LHC Optics Measurement with Proton Tracks Detected by the Roman Pots of the TOTEM Experiment

Precise knowledge of the beam optics at the LHC is crucial to fulfil the physics goals of the TOTEM experiment, where the kinematics of the scattered protons is reconstructed with the near-beam telescopes -- so-called Roman Pots (RP). Before being detected, the protons' trajectories are influenced by the magnetic fields of the accelerator lattice. Thus precise understanding of the proton transport is of key importance for the experiment. A novel method of optics evaluation is proposed which exploits kinematical distributions of elastically scattered protons observed in the RPs. Theoretical predictions, as well as Monte Carlo studies, show that the residual uncertainty of this optics estimation method is smaller than 0.25 percent.


Introduction
The TOTEM experiment [1] at the LHC is equipped with near beam movable insertions -called Roman Pots (RP) -which host silicon detectors to detect protons scattered at the LHC Interaction Point 5 (IP5) [2]. This paper reports the results based on data acquired with a total of 12 RPs installed symmetrically with respect to IP5. Two units of 3 RPs are inserted downstream of each outgoing LHC beam: the "near" and the "far" unit located at s = ±214.63 m and s = ±220.00 m, respectively, where s denotes the distance from IP5. The arrangement of the RP devices along the two beams is schematically illustrated in figure 1.
Each unit consists of 2 vertical, so-called "top" and "bottom", and 1 horizontal RP. The two diagonals top left of IP-bottom right of IP and bottom left of IP-top right of IP, tagging elastic candidates, are used as almost independent experiments. The details of the set-up are discussed in [3]. Each RP is equipped with a telescope of 10 silicon microstrip sensors of 66 µm pitch which provides spatial track reconstruction resolution σ(x, y) of 11 µm [4]. Given the longitudinal distance between the units of ∆s = 5.372 m the proton angles are measured by the RPs with an uncertainty of 2.9 µrad.
During the measurement the detectors in the vertical and horizontal RPs overlap, which enables a precise relative alignment of all the three RPs by correlating their positions via common particle tracks. The alignment uncertainty better than 10 µm is attained, the details are discussed in [4,5].
The proton trajectories, thus their positions observed by RPs, are affected by magnetic fields of the accelerator lattice. The accelerator settings define the machine optics which can be characterized with the value of β * at IP5. It determines the physics reach of the experiment [3]: runs with high β * = 90 -2500 m are characterized by low beam divergence allowing for precise scattering angle measurements while runs of low β * = 0.5 -11 m, due to small interaction vertex size, provide higher luminosity and thus are more suitable to study rare processes. In the following sections we will analyze two representatives of these LHC runs, corresponding to machine optics with β * = 3.5 m and 90 m, respectively [2,6].
In order to reconstruct the kinematics of proton-proton scattering precisely, an accurate model of proton transport is indispensable. TOTEM has developed a novel method to evaluate the optics of the machine by using angle-position distributions of elastically scattered protons observed in the RP detectors. The method, discussed in detail in the following sections, has been successfully applied to data samples recorded in 2010 and 2012 [8][9][10][11][12].
Section 2 introduces the so-called transport matrix, which describes the proton transport through the LHC lattice, while machine imperfections are discussed in section 3. The proposed novel method for optics evaluation is based on the correlations between the transport matrix elements. These correlations allow the estimation of those optical functions which are strongly correlated to measurable combinations and estimators of certain elements of this transport matrix. Therefore, it is fundamental to study these correlations in detail, which is the subject of section 4. The corresponding eigenvector decomposition of the transport matrix is used to gain insight into the magnitude of the reduction of uncertainties in the determination of LHC optics that can be obtained from using TOTEM data and provides the theoretical baseline of the method.
Section 5 brings the theory to practice, by specifying the estimators, obtained from elastic track distributions measured in RPs. Finally, the algorithm that we applied to estimate the LHC optics from TOTEM data is described and discussed in section 6. The uncertainty of this novel method of LHC optics determination was estimated with Monte Carlo simulations, that are described in detail in section 7. The trajectory of protons produced with transverse positions ‡ (x * , y * ) and angles (Θ * x , Θ * y ) at IP5 is described approximately by a linear formula

