass energy spread ("monochromatization") [19].
The strength of the synchrotron radiation is characterized by the parameter Υ, defined as [10,11] GT the Schwinger critical field, ω c the critical photon energy as defined by Sands [1], and E e the electron energy before radiation.
For the collision of 3-dimensional Gaussian bunches with rms sizes σ * x , σ * y and σ z , possibly under a small horizontal crossing angle θ c , the average Υ is [11] Υ ≈ 5 6 where α denotes the fine structure constant (α ≈ 1/137), and r e ≈ 2.8 × 10 −15 m the classical electron radius. For all proposed high-energy circular e + e − colliders, Υ 1 and σ * x σ * y . In this case we can approximate the average number of photons per collision as [11][12][13] n γ ≈ 12 where Φ piw ≡ θ c σ z /(2σ * x ) is a geometric reduction factor, also known as the "Piwinski angle". The average relative energy loss is [11] The average photon energy normalized to the beam energy, < u >, is given by the ratio of δ B and n γ : The quantum excitation, which gives rise to energy spread and emittance, is the product of the mean square photon energy and the mean emission rate [1]. In the case of beamstrahlung, the mean rate is simply given by n γ divided by the average time interval between collisions, e.g., half the revolution period in case of two interaction points. Introducing y ≡ ω/E e and the emission rate spectrum (photons emitted per second per energy interval) is described by the function [11,20] which in the classical regime (Υ → 0) reduces to [1] The number of photons radiated per unit time is obtained by integrating over ω: In the classical radiation regime and for a constant bending radius ρ, the mean square photon energy < u 2 > is related to the average photon energy < u > via [1] where < u >∝ 1/ρ. Using the function dW γ /dω of (6), we can numerically determine the exact ratio for a constant value of the critical photon energy or of the bending radius. The result, shown in Fig. 1, demonstrates that the error of the classical relation (9) is smaller than 1% for Υ values up to several times 10 −3 [18,21,22]. The classical formulae for synchrotron radiation would also be modified for an interaction length (≈ σ * x /θ c ) shorter than the classical formation length ρ/γ [23,24], with ρ the local bending radius. This "short-magnet" regime is characterized by an "undulator parameter" K max ≡ 2r e N b /(σ x θ c ) < 1, while the classical radiation spectrum applies for K max > 1. As we will see below, for all the cases of interest K max ≥ 3, so that the effect of short-magnet radiation can be neglected.
In the case of a real bunch collision, the relation between < u > and < u 2 > is further modified, however, for another reason: The local bending radius is not constant, but varies with the transverse and longitudinal position of the colliding particle, and with the time during the collision [25,26]. Indeed, while at constant bending radius ρ we have [1] u = 4/(5 √ 3)hcγ 3 /ρ, and < u 2 > well represented by (9), in general (9) must be modified as where the correction factor Z c is related to the variation of 1/ρ in time and space: where ... denotes the bunch average in space (x, y, s) and time t during a collision.
