Increased vertical e ± beam size and polarisation ?

In order to maintain the stability of the proton beam in HERA at high electron and positron currents the vertical electron and positron beam sizes must be increased. This article discusses the consequent implications for the electron and positron spin polarisation in HERA. 1 Preamble “Spin”, the quantised intrinsic angular momentum carried by many elementary particles, is a profound and mysterious attribute of elementary particles and its existence has far reaching ramifications. For example, a collection of fermions, whose intrinsic angular momenta come in multiples of 1 2 ~, has an anti–symmetric quantum state. This forces electrons in atoms to occupy different states and that in turn causes heavy atoms to be larger than light atoms. On the other hand, bosons, whose intrinsic angular momenta come in multiples of ~ have symmetric quantum states so that there is no quantum statistical restriction on their density. The examples of the significance of spin are endless and of course spin is essential for a proper understanding of the high energy interactions between electrons/positrons and protons studied at HERA. The macroscopic effects resulting from electron spin are usually the result of effects taking place at the atomic level. For example, the materials used in every day life would look very different if their various atomic constituents were all of the same size and with all electrons in the lowest energy state. But high energy electron/positron storage rings provide a counter example. As we will see, the beams in these large devices can become spin polarised, i.e. the spin vectors have a tendency to line up in the same direction. Since it concerns spin, which comes in units of ~, this is a purely quantum mechanical effect: electron/positron storage rings facilitate quantum spin effects “in the large”. Of course, the non–zero electron/positron beam sizes resulting from the quantised nature of the synchrotron radiation is another example. The potential for the natural build up of spin polarisation (Section 2) in the electron/positron (e±) beam of the HERA collider was recognised from its very beginnings in the early 1980s. Polarisation was seen as a very useful tool for making discoveries through 1 D R A FT the spin dependence of high energy e±–proton interactions [1]. It was therefore a topic in the HERA Proposal and HERA was designed and built with polarisation in mind. Then in 1994 the e± ring of HERA became the first and only high energy e± storage ring to demonstrate high energy longitudinal spin polarisation [2] when it supplied longitudinally polarised e± to the HERMES experiment with the help of the pair of spin rotators at the East interaction region. For this the detector solenoids at H1 and ZEUS (where the polarisation was vertical) were compensated locally with anti–solenoids and a reasonably good strong spin match (Section 4) could be achieved in spite of the fact that the horizontal dispersion in the East straight section did not allow a perfect “synchrotron spin match”. The depolarisation resulting from misalignments was held in check with harmonic closed orbit spin matching (Section 4). However, pairs of rotators for H1 and ZEUS were only installed in 2001 during the HERA Luminosity Upgrade. The higher luminosity derives from two modifications: 1) the beta functions at the North and South interaction points (IPs) have been reduced, 2) the horizontal emittance has been reduced by having stronger quadrupoles in the arcs and by shifting the horizontal damping constant by running with a shifted frequency in the rf cavities. But with the Upgrade, the anti–solenoids had to be removed to allow the focusing magnets to be moved closer to the IPs in order to achieve the smaller beta functions. Now the design orbit is curved in and around the solenoids and the magnetic fields in this region are very complicated owing to the overlap of solenoid and combined function fields. Thus in H1 at least, the polarisation is not everywhere parallel to the solenoid axis. This means that the rotators must be slightly retuned to return the polarisation axis to vertical in the arcs and thereby prevent the depolarisation which would result from having a tilted polarisation axis. Moreover, the North and South straight sections now have skew quadrupoles in order to compensate the horizontal–vertical coupling caused by the uncompensated solenoids. Then although the North and South interaction regions can be made spin transparent (Section 4) in the absence of solenoids and skew quadrupoles [3], the fields of these magnets break the spin transparency and strong extra depolarising resonance effects (Sections 2 and 3) appear. See [4] for details. In addition, the increased quadrupole strengths in the arcs lead to potentially larger closed orbit distortions. This in turn meant that the harmonic bump layout had to be improved. Currently 16 closed bumps are used [3]. So although HERA was originally layed out for polarisation, the Luminosity Upgrade brings clear disadvantages for the polarisation. Nevertheless, after careful orbit correction and with the H1 and ZEUS solenoids and the skew quadrupoles running, a positron polarisation of about 51% was attained simultaneously at three IPs at the routine energy of about 27.5 GeV in March 2003. Moreover, the positrons were in collision with about 20 mA of 920 GeV protons. The theoretical maximum for this configuration is 83%. But in the