Magnetic Field Effects on In-Medium Υ Dissociation

. The electromagnetic ﬁelds during relativistic heavy ion collisions are calcu-lated using a simple model which characterises the emerging quark-gluon medium by its conductivity only. An estimate of the average magnetic ﬁeld strength experienced by the b¯b mesons produced in the collision is made. In a su ﬃ ciently strong magnetic ﬁeld, the individual spins of b¯b mesons can align with the ﬁeld leading to quantum mixing of the singlet and triplet spin conﬁgurations. The extent of this intermixture, however, is found to be negligible at ﬁeld strengths occurring in heavy ion collisions at LHC energies.


Introduction
The suppression of Υ meson production in heavy ion collisions can be used as a probe for the properties of the quark-gluon medium expected to be formed during these events. Being part of the more massive bb meson family, the Υ is less prone to recombination than the relatively lightweight J/ψ meson which might offer the opportunity for a cleaner examination of the underlying suppression mechanisms.
We have interpreted recent LHC data [1] for the Υ(1S) state using a model with collisional damping, screening of the bb binding potential, gluon-induced dissociation, and hydrodynamic expansion of the fireball [2]. Concerning the Υ(2S) state, however, the model underestimates suppression in peripheral collisions which indicates the need for further suppression mechanisms to be included. Promising candidates are magnetic field effects which are expected to possess the required centrality dependence.
Following the calculation of the electromagnetic fields generated in heavy ion collisions, we will make a first attempt to investigate their effect on the bb mesons produced in the collision by studying the magnetically induced mixing of the spin singlet and triplet states.

Electromagnetic fields in heavy ion collisions
We start by examining a single point charge e moving with constant velocity u through a conducting quark-gluon medium. For a resting observer at position r = xê x + yê y + zê z and time t, the charge's worldline is given by These equations can easily be solved in Fourier space but a re-transform to real space is infeasible since the dispersion relations of the quark-gluon medium's permittivity ε, permeability µ, and conductivity σ are unknown to date. To circumvent this problem, we adopt a very simple model for the medium's electromagnetic properties [3]: We neglect the medium's permittivity and permeability, i. e. set ε ≡ µ ≡ 1, and characterise it by its finite conductivity only. The conductivity, in turn, is set to the constant value σ = 5.8 MeV [3,4]. Using this simple model, exploiting cylindrical symmetry, and considering only high-relativistic settings , we arrive at the approximate solution where we use the shorthand t eff = t − z/v and cylindrical coordinates ρ = x 2 + y 2 , ϕ = atan(y/x). One can think of a heavy ion nucleus as a nearly continuous distribution n Z of point charges e normalised to the number of protons Z. At LHC beam rapidities, nuclei are Lorentz-contracted to almost 2-dimensional slabs. Thus, we model n Z as being localised in z-direction while choosing a Woods-Saxon shape in the (x, y)-plane, where R and a denote the radius and diffuseness of the nucleus, respectively. Due to the linearity of Maxwell's equations, the nucleus' electromagnetic fields E Z , B Z are given by the convolution of n Z with the point charge fields E 1 , B 1 , Since n Z is radially symmetric, E Z , B Z and E 1 , B 1 share the same cylindrical symmetries and vanishing components. As can be seen in Fig. 1, the general shape of the field components as functions of ρ and t eff is not substantially altered by the convolution. The kink which can be seen in the plots of |E z 1 | and |E z Z | is the result of a change in sign occurring in the longitudinal field components when the point charge (or nucleus) passes by the observer. Due to time dilation, the observer perceives the sign  change at t eff > 0 for ρ > 0. The exact point in time (and space) is slightly shifted for the convoluted fields because of the now finite-sized charge distribution. By summing the fields of two antiparallel moving nuclei with velocities ±v beam , shifted in x-direction by the impact parameter b = bê x , we finally obtain the total electromagnetic field in a heavy ion collision,

Effective average field strength for bb mesons
The electromagnetic field representations derived so far are only valid for observers resting in the coordinate system (t, x, y, z) which can be identified with the laboratory frame in heavy ion collisions.
To examine field effects on moving observers, we need to change to their rest frames and transform the fields accordingly. If we neglect any transverse momentum, all bb mesons produced in the initial collision share the following properties: • A constant velocity u, • movement parallel to the heavy ion nuclei, u = uê z , and • a starting position in the (z = 0)-plane at t = 0, the time of the initial collision.
The restframe of this kind of observers coincides with the longitudinally co-moving frame (LCF) [5], which is reached through the coordinate transform (t, x, y, z) → (τ, x, y, ϑ) with For the observers, the coordinate τ is always equal to their proper time √ 1 − u 2 t = t/γ u and the coordinate ϑ corresponds to their rapidity artanh(u). Converting natural to SI units, 1 fm −2 corresponds to approximately 2 × 10 14 Tesla.
An electromagnetic field E lab , B lab in the laboratory frame translates into the new coordinates as From our moving observers' perspective, this is equivalent to a Lorentz boost, since In the following, we will need an estimate of the magnetic field strength experienced by bb mesons produced in heavy ion collisions. For this purpose, we average the magnetic field magnitude with respect to the bb meson distribution n bb , where we have transformed B from Eq. (12) according to (14). We assume n bb to be proportional to the number of nucleon-nucleon collisions T AA in real space and Gaussian-shaped in rapidity-space, where σ ϑ ≈ 2.7 based on a fit of recent CMS data on LHC bb production [6]. In Fig. 2, the result is shown as function of the impact parameter b and selected values of τ. The portrayed time frame 0.5 fm/c ≤ τ ≤ 3.0 fm/c is approximately equal to the lifetime of the quark-gluon medium, starting from the moment of bb meson formation. For fixed τ, the magnetic field strength reaches its maximum at b ≈ 2R when the two nuclei barely touch. For smaller impact parameters, the individual magnetic fields interfere destructively, whereas for large b the nuclei field strengths decrease due to the large distances (cf. Fig. 1). With increasing proper time, B decreases monotonically. of the Ψ 0 t and η b spin states | Ψ 0 t |η b | 2 = |sin(α)| 2 is plotted as a function of the impact parameter b and proper time τ. Being a monotonous function of B , the general shape of the overlap is similar to Fig. 2. However, even for the (b, τ) configuration yielding the highest magnetic field strength, the overlap is 10 −5 and therefore negligible. In consequence, the magnetically perturbed states Ψ s , Ψ 0 t in relativistic heavy ion collisions are expected to behave virtually identically to their unperturbed counterparts η b , Υ 0 .

Outlook
Due to exposure to strong magnetic fields, the bb mesons in relativistic heavy ion collisions might behave differently from the ones produced in pp events where almost no electromagnetic fields are present. The magnetically perturbed spin singlet state |Ψ s and m S = 0 triplet state |Ψ 0 t are each composed of a superposition of the two unperturbed spin states |η b and |Υ 0 . Hence, properties of the perturbed states related to spin, such as the decay widths between excited states and the final decay into dimuon pairs, can differ compared to their unperturbed counterparts. However, the extent of this effect is found to be negligible at LHC energies. Therefore, future investigation will focus on other possible interactions between Υ mesons and electromagnetic fields. Especially the fields' impact on bb formation and thereby the initial meson populations could prove important since the highest field strengths occur immediately after the collision where meson formation takes place. In addition, photon-gluon induced dissociation of bb pairs and dissociation due to increased potential energy may significantly affect in-medium decay widths for the excited bb states.