1 Introduction

Colliders have developed immensely in the last decades; they have made a forefront contribution in exploring and discovering new physics [1]. The collider performance is defined by its luminosity \(\mathscr {L}\) and the available energy in the centre of mass of colliding beams \(E^{CoM}\), where the luminosity characterizes the intensity of particles collisions at the interaction point (IP) and the centre of mass energy shows the ability of the collider to produce heavier particles or probe smaller scales [2]. These two parameters are tuned in experiments to discover rare and new events. The probability of discovering new events can be increased in two ways: 1. Increase the event cross-section, which manifests the probability of a particular class of events to take place, which can be accomplished by increasing the energy of the colliding beams, which demands hard facility upgrades; 2. Increase the luminosity through the optimization of the collision conditions, which is more accessible.

Luminosity is often measured with dedicated detectors–luminometers. To perform precise measurements, these detectors must be calibrated properly. Several luminosity-calibration methods are used for hadron colliders and are summarized in [2]. A purely experimental technique based on the beam separation scan, which is the so-called van-der-Meer (vdM) scan, was successfully used at ISR [3, 4], RHIC [5,6,7], and LHC [8,9,10]. In a vdM scan, two beams are swept across each other, and the response of the luminometer under test is measured as a function of the distance between beam orbits (beam separation). The response rates are then fitted by Gaussian or double Gaussian fit models.

Regularly, the models used to extract observables from vdM scan data imply a Gaussian distribution of particles in colliding beams, which restricts the precision of calibration. For the high-energy colliders, the actual particle densities deviate from the exact Gaussian. In [11, 12], it was found that the colliding bunches have non-Gaussian tails. The non-Gaussian shape of the tails can be attributed to the different effects they experience due to intra-beam effects such as intra-beam scattering, synchrotron radiation, quantum excitation, and due to the mutual interaction of the two colliding beams such as beam-beam effect, e-cloud, and luminosity burn-off. These combined effects change the tail-to-core population of the colliding bunches. Therefore, the effect of the non-Gaussian tails on the absolute luminosity should be investigated for precise luminosity calibration. Some artificial corrections (like a second Gaussian [10] or deformation of the scan curve with a polynomial [9]) are often introduced to take non-Gaussian tails into account. In [12, 13], it was observed that the q-Gaussian distribution [14] provides a more realistic model for the actual bunch profile for LHC and the HL-LHC upgrade; therefore, it provides a more natural base to study the effect of their non-Gaussian tails on the main luminosity parameters. In [15], the impact of non-Gaussian tails on the absolute luminosity and emittance evolution was studied using a q-Gaussian model for bunch profile to investigate the modification of bunch shape due to the combined effects of intra-beam scattering and synchrotron radiation.

The aim of this work is to estimate the influence of the non-Gaussian tails on the luminosity calibration by assuming q-Gaussian bunches, as well as to look for ways to obtain more precise vdM scan models. The work is structured as follows: In Sect. 2, the luminosity concept is introduced, and the main parameters of the vdM scan are discussed. In Sect. 3, a full description of the q-Gaussian distribution is given. After that, the analytical formulas for the overlap integral and convolved beam sizes for q-Gaussian bunches are derived, and their deviation from that of Gaussian bunches is estimated. Then, the effect of the tilt angle in the transverse plane of the colliding bunches is considered, and its effect on the overlap integral is estimated. In Sect. 4, the vdM scan performed with q-Gaussian bunches is considered. First, the “toy” vdM scan is modeled where the beam overlap is calculated based on the theory developed in Sect. 3, three different fit models (Gaussian, Double Gaussian and q-Gaussian) are applied to this “toy” dataset and the best fit is determined at these conditions when scan shape has non-Gaussian tails. Then, the real experimental response rate data (from LHC CMS vdM program [10]) is tested to check if q-Gaussian fit model really works better than current Gaussian-based models.