Proton transport model
where d = (x, Θ x , y, Θ y , ∆p/p) T , p and ∆p denote the nominal beam momentum and the proton longitudinal momentum loss, respectively. The single pass transport matrix is defined by the optical functions [13]. The horizontal and vertical magnifications v x,y = β x,y /β * cos ∆µ x,y and the effective lengths L x,y = β x,y β * sin ∆µ x,y (4) ‡ The ' * ' superscript indicates that the value is taken at the LHC Interaction Point 5.
are functions of the betatron amplitudes β x,y and the relative phase advance and are of particular importance for proton kinematics reconstruction. The D x and D y elements are the horizontal and vertical dispersion, respectively. Elastically scattered protons are relatively easy to distinguish due to their scattering angle correlations. In addition, these correlations are sensitive to the machine optics. Therefore, elastic proton-proton scattering measurements are ideally suited to investigate the optics the LHC accelerator.
In case of the LHC nominal optics the coupling coefficients are, by design, equal to zero m 13 , ..., m 42 = 0 .
Moreover, for elastically scattered protons the contribution of the vertex position (x * , y * ) in (1) is canceled due to the anti-symmetry of the elastic scattering angles of the two diagonals. Also, those terms of (1) which are proportional to the horizontal or vertical dispersions D x,y vanish, since ∆p = 0 for elastic scattering. Furthermore, the horizontal phase advance ∆µ x = π at 219.59 m, shown in figure 2, and consequently the horizontal effective length L x vanishes close to the far RP unit, as it is shown in figure 3. Therefore, dL x /ds is used for the reconstruction of the kinematics of proton-proton scattering.
In summary, the kinematics of elastically scattered protons at IP5 can be reconstructed on the basis of RP proton tracks using (1): The vertical effective length L y and the horizontal magnification v x are applied in (7) due to their sizeable values, as shown in figures 4 and 5. As the values of the reconstructed angles are inversely proportional to the optical functions, the errors of the optical functions dominate the systematic errors of the final, physics results of TOTEM RP measurements.
The proton transport matrix T (s; M), calculated with MAD-X [14], is defined by the machine settings M, which are obtained on the basis of several data sources: the magnet currents are first retrieved from TIMBER [15] and then converted to magnet strengths with LSA [16], implementing the conversion curves measured by FIDEL [17]. The WISE database [18] contains the measured imperfections (field harmonics, magnet displacements and rotations) included in M.

Machine imperfections
The real LHC machine [2] is subject to additional imperfections ∆M, not measured well enough so far, which alter the transport matrix by ∆T : The most important transport matrix imperfections are due to: -the magnet current-strength conversion error: σ(k)/k ≈ 10 −3 , -the beam momentum offset: σ(p)/p ≈ 10 −3 .
Their impact on the important optical functions L y and dL x /ds is presented in table 1.
It is clearly visible that the imperfections of the inner triplet (the so called MQXA and MQXB magnets) are of high influence on the transport matrix while the optics is less sensitive to the strength of the quadrupoles MQY and MQML.
Generally, as indicated in table 1, for high-β * optics the magnitude of ∆T is sufficiently small from the viewpoint of data analysis.
However, the sensitivity of the low-β * optics to the machine imperfections is significant and cannot be neglected. The proton reconstruction is based on (7). Thus it is necessary to know the effective lengths L x,y and their derivatives with an uncertainty better than 1-2 % in order to measure the total cross-section σ tot with the aimed uncertainty of [19]. The currently available ∆β/β beating measurement with an error of 5−10 % does not allow to estimate ∆T with the uncertainty, required by the TOTEM physics program [20]. However, as it is shown in the following sections, ∆T can be determined well enough from the proton tracks in the Roman Pots, by exploiting the properties of the optics and those of the elastic pp scattering, so that the aimed 1% relative uncertainty in the determination of the total pp cross-section becomes within the reach of TOTEM.

Correlations in the transport matrix
The transport matrix T defining the proton transport from IP5 to the RPs is a product of matrices that describe the magnetic field of the lattice elements along the proton trajectory. The imperfections of the individual magnets alter the cumulative transport function. It turns out that independently of the origin of the imperfection (strength of   any of the magnets, beam momentum offset) the transport matrix is altered in a similar way, as can be described quantitatively with eigenvector decomposition, discussed in section 4.1.

Correlation matrix of imperfections
Assuming that the imperfections discussed in section 2 are independent, the covariance matrix describing the relations among the errors of the optical functions can be calculated: where T r is the relevant 8-dimensional subset of the transport matrix which is presented as a vector for simplicity. The optical functions contained in T r differ by orders of magnitude and, are expressed in different physical units. Therefore, a normalization of V is necessary and the use of the correlation matrix C, defined as is preferred. An identical behaviour of uncertainties for both beams was observed and therefore it is enough to study the Beam 1. In case of the β * = 3.5 m optics the following error correlation matrix is obtained: The non-diagonal elements of C, which are close to ±1, indicate strong correlations between the elements of ∆T r . Consequently, the machine imperfections alter correlated groups of optical functions.
Since the two largest eigenvalues λ 1 = 4.9 and λ 2 = 2.3 dominate the others, the correlation system is practically two dimensional with the following two eigenvectors Therefore, contributions of the individual lattice imperfections cannot be evaluated. On the other hand, as the imperfections alter approximately only a two-dimensional subspace, a measurement of a small set of weakly correlated optical functions would theoretically yield an approximate knowledge of ∆T r .