future, much higher proton and e± currents are foreseen. Apart from the direct depolarising effects of the magnetic fields of the high proton currents, this can degrade the e± beams owing to the large beam–beam tune shifts. In addition the higher e± currents combined with the small natural beam height of these beams [5] can cause severe disruption to the proton beam. The first step towards ensuring that the beam–beam forces are not too troublesome for each beam is to arrange that the e± and proton beams have the same sizes at the IPs. Thus the e± beam height must be increased. This could probably be achieved 2 D R A FT by re–introducing coupling with skew quadrupoles. But then it would be necessary to keep the e± beam ellipse non–tilted at the IPs and it would be bad for spin transparency, again causing enhancement of depolarising resonances. There would be other problems too which I come to later. Alternatively one could run near a transverse coupling resonance. But even if this did not have implications for the general beam dynamics (e.g. the dynamic aperture), it would introduce uncontrolled depolarisation effects. A third way to increase the e± beam height at the IPs would be either to increase the vertical beta function there or to increase the vertical emittance. Since increasing the vertical beta function would bring an intolerable increase in the vertical e± beam–beam tune shift, the vertical emittance must be increased instead. The cleanest way to do this is to increase the vertical dispersion by introducing extra vertical curvature into the closed orbit by means of extra vertical bumps. To get a perfect match of the two vertical beam sizes an emittance ratio, y/ x, of about 0.17 would be needed [4]. However, as will become clear below, poorly chosen vertical bumps can be far from helpful for attaining high polarisation. This article addresses these issues and it is structured as follows. I begin with an overview of the theory and phenomenology of radiative polarisation. I then discuss the calculation of the rate of depolarisation in the approximation that both the orbital and the spin motion are linearised. This formalism is then used to discuss the reduction of the rate of depolarisation by various forms of spin matching. Then after a short section on higher order effects I am in a position to discuss the implications of the various ways of increasing the e± beam height at the IPs. The section on spin matching is adapted from [6] since that is the most complete account available. This section contains everything needed for a discussion on spin matching in the linear approximation. 2 Spin polarisation – an overview Relativistic e± circulating in the (vertical) guide field of a storage ring emit synchrotron radiation and a tiny fraction of the photons can cause spin flip from up to down and vice versa. However, the up–to–down and down–to–up rates differ, with the result that in ideal circumstances the beam can become spin polarised anti–parallel (parallel) to the field, reaching a maximum polarisation, Pst, of 8 5 √ 3 = 92.4%. This, the Sokolov–Ternov (S–T) polarising process, is very slow on the time scale of other dynamical phenomena occurring in storage rings, and the inverse time constant for the exponential build up is [7]: τ−1 st = 5 √ 3 8 reγ 5 ~ me|ρ| (1) where re is the classical electron radius, γ is the Lorentz factor, ρ is the radius of curvature in the magnets and the other symbols have their usual meanings. The time constant is usually in the range of a few minutes to a few hours. However, even without radiative spin flip, the spins are not stationary but precess in the external fields. In particular, the centre–of–mass spin expectation value ~ S (the “spin”) of a relativistic charged particle travelling in electric and magnetic fields is governed by the Thomas–BMT equation d~ S/ds = ~ Ω × ~ S where s is the distance around the ring [8, 9]. The vector ~ Ω depends on the electric and magnetic fields, the energy and the velocity. Thus it 3 D R A FT depends on s and on the position of the particle u ≡ (x, px, y, py, σ, δ) in the 6–D phase space of the motion. The coordinate δ is the fractional deviation of the energy from the energy of a synchronous particle (“the beam energy”) and σ is the distance from the centre of the bunch. x, y are the horizontal and vertical positions of the particle relative to the reference trajectory and px = x ′, py = y ′ (except in solenoids) are their conjugate momenta. In a simplified picture the majority of the photons in the synchrotron radiation do not cause spin flip but tend instead to randomise the e± orbital motion in the (inhomogeneous) magnetic fields. Then, if the ring is insufficiently well geometrically aligned and/or if it contains special magnets like the “spin rotators” needed to produce longitudinal polarisation at a detector, the spin–orbit coupling embodied in the T–BMT equation can cause spin diffusion, i.e. depolarisation. Compared to the S–T polarising effect the depolarisation tends to rise very strongly with beam energy. The equilibrium polarisation is then less than 92.4% and will depend on the relative strengths of the polarisation and depolarisation processes. Estimates of the equilibrium polarisation attainable are based on the Derbenev–Kondratenko (D–K) formalism [10, 11]. This implicitly asserts that the value of the equilibrium polarisation in an e± storage ring is the same at all points in phase space and is given by Pdk = ∓ 8 5 √ 3 ∮