2 Luminosity concept and Van-der-Meer scan

2.1 Luminosity and overlap integral

The luminosity depends only on the properties of colliding beams. The luminosity of a single bunch-crossing is defined as [16]:

$$\begin{aligned} \mathscr {L}({\varvec{\varDelta }}_r,t)=N_{1} N_{2} K\int \rho _1^{lab}({\textbf {r}}-{\varvec{\varDelta }}_r,t)\rho _2^{lab}({\textbf {r}},t)\,d^3{\textbf {r}}\,dt, \nonumber \\ \end{aligned}$$
(1)

where \(N_{1,2}\) are the number of particles in the colliding bunches, which move with velocity \({\textbf {v}}_1\) and \({\textbf {v}}_2\) correspondingly, K is Moller kinematic relativistic factor \(K= \sqrt{({\textbf {v}}_1-{\textbf {v}}_2)^2-\frac{({\textbf {v}}_1\times {\textbf {v}}_2)^2}{c^2}}\) [17], \(\rho _{1,2}^{lab}({\textbf {r}},t)\) are the normalized particle distribution densities in the colliding bunches in the lab frame and \({\varvec{\varDelta }}_r\) is the separation between their centres. The multiplication of the Moller kinematic relativistic factor with the integral in Eq. (1) represents the reciprocal of the effective area of the luminous region at the interaction point, \(\varOmega =1/A_{eff}\), which is the so-called overlap integral [18]. And its invariant form can be obtained from [16, 19] as:

$$\begin{aligned} \varOmega ({\varvec{\varDelta }}_{r_{\bot }})=\frac{1}{\gamma _\bot }\int \rho _1^{lab,\bot }({\textbf {r}}_\bot -{\varvec{\varDelta }}_{r_{\bot }})\rho _2^{lab,\bot }({\textbf {r}}_\bot )\,d{\textbf {r}}_\bot , \end{aligned}$$
(2)

where \(\rho _{1,2}^{lab,\bot }({\textbf {r}}_\bot )\) are the spatial transverse particle distributions of the colliding bunches in the lab frame, \({\varvec{\varDelta }}_{r_\bot }\) is the transverse beam separation and \(\gamma _\bot \) is the Lorentz relativistic factor due to transverse boost.

2.2 Van-der-meer scan

In principle, luminosity is measured using detectors reacting on the flux of collision products. They can be designed based on different effects and principles [9, 10]. In this paper we are not interested in the specific process used for luminosity measurements, the measured quantity is called “response rates” and notated with R.

The van-der-Meer (vdM) scan was proposed by S. Van Der Meer [4]. It is based on separating two beams across each other in the transverse plane, while the response rate R of their interaction is monitored as a function of the transverse separation between their orbits \(\varDelta \). If the colliding bunches have factorizable densities, \(\rho ^{lab}({\textbf {r}}_\bot )=\rho _x(x) \rho _y(y)\), or in other words, if their overlap integral is factorizable, \(\varOmega ({\varvec{\varDelta }}_{r_\bot })=\varOmega _x(\varDelta _x)\ \varOmega _y(\varDelta _y)\), and response rate R has no correlation between separations in the horizontal “x” and vertical “y” directions, therefore two one-dimensional vdM scans are performed separately in each direction, and the convolved bean sizes (i.e. RMS widths of the resulting scan curves) are found as:

$$\begin{aligned} \varSigma _x=\frac{\int R(\varDelta _x,0)\, d\varDelta _x}{C_x\, R(0,0)};\quad \varSigma _y=\frac{\int R(0,\varDelta _y)\, d\varDelta _y}{C_y\, R(0,0)}, \end{aligned}$$
(3)

where \(C_{x,y}\) are constants and depend on the particle densities. The scan curve shape in the horizontal and vertical directions, and the maximum overlap integral and luminosity are given by

$$\begin{aligned} \varOmega (0,0)=\frac{1}{C_xC_y\varSigma _x\varSigma _y};\quad \mathscr {L}(0,0)=fN_1N_2\varOmega (0,0), \end{aligned}$$
(4)

consequently, the calibration constant, which is the so-called the visible cross-section \(\sigma ^{vis}\), that relates the measurable response rates R of a luminometer recorded by a given luminosity algorithm to the absolute luminosity \(\mathscr {L}\), is determined as

$$\begin{aligned} \sigma ^{vis}=\frac{R(0,0)}{fN_1N_2\varOmega (0,0)}. \end{aligned}$$
(5)

where the visible cross-section \(\sigma ^{vis}\) is the ratio of measured rates to corresponding beam overlap. If the particle densities are not factorizable or, more specifically, their overlap integral has \(x-y\) coupling, the vdM scan is not precise.