Error estimation of the method
Let us assume for the moment that we can precisely reconstruct the contributions to ∆T r of the two most significant eigenvectors while neglecting that of the others. The error of such reconstructed transport matrix can be estimated by evaluating the contribution of the remaining eigenvectors: where and N = (ν 1 , ..., ν 8 ) is the basis change matrix composed of eigenvectors ν i corresponding to the eigenvalues λ i . The relative optics uncertainty before and after the estimation of the most significant eigenvectors is summarized in  Table 2. Nominal values of the optical functions T r,i and their relative uncertainty before ( V i,i / |T r,i |) and after (δ∆T r,i / |T r,i |) the determination of the two most significant eigenvectors (β * = 3.5 m, Beam 1).
limit ourselves only to the first two most significant eigenvalues, the uncertainty of optical functions due to machine imperfections drops significantly. In particular, in case of dL x /ds and L y a significant error reduction down to a per mil level is observed. Unfortunately, due to ∆µ x = π (figure 2), the uncertainty of L x , although importantly improved, remains very large and the use of dL x /ds for proton kinematics reconstruction should be preferred. In the following sections a practical numerical method of inferring the optics from the RP proton tracks is presented and its validation with Monte Carlo calculations is reported.

Optics estimators from proton tracks measured by Roman Pots (β * =3.5 m optics)
The TOTEM experiment can select the elastically scattered protons with high purity and efficiency [8,9]. The RP detector system, due to its high resolution (σ(x, y) ≈ 11 µm, σ(Θ x,y ) ≈ 2.9 µrad), can measure very precisely the proton angles, positions and the angle-position relations on an event-by-event basis. These quantities can be used to define a set of estimators characterizing the correlations between the elements of the transport matrix T or between the transport matrices of the two LHC beams. Such a set of estimatorsR 1 , ...,R 10 (defined in the next sections) is exploited to reconstruct, for both LHC beams, the imperfect transport matrix T (M) + ∆T defined in (8).

Correlations between the beams
Since the momentum of the two LHC beams is identical, the elastically scattered protons will be deflected symmetrically from their nominal trajectories of Beam 1 and Beam 2: which allows to compute ratios R 1,2 relating the effective lengths at the RP locations of the two beams. From (1) and (18) we obtain: where the subscripts b 1 and b 2 indicate Beam 1 and 2, respectively. Approximations present in (19) and (20)  The width of the distributions is determined by the beam divergence and the vertex contribution, which leads to 0.5% uncertainty on the eigenvector's slope parameter.

Single beam correlations
The distributions of proton angles and positions measured by the Roman Pots define the ratios of certain elements of the transport matrix T , defined by (1) and (2). First of all, dL y /ds and L y are related by The corresponding estimatorsR 3 andR 4 can be calculated with an uncertainty of 0.5% from the distributions as presented in figure 8. Similarly, we exploit the horizontal dependencies to quantify the relations between dL x /ds and L x . As L x is close to 0, see figure 3, instead of defining the ratio we rather estimate the position s 0 along the beam line (with the uncertainty of about 1 m), for which L x = 0. This is accomplished by resolving for s 0 , where s 1 denotes the coordinate of the Roman Pot station along the beam with respect to IP5. Obviously, dL x (s)/ds is constant along the RP station as no magnetic fields are present at the RP location. The ratios L x (s 1 )/ dLx(s 1 ) ds for Beam 1 and 2, similarly to the vertical constraints R 3 and R 4 , are defined by the proton tracks: which is illustrated in figure 9. In this way two further constraints and the corresponding estimators (for Beam 1 and 2) are obtained:

Coupling / rotation
In reality the coupling coefficients m 13 , ..., m 42 cannot be always neglected, as it is assumed by (6). RP proton tracks can help to determine the coupling components of the transport matrix T as well, where it is especially important that L x is close to zero at the RP locations. Always based on (1) and (2), four additional constraints (for each of the two LHC beams and for each unit of the RP station) can be defined: The subscripts "near" and "far" indicate the position of the RP along the beam with respect to the IP. Geometrically R 7,...,10 describe the rotation of the RP scoring plane about the beam axis. Analogously to the previous sections, the estimatorsR 7,...,10 are obtained from track distributions as presented in figure 10 and an uncertainty of 3% is achieved.

