#### Abstract

The dynamic alignment method of the forward proton detectors proposed by the CDF collaboration is reviewed. The applicability of the method at the LHC is discussed.

#### 1. Introduction

A typical collision of protons at the LHC consists of parton-parton interaction. This causes the colour charge flow between the protons and leads to the creation of colour dipoles. In consequence, the dipoles radiate, filling the detector with particles. However, in a fraction of events the protons interact coherently, either electromagnetically—by exchanging a photon, or strongly—via an exchange of a colour singlet object named Pomeron. From such a collision one or both protons may emerge intact and may lose a part of its/their energy and be scattered at very small angles into the accelerator beam pipe. Exchange of a colour singlet object may result in a suppression of particle radiation in some regions of the phase space—the so-called rapidity gap.

When both protons stay intact, and the two emitted photons or Pomerons interact with each other producing a state in the central part of the rapidity space, the event is called the central exclusive production (CEP) [1]. Various objects can be produced in such a process: particles (, Higgs boson, ) or systems of particles (these may be: a pair of leptons, photons, jets, SUSY particles, etc.). At hadron colliders, such processes provide a unique opportunity to measure all particles of the final state. This leads to kinematic constraints, and results in a good resolution of the centrally produced system mass reconstruction in a wide range of masses [2]. The tagging of protons emerging from such an interaction can be done with detectors that are inserted into the beam pipe far away from the interaction point (typically several dozens or even hundreds of metres).

At the LHC, such detectors are installed around the ATLAS interaction point (IP1)—ALFA (absolute luminosity for ATLAS) and the CMS interaction point (IP5)—TOTEM (total cross-section, elastic scattering, and diffraction dissociation). However, both detector systems are designed to work during dedicated LHC runs with a special machine tune (the so-called high optics), when the luminosity is a few orders of magnitude smaller than the nominal cm^{−2} s^{−1}. Therefore, ATLAS and CMS collaborations plan to install additional detectors that will be able to work in the standard LHC tune environment: the AFP (ATLAS Forward-Proton) detectors and the HPS (high precision spectrometer).

Forward-proton detectors can measure the position and direction of the scattered proton trajectory. From that measurement, the fractional momentum loss ( is the initial proton energy, is the scattered proton energy) and the four-momentum transfer can be reconstructed (actually, denotes hereafter its absolute value). A crucial element of the reconstruction resolution is a proper alignment of the detectors, that is, precise knowledge of their positions. One should note that the positions of the detectors (their distance from the beam centre) have to be adjusted according to the actual beam conditions—the detectors must be movable. For instance, at early stages of the run, when the beam can be unstable, the detectors are situated in their home positions. Later, they are inserted into the beam pipe and placed in the immediate vicinity of the beam. As the positions of the detectors need to be established on the run-by-run basis, one needs a data-driven method of aligning forward-proton detectors.

#### 2. Dynamic Alignment Method

The dynamic alignment method [3, 4] has been used by CDF for their RPS (Roman pot spectrometer) detector. For a given time period, a sample of single diffractive events was collected. Then, the four-momentum transfer, (henceforth primed variables denote the reconstructed values and unprimed variables denote the true values of the observables), was reconstructed assuming different detectors positions. In particular, one is interested in the value of the distribution at Naturally, the value of the reconstructed four-momentum transfer is affected by the assumed positions of the detectors. Thus, is a function of the detector misalignment. The dynamic alignment method assumes that reaches the maximum for perfectly aligned detectors. Below, a simplified justification of this method is given.

As mentioned before, the detectors measure the trajectories of scattered particles, that is, the trajectory position and its elevation angles (slopes) . In the simplest case there are two detector stations per beam placed at the distances of and from the IP. Since both measure the proton trajectory positions at and at , and are spaced by the distance then the trajectory parameters at the mid point, , are:

The misalignment of such detectors has four degrees of freedom for both stations and both directions: , , , and . In fact, one should include also a possible misalignment of the detector position and skewness of the coordinate system, but in a real experimental environment they are negligible. It will prove to be helpful to use the global and the relative misalignments, instead of .

A large fraction of protons tagged in forward detectors originates from single diffractive processes (i.e., single diffractive dissociation). The four-momentum transfer distribution of these events is exponential as follows where is the single diffraction cross-section and is the nuclear slope. The scattered proton momentum (at the IP) is unfolded from the measured values of . In particular: Next, these values are used to calculate the four-momentum transfer. However, if the detectors are misaligned, the unfolded momentum is affected as follows: where

As and , one can obtain the reconstructed four-momentum transfer distribution (assuming that and are proton momentum independent): where , , and are such that Expanding into a power series, one obtains This shows that reaches its maximal value for the perfect alignment.