In practice, the vdM scan is performed under special conditions where the beam parameters are optimized to determine the calibration constant with high precision. The vdM scan formalism is valid for arbitrary particle densities and crossing angles [16].

3 Theory

3.1 Q-Gaussian probability distribution function

The q-Gaussian distribution has diverse applications in generalized statistical theory, laser, plasma, and astronomy, see [20,21,22,23,24], and it has been used to investigate emittance evolution and beam profile modeling for LHC [13, 15]. The q-Gaussian distribution is known for its remarkable ability to represent a variety of distributions, from bounded finite distributions such as rectangular distribution at “\(q\rightarrow -\infty \)” and parabolic distribution at “\(q\rightarrow 0\)” to heavy-tailed infinite distributions such as Student’s t-distribution, where q controls the tails population. The q-Gaussian distribution is shown in Fig. 1a and is defined as:

$$\begin{aligned} QG(u;q,\beta ^{qG})=\frac{\sqrt{\beta ^{qG}}}{C^{qG}}e_q(-\beta ^{qG}u^2), \end{aligned}$$
(6)

where \(\beta ^{qG}\) is a real positive number, \(e_q\) is q-exponential and \(C^{qG}\) is the normalization constant, and they are defined as:

$$\begin{aligned} e_q(-\beta ^{qG}u^2)= {\left\{ \begin{array}{ll} \exp (-\beta ^{qG}u^2) &{} \text {if } q = 1\\ {[}1-(1-q)\beta ^{qG}x^2]_{+}^{\frac{1}{1-q}} &{} \text {if } q \ne 1\\ \end{array}\right. }, \end{aligned}$$
(7)

and

$$\begin{aligned} C^{qG}= {\left\{ \begin{array}{ll} \frac{2}{(3-q)\sqrt{1-q}} Beta(\frac{1}{1-q},\frac{1}{2}) &{} \text {if } q< 1\\ \sqrt{\pi } &{} \text {if } q = 1\\ \frac{1}{\sqrt{q-1}} Beta(\frac{1}{q-1}-\frac{1}{2},\frac{1}{2}) &{} \text {if } 1<q<3 \end{array}\right. }. \end{aligned}$$
(8)

For \(q<1\), the core is blown up, and the distribution becomes finite with light tails with \(u\in \left[ -\frac{1}{\sqrt{(1-q)\beta ^{qG}}},\frac{1}{\sqrt{(1-q)\beta ^{qG}}}\right] \). For \(q>1\), the tail density increases, and the distribution becomes heavy-tailed. At \(q=1\), the standard Gaussian distribution is restored. The standard deviation of the q-Gaussian distribution is dependent on q and \(\beta ^{qG}\), and is given by

$$\begin{aligned} \sigma ^{qG}= {\left\{ \begin{array}{ll} \frac{1}{\sqrt{(5-3q)\beta ^{qG}}} &{} \text {if } q< \frac{5}{3}\\ \infty &{} \text {if } \frac{5}{3}\le q< 2\\ Undefined &{} \text {if } 2 \le q<3 \end{array}\right. }. \end{aligned}$$
(9)

The standard deviation \(\sigma ^{qG}\) represents the RMS bunch dimension; therefore, in the q-Gaussian bunch model, the range of the tail weight q is limited to \(q<\frac{5}{3}\).

Fig. 1
figure 1

The bunch profile of q-Gaussian bunches with dimension \(\sigma _{u}^{qG}=100\ \upmu \)m and tail densities \(q=0.8\), 1.0 and 1.2 (a) and their overlap integral during the separation scan over separation \(\varDelta _u\) in the range 0 to 6 \(\sigma _u^{qG}\) (b)