Optical functions estimation
The machine imperfections ∆M, leading to the transport matrix change ∆T , are in practice determined with the χ 2 minimization procedure: defined on the basis of the estimatorsR 1 ...R 10 , where the arg min function gives the phase space position where the χ 2 is minimized. As it was discussed in section 4.1, although the overall alteration of the transport matrix ∆T can be determined precisely based on a few optical functions' measurements, the contributions of individual imperfections cannot be established. In terms of optimization, such a problem has no unique solution and additional constraints, defined by the machine tolerance, have to be added. Therefore, the χ 2 function is composed of the part defined by the Roman Pot tracks' distributions and the one reflecting the LHC tolerances: The design part where k i and φ i are the nominal strength and rotation of the ith magnet, respectively. Thus (28) defines the nominal machine (k i , φ i , p i ) as an attractor in the phase space. Both LHC beams are treated simultaneously. Only the relevant subset of machine imperfections ∆M was selected. The obtained 26-dimensional optimization phase space includes the magnet strengths (12 variables), rotations (12 variables) and beam momentum offsets (2 variables). Magnet rotations are included into the phase space, otherwise only the coupling coefficients m 13 , ..., m 42 could induce rotations in the (x, y) plane (25), which could bias the result. The measured part contains the track-based estimatorsR 1 ...R 10 (discussed in detail in section 5) together with their uncertainty. The subscript "MAD-X" defines the corresponding values evaluated with the MAD-X software during the χ 2 minimization. Table 3 presents the results of the optimization procedure for the β * = 3.5 m optics used by LHC in October 2010 at beam energy E = 3.5 TeV.
The obtained value of the effective length L y of Beam 1 is close to the nominal one, while Beam 2 shows a significant change. The same pattern applies to the values of dL x /ds. The error estimation of the procedure is discussed in section 7.  Table 3. Selected optical functions of both LHC beams for the β * = 3.5 m optics, obtained with the estimation procedure, compared to their nominal values.

Monte Carlo validation
In order to demonstrate that the proposedR i optics estimators are effective the method was validated with Monte Carlo simulations.
In each Monte Carlo simulation the nominal machine settings M were altered with simulated machine imperfections ∆M within their tolerances using Gaussian distributions. The simulated elastic proton tracks were used afterwards to calculate the estimatorsR 1 ...R 10 . The study included the impact of -magnet strengths, -beam momenta, -magnet displacements, rotations and harmonics, -settings of kickers, -measured proton angular distribution.
The error distributions of the optical functions ∆T obtained for β * = 3.5 m and E = 3.5 TeV are presented in figure 11 and  Table 4. The Monte-Carlo study of the impact of the LHC imperfections ∆M on selected transport matrix elements dL x /ds and L y for β * = 3.5 m at E = 3.5 TeV. The LHC parameters were altered within their tolerances. The relative errors of dL x /ds and L y (mean value and RMS) characterize the optics uncertainty before and after optics estimation.  1.8 · 10 −2 1.5 −7 · 10 −2 0.21 Table 5. The Monte-Carlo study of the impact of the LHC imperfections ∆M on selected transport matrix elements dL x /ds and L y for β * = 90 m at E = 4 TeV. The LHC parameters were altered within their tolerances. The relative errors of dL x /ds and L y (mean value and RMS) characterize the optics uncertainty before and after optics estimation. First of all, the impact of the machine imperfections ∆M on the transport matrix ∆T , as shown by the MC study, is identical to the theoretical prediction presented in table 2. The bias of the simulated optics distributions is due to magnetic field harmonics as reported by the LHC imperfections database [18]. The final value of mean after optics estimation procedure contributes to the total uncertainty of the method.
The errors of the reconstructed optical functions are significantly smaller than evaluated theoretically in section 4.2. This results from the larger number of design and measured constraints (27), employed in the numerical estimation procedure of section 6. In particular, the collinearity of elastically scattered protons was exploited in addition.
Finally, the achieved uncertainties of dL x /ds and L y are both lower than 2.5 for both beams.

Conclusions
TOTEM has proposed a novel approach to estimate the optics at LHC. The method, based on the correlations of the transport matrix, consists of the determination of the optical functions, which are strongly correlated to measurable combinations of the transport matrix elements.
At low-β * LHC optics, where machine imperfections are more significant, the method allowed us to determine the real optics with a per mil level uncertainty, and also permitted to assess the errors of the transport matrix errors from the tolerances of various machine parameters. In case of high-β * LHC optics, where the machine imperfections have smaller effect on the optical functions, the method remains effective and reduces the uncertainties to the desired per mil level. The method has been validated with the Monte Carlo studies both for high-and low-β * optics and was successfully used in the TOTEM experiment to calibrate the optics of the LHC accelerator directly from data in physics runs for precision TOTEM measurements of the total pp cross-section.