At this point some important remarks have to be made. Firstly, in order to reconstruct the distribution at , such events must fall into the acceptance of the detector. As will be discussed later, it is not always the case. Secondly, the derivation presented above assumes that and are constant (not dependent on the proton momentum), that is, higher derivatives of and are zero. This assumption is needed only to obtain the analytic formula for , but is not crucial for the method. However, there are restrictions on the possible variation of and —on the average, they must be substantially different from zero; otherwise, the net effect cancels. Thirdly, the fact that reaches the maximum for the perfect alignment does not mean that by the maximisation of one aligns the detectors. This is because maximisation of is equivalent to the request of and . There is no unique solution of these two equations, as there are four unknown misalignment degrees of freedom. However, when the matrix of partial derivatives of and : has two rows that are negligible and the remaining two rows form a matrix with a non-vanishing determinant, the method provides a direct alignment for variables corresponding to non-negligible rows. Otherwise, one needs another method that will provide the two missing constraints.

Finally, one must remember that there are additional experimental effects that influence the measurement. Such factors (e.g., spatial resolution of the detectors and beam angular spread) cause random smearing of the reconstructed four-momentum transfer, leading to statistical errors of the obtained distribution. Thus, for a given number of collected single diffractive events, there is a limit on the alignment precision.

#### 3. Alignment at the LHC

In this section, the dynamic alignment method applicability at the LHC is presented; the forward-proton detectors in the ATLAS experiment are considered—the ALFA [5] and the AFP detectors [6].

The ALFA detector consists of two Roman pot stations placed on each side of the IP at the distances of 237.4 and 241.5 metres. Each pot allows vertical insertion of position sensitive and triggering detectors into the beam pipe. The detectors use scintillating fibres to measure the scattered proton position.

The main purpose of ALFA is to measure the elastic proton-proton scattering in the Coulomb amplitude dominance region. This is needed for precise calculations of the scattering amplitude and hence a precise determination of the LHC luminosity. For that purpose a special high machine optics will be used. In consequence, single diffractive events with will not be within the detectors acceptance for all possible proton momentum losses [7]. Therefore, the dynamic alignment method cannot be used for the ALFA detectors.

The AFP detectors are currently at the well-advanced R&D stage and are planned to be installed during the 2013/14 LHC shutdown [6]. Similarly as in the ALFA experiment, two detector stations per outgoing beam are planned—they will be positioned symmetrically at 206 and 214 metres away from the ATLAS IP. However, instead of the Roman Pots, the Hamburg movable beam pipe mechanism [8, 9] will be used to position the detectors in vicinity of the proton beam. Each station will consist of six silicon-pixel detector planes for the proton position measurement. This will provide the spatial resolutions of 10 m and 30 m in and directions, respectively. In addition, the stations at 214 metres station will be equipped with a fast-timing detector (with a resolution of several picoseconds) necessary for the pile-up background reduction.

As has been discussed in Section 2, the derivatives of the and unfolding functions ( and ) with respect to , , , and are crucial for the method. Therefore, they were calculated for single diffractive events generated with Pythia 6.4 [10], transported with FPTrack [11] to the AFP positions, and then unfolded as in [2]. The ranges of these derivatives are given in Table 1.

In the proton transport, the Coulomb multiple scattering and the spatial resolution effects were taken into account. The multiple scattering in the first station material resulted the average angular deflection of the trajectory of about 0.5 rad which lead to the trajectory displacement at the second station smaller than the apparatus spatial resolution. The multiple scattering due to the dead material in the second station was neglected since its effects were negligible for the measurement and unfolding accuracies.

It can be seen that in the first column the most significant element is the , and in the second column the most important is the . Therefore, for both columns the and rows can be neglected. The remaining and rows form a matrix with a nonzero determinant, which depends only on the most significant elements in each column. This shows that the dynamic alignment method will work for the AFP detectors. However, contrary to the situation of the CDF detectors, the method is sensitive to the relative alignment and not to the global one.

This is illustrated in Figures 1 and 2, where the reconstructed four-momentum transfer distributions are presented for different values of and : *μ*m, *μ*m, and *μ*m. Indeed, the simulation confirms the conclusions drawn from the analysis of and derivatives. The four-momentum-transfer distribution at decreases when a relative misalignment is introduced. Such a behaviour is not observed in case of the global misalignment. This results from the fact that and columns in Table 1 are practically negligible. Also, as can be seen in Figures 1 and 2, the sensitivity to the relative misalignment in the vertical direction is smaller than that for the horizontal one.

A comment on the practical significance of the above discussed method is in place here. As it was mentioned the spatial resolution of the detector is 10 m × 30 m. Obviously these numbers define the required alignment precision needed to fully exploit the physics potential of the AFP detectors. It has been estimated that a relatively small integrated luminosity of 30 is sufficient to obtain the required precision of the relative alignment of the detectors using the presented method. This corresponds to the data collection time ranging from few seconds to few minutes depending on the instantaneous luminosity of the LHC.

#### 4. Conclusions

The dynamic alignment method for the forward-proton-tagging detectors proposed by the CDF Collaboration was reviewed. A simplified mathematical justification of this method was given and its applicability to the detectors in the ATLAS Experiment was discussed. It was shown that the method works differently than at the Tevatron, where it allowed the *global* alignment determination. At the LHC, the method can be successfully used for the *relative* alignment of the AFP detectors, whereas it will not work for the ALFA stations.