Full size image

3.2 Overlap integral of q-Gaussian bunches

Let’s assume two bunches, 1 and 2, with transverse particle densities \(\rho _{1,2}(x,y)\) in the lab frame, the two bunches collide head-on (i.e. there is no crossing angle \(\phi =0\)) and the bunches have factorizable particle densities in horizontal and vertical directions. If the two bunches are separated in opposite directions with separation \(\frac{\varDelta _u}{2}\), the one-dimensional overlap integral \(\varOmega _u\) \((u=x,y)\) can be written as:

$$\begin{aligned} \varOmega _u(\varDelta _u)=\int \rho _1 \left( u-\frac{\varDelta _u}{2}\right) \rho _2\left( u+\frac{\varDelta _u}{2}\right) \, du. \end{aligned}$$
(10)

If the two bunches have q-Gaussian particle densities, and they have equal dimensions and tail densities in their respective direction (i.e. \(\sigma _{1u}=\sigma _{2u}=\sigma _{u}\) and \(q_{1u}=q_{2u}=q\) ), Eq. (10) becomes

(11)

where \(\beta ^{qG}\) is determined from Eq. (9). For \(q=1\), the q-Gaussian is equivalent to the normal Gaussian, and \(\varOmega _u^{qG}(\varDelta _u,1)\) is given by

$$\begin{aligned} \varOmega _u^{qG}(\varDelta _u;1)=\frac{\sqrt{\beta ^{qG}}}{\sqrt{2 \pi }} \exp \left( -\beta ^{qG}\frac{\varDelta _u^2}{2}\right) . \end{aligned}$$
(12)

For \(q\ne 1\), using \(e_q\) definition from Eq. (7) can be written as:

(13)

For finite light-tailed bunches with \(q<1\), Eq. (13) becomes:

(14)

Since the bunches have finite tails, the overlap integral is finite over the region where the densities of the two bunches can overlap, hence the integration limits \(u_{1}\) and \(u_{2}\) are given by

$$\begin{aligned}&\{u_1,u_2\} \\ {}&\quad = {\left\{ \begin{array}{ll} \bigg \{-\frac{1}{\sqrt{(1-q)\beta ^{qG}}} -\frac{\varDelta _u}{2},\frac{1}{\sqrt{(1-q)\beta ^{qG}}} +\frac{\varDelta _u}{2}\bigg \}, \\ \text {if } -\frac{2}{\sqrt{(1-q)\beta ^{qG}}}< \varDelta _u<0 \\ \bigg \{-\frac{1}{\sqrt{(1-q)\beta ^{qG}}} +\frac{\varDelta _u}{2},\frac{1}{\sqrt{(1-q)\beta ^{qG}}} -\frac{\varDelta _u}{2}\bigg \}, \\ \text {if } 0 \le \varDelta _u < \frac{2}{\sqrt{(1-q)\beta ^{qG}}} \end{array}\right. }. \end{aligned}$$

By solving Eq. (14), The general form of the overlap integral of light-tailed q-Gaussian bunches with equal bunch sizes and tail densities \(\varOmega _u^{qG}\left( \varDelta _u;q<1\right) \) is obtained as:

(15)

where\( _2F_1\) is the Gaussian hypergeometric function [25]. For a detailed derivation of Eq. (15), see Appendix A.

For infinite heavy-tailed bunches with \(q>1\), Eq. (13) becomes:

(16)

by solving Eq. (16), The general form of the overlap integral of heavy-tailed q-Gaussian bunches with equal bunch sizes and tail densities \(\varOmega _u^{qG}\left( \varDelta _u;q>1\right) \) is obtained as:

$$\begin{aligned}&\varOmega _u^{qG}\left( \varDelta _u;q>1\right) =\frac{\sqrt{\beta ^{qG}}}{\sqrt{q-1}\ {C^{qG}}^2} Beta\left( \frac{1}{2},\frac{5-q}{2q-2}\right) \nonumber \\&\quad \times _2F_1\left( \frac{1}{q-1},\frac{5-q}{2q-2};\frac{q+1}{2q-2};(1-q)\beta ^{qG}\frac{\varDelta _u^{2}}{4} \right) . \end{aligned}$$
(17)

For a detailed derivation of Eq. (17), see Appendix A.

The crossing angle effect can be considered in Eqs. (12), (15) and (17) through the modification of the transverse bunch size as in [19].