To treat the case of a nonzero crossing angle, we consider the collision in a co-moving (boosted) frame [27], where the collision is "head-on", but both bunches are tilted by an opposite angle of magnitude θ c /2. In first approximation, for the future circular colliders considered, we may ignore the disruption effects [11], and we also neglect the rms angular beam divergence compared with the crossing angle θ c . We do take into account the vertical hourglass effect by considering a vertical rms beam size which changes with longitudinal position s as where ε y denotes the vertical rms emittance and β * y the vertical beta function at the focal point. Under these assumptions, the inverse local bending radius ρ at transverse coordinates (x, y), and longitudinal coordinates s (along the beam line) and z (co-moving, along the bunch, with z = 0 referring to the centre of the bunch, and z = (s − ct); where t is time and c the speed of light) can be approximated as [28] 1 ρ(x, y, s, z) = F x − s θ c 2 , y, σ y (s) where F(x, y, σ y (s)) may be expressed in terms of the complex error function w as [29] F(x, y, σ y (s)) = Including the crossing angle and the hourglass effect, the average inverse bending radius is obtained as a quadruple integral of (14) over the four dimensions: which can be evaluated numerically. Similarly, we write Using Eqs. (16), and (17) we compute the correction factor Z c , which is illustrated in Fig. 2 as a function of the transverse beam-size aspect ratio at the collision point, for different values of crossing angle, holding the beta function β * y mm, the vertical rms beam size σ * y , and the bunch length σ z constant.  Putting everything together, the quantum excitation [1] from beamstrahlung emitted in a single collision can be written Balancing the sum of the excitation due to beamstrahlung and due to arc synchrotron radiation against the radiation damping from the arcs alone (the average energy loss and, hence, the damping effect due to beamstrahlung is negligible [17]) yields the total equilibrium emittance ε x,tot and relative rms momentum spread δ tot as where τ x and τ E denote the usual horizontal and longitudinal radiation damping times [1], respectively, T rev the revolution period, and n IP the number of interaction points. The terms with subindex SR refer to the standard equilibrium parameters without beamstrahlung. The dispersion invariant H * x is defined as [1] H where β * x , α * x , D * x and D x * denote optical beta and alpha function (Twiss parameters), the dispersion and slope of the dispersion at the collision point, respectively.
The beamstrahlung parameters (Υ, δ B , < u > and ρ) strongly depend on the bunch length. The "total" (equilibrium) bunch length is related to the total energy spread via the classical relation [1] where Q s denotes the synchrotron tune, C the circumference, and α C the momentum compaction factor. In the case of zero IP dispersion, beamstrahlung excites the beam particles only longitudinally, and the total energy spread follows from the self-consistency relation [16,17] where we have introduced the coefficient in which the correction factor (12) enters. Table I presents example parameters from the FCC-ee and CEPC designs. The strong impact of beamstrahlung is evident when comparing the rms bunch length and momentum spread due to standard arc synchrotron radiation, σ zSR and σ δSR , and the corresponding values in collision, σ zBS and σ δBS . Beamstrahlung increases the bunch length and momentum spread by a factor ranging from about 2 to 4, depending on the beam energy.  [3] and three operation modes of FCC-ee [2], illustrating the effect beamstrahlung on the rms relative momentum spread, σ δBS , and on the rms bunch length, σzBS, according to Eqs. (22) and (23), for nIP = 2 identical IPs. The analytically computed values can be compared with the result of beam-beam tracking simulations for FCC-ee, namely the values σ δSIM and σzSIM) shown underneath [2]. Parameters calculated in this letter are shown in bold. The last two rows indicate the tolerances on spurious IP dispersion for a transverse emittance growth of less than 10%, based on Eqs. (27 ) and (28), respectively. The FCC-ee simulation values (subindex "SIM") are from D. Shatilov [2]. In the presence of nonzero IP dispersion, also the transverse emittance increases due to the beamstrahlung. Considering a small spurious horizontal dispersion at the interaction point (IP), and assuming that |D * x |σ δ,tot β * x ε x , ε x,tot is no longer constant, but determined by the additional equation The spurious dispersion at the IP should not be so large as to lead to significant emittance blow up ∆ x(y) / x,(y) From Eqs. (25) and (26) we derive the corresponding tolerances for the IP dispersion, namely and |H * y | < σ 2 δ,tot β * x 3/2 ε 3/2 x,SR ε y,SR 2V ∆ε y ε y .
The resulting tolerances on the two dispersion invariants, for a maximum blow up of 10% are shown in the last two rows of Table I. In conclusion, beamstrahlung greatly affects the equilibrium beam distribution in future circular Higgs (or electroweak) factories, in particular momentum spread and bunch length, which must be taken into account when designing the next generation of lepton colliders, in addition to the constraints reported in [12,14]. The beamstrahlung effect also introduces new tolerances on the IP optics parameters.