At the limit q tends to 1, the q-Gaussian bunches tend to Gaussian; therefore, the overlap integral of the q-Gaussian bunches should have the same tendency. Thus, the limit of Eqs. (15) and (17) is evaluated as in Eq. (18), and it shows the fulfillment of the tendency. See Appendix B for details.

$$\begin{aligned} \lim _{q \rightarrow 1}\left( \varOmega _u^{qG}\left( \varDelta _u;q<1\right) \right)&=\lim _{q \rightarrow 1}\left( \varOmega _u^{qG}\left( \varDelta _u;q>1\right) \right) \nonumber \\&=\varOmega _u^{qG}\left( \varDelta _u;q=1\right) . \end{aligned}$$
(18)
Fig. 2
figure 2

The deviation map of the one-dimensional overlap integral of q-Gaussian bunches \(\varOmega _u^{qG} \) from that of the Gaussian \(\varOmega _u^{G}\) for bunches with equal dimensions \(\sigma _{u}^{qG}=100\ \upmu \)m with tail densities q in the range 0.8 to 1.2 for vdM separation scan with separation \(\varDelta _u\) in the range 0 to 4 \(\sigma _u^{qG}\)

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3.3 Difference between regular Gaussian and Q-Gaussian beams of equal RMS beam size

Since the actual particle densities in the colliding bunches have non-Gaussian tails, the impact of the non-Gaussian tails on the overlap integral should be investigated. In order to do so, the separation scan for q-Gaussian bunches is modeled by using the analytical formula for the overlap integral \(\varOmega ^{qG}\) in Eqs. (15) and (17) with bunch parameters from van-der-Meer scans performed at CMS experiment (CERN). The vdM scan special conditions are \(\sigma _u=100\ \upmu \)m with no crossing angle [10]. Three beams of the same RMS beam size \(\sigma _u^{qG}=100\ \upmu \)m and different tail densities \(q=0.8\), 1.0 and 1.2 were considered to represent bunches with light tails, Gaussian, and heavy tails, respectively. The bunch profiles are shown in Fig. 1a, and their respective overlap integral \(\varOmega ^{qG}\) over separation \(\varDelta _u\) in the range of 0 to 6 \(\sigma _u^{qG}\) are shown in Fig. 1b. At zero separation (\(\varDelta _u=0\)), the heavy-tailed bunches have the highest overlap integral. Remarkable deviations of overlap integral of the q-Gaussian bunches \(\varOmega ^{qG}\) from that of Gaussian \(\varOmega ^{G}\) are also observed in Fig. 1b; these deviations are dependent on the separation. Thus to estimate this dependency, a deviation map was constructed, as shown in Fig. 2, where a beam separation scan with the separation \(\varDelta _u\) in the range 0 to 4 \(\sigma _u^{qG}\) is modeled, and the deviation of the overlap integral of q-Gaussian bunches to the Gaussian with identical bunch dimensions is estimated for different tail densities q in the range 0.8 to 1.2. This tail density range corresponds to a tail population that differs from Gaussian by up to \(20\%\).

The deviation map shows that the dependency of deviation on the separation \(\varDelta _u\) is divided into 3 regions: region 1- form zero separation to point “a”, the overlap integral is higher for heavy-tailed beams (lower for light-tailed beams), and it decreases (increases) as the separation increase until it equals to the overlap integral of Gaussian beams at point “a”; region 2- from point “a” to point “b”, the overlap integral is lower for heavy-tailed beams (higher for light-tailed beams), and it decreases (increases) as the separation increases until it reaches a minimum (maximum), then, it increases (decreases) with further separation until it equals to that of the Gaussian at point “b”; region 3- from point “b”, the overlap integral is higher for heavy-tailed beams (lower for light-tailed beams), and it increases (decreases) as the separation increases. The limits of these deviations are summarized in Table 1. The positions of point “a” and point “b” are different for different tail densities q values, and they fall in the range 1.037–1.065 \(\sigma _u^{qG}\) and 3.269–3.416 \(\sigma _u^{qG}\), respectively. This sensitivity of the overlap integral to the tail density of the colliding bunches justifies the need for precise consideration of non-Gaussian beam shape in luminosity modeling.

Table 1 The deviation limits of the overlap integral of the one-dimensional overlap integral of q-Gaussian bunches \(\varOmega _u^{qG} \) from that of the Gaussian \(\varOmega _u^{G}\) at different dependency regions for the vdM separation scan with separation \(\varDelta _u\) in the range 0 to 4 \(\sigma _u^{qG}\) for bunches with equal dimensions \(\sigma _{u}^{qG}=100\ \upmu \)m with tail density q in the range 0.8 to 1.2
Full size table

3.4 Convolved beam size of q-Gaussian bunches

Since the response rate R is proportional to the overlap integral \(\varOmega \), the one-dimensional convolved beam size of two q-Gaussian bunches \(\varSigma _u^{qG}\) is found from Eq. (3) as

$$\begin{aligned} \varSigma _u^{qG}(q)=\frac{\int \varOmega _u^{qG}(\varDelta _u;q)\,d \varDelta _u}{C_u\ \varOmega _u^{qG}(0;q)}, \end{aligned}$$
(19)

where the constant \(C_u\) is taken as \(\sqrt{(5-3q)}\ C^{qG}\) for q-Gaussian bunches. Since for any arbitrary normalized particle densities \(\int \varOmega _u(\varDelta _u) \,d\varDelta _u=1\) [16], therefore for \(\sigma _{1u}=\sigma _{2u}=\sigma _u\) and \(q_{1u}=q_{2u}=q\), the convolved beam size \(\varSigma _u^{qG}(q)\) is obtained as:

$$\begin{aligned} \varSigma _u^{qG}(q)={\left\{ \begin{array}{ll} \frac{Beta\left( \frac{1}{2},\frac{3-q}{2q-2}\right) }{Beta\left( \frac{1}{2},\frac{5-q}{2q-2}\right) }\ \sigma _u^{qG} &{} \text {if } 1<q<3 \\ \sqrt{2}\ \sigma _u^{qG} &{} \text {if } q=1 \\ \frac{Beta\left( \frac{1}{2},\frac{2-q}{1-q}\right) }{Beta\left( \frac{1}{2},\frac{3-q}{1-q}\right) }\ \sigma _u^{qG} &{} \text {if } q<1 \end{array}\right. }. \end{aligned}$$
(20)

Similarly, the convolved beam size of q-Gaussian bunches with different bunch dimensions \(\sigma _{1u}\ne \sigma _{2u}\) is presented in Appendix C.

Similar to the overlap integral, the convolved beam size of q-Gaussian bunches tends to that of Gaussian at the limit of q tends to 1.

$$\begin{aligned} \lim _{q \rightarrow 1}\left( \varSigma _u^{qG}(q<1)\right) =\lim _{q \rightarrow 1}\left( \varSigma _u^{qG}(q>1)\right) =\sqrt{2}\ \sigma _u^{qG}. \nonumber \\ \end{aligned}$$
(21)

Since, for precise luminosity calibration, the convolved beam size should be precisely defined from vdM scan curve, it is essential to investigate the effect of non-Gaussian tail populations on the convolved beam size. The dependence of the one-dimensional convolved beam size of q-Gaussian bunches \(\varSigma _u^{qG}\) on the tail density q is modeled, using Eq. (20), for bunches with vdM scan special conditions of \(\sigma _u^{qG}=100\) \(\upmu \)m with no crossing angle, see [10]. The tail density q in the range 0.8 to 1.2 is investigated. Figure 3a shows that for a certain bunch dimension, the convolved beam size increases as the tail density increases. Figure 3b shows the deviation of convolved beam size of q-Gaussian bunches \(\varSigma _u^{qG}\) from that of Gaussian \(\varSigma _u^{qG}\): it was found that a difference of \(20\%\) in tails population from Gaussian leads to a deviation up to \(4.25\%\) for heavy-tailed bunches and down to \(-3.35\%\) for light-tailed bunches.

Fig. 3
figure 3

The convolved beam size of q-Gaussian bunches \(\varSigma _u^{qG}\) for bunches with equal dimensions \(\sigma _u^{qG}=100\) \(\upmu \)m and tail densities q in the range 0.8 to 1.2 collide head-on (a), and its deviation from that of Gaussian bunches (b)

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3.5 The impact of the tilt angle

When two bunches collide, their overlap integral depends on their transverse particle distributions in the lab frame, as in Eq. (2), In general, their respective horizontal and vertical axes in the transverse planes do not usually coincide, which leads to a small angle between the bunches transverse planes, which is the so-called tilt angle. The tilt angle is usually small, and it leads to transverse densities that have \(x-y\) coupling in the lab frame as shown in Fig. 4. Thus, the effect of the tilt angle on the overlap integral is investigated for bunches with Gaussian and non-Gaussian tails, and its impact on the vdM scan is considered.

Fig. 4
figure 4

The longitudinal (a) and transverse (b) planes for bunch 1 (red) frame of reference \(X'Y'Z'\); bunch 2 (blue) frame of reference \(X''Y''Z''\) and lab frame of reference XYZ, where the transverse planes of bunch 1 and bunch 2 are rotated in opposite directions with a tilt angle \(\theta \)

Full size image

3.5.1 Gaussian tails

Let’s assume two bunches with transverse densities \(\rho _{1,2}\) are colliding at zero crossing angle, where the bunches have equal dimensions in their respective horizontal and vertical directions such that (\(\sigma _{1x}=\sigma _{2x}=\sigma _x\) and \(\sigma _{1y}=\sigma _{2y}=\sigma _y\)), let’s define three frames of reference: bunch 1 frame of reference \(X'Y'Z'\); bunch 2 frame of reference \(X''Y''Z''\) and lab frame of reference XYZ. In case of a tilt angle \(\theta \) between \(X'Y'\)and \(X''Y''\) such that the \(X'Y'\) and \(X''Y''\) are rotated with an angle \(\frac{\theta }{2}\) in opposite directions around XY, as shown in Fig. 4, the bunches coordinates are defined in the lab frame as:

$$\begin{aligned} x'= & {} x\cos \left( \frac{\theta }{2} \right) + y\sin \left( \frac{\theta }{2}\right) ;\nonumber \\ y'= & {} - x\sin \left( \frac{\theta }{2} \right) + y\cos \left( \frac{\theta }{2}\right) ;\nonumber \\ x''= & {} x\cos \left( \frac{\theta }{2} \right) - y\sin \left( \frac{\theta }{2}\right) ;\nonumber \\ y''= & {} - x\sin \left( \frac{\theta }{2} \right) - y\cos \left( \frac{\theta }{2}\right) . \end{aligned}$$
(22)

If there are horizontal and vertical separations \(\varDelta _x\) and \(\varDelta _y\), respectively, the overlap integral \(\varOmega _\theta ^G\) is found as

$$\begin{aligned} \varOmega _{\theta }^{G}&=\frac{1}{4 \pi ^2 \sigma _x^2 \sigma _y^2} \iint \rho _1(x-\varDelta _x,y-\varDelta _y,\theta )\nonumber \\&\qquad \times \rho _2(x,y,\theta ) \, dx \, dy\nonumber \\&\quad =\varOmega _{\theta ,x}(\varDelta _x,\theta )\, \varOmega _{\theta ,y}(\varDelta _y,\theta ), \end{aligned}$$
(23)

where \(\varOmega _{\theta ,x}^G\) and \(\varOmega _{\theta ,y}^G\) are the overlap integral in the horizontal and vertical direction, respectively, and defined as:

$$\begin{aligned}{} & {} \varOmega _{\theta ,x}(\varDelta _x,\theta )=\frac{1}{\sqrt{2\pi }\sqrt{(\sigma _x^2+\sigma _y^2)+(\sigma _x^2-\sigma _y^2)\cos (\theta )}}\nonumber \\{} & {} \quad \times \exp \left( \frac{-\varDelta _x^2}{2\left( (\sigma _x^2+\sigma _y^2)+(\sigma _x^2-\sigma _y^2)\cos (\theta )\right) }\right) ;\nonumber \\{} & {} \varOmega _{\theta ,y}(\varDelta _y,\theta )=\frac{1}{\sqrt{2\pi }\sqrt{(\sigma _x^2+\sigma _y^2)-(\sigma _x^2-\sigma _y^2)\cos (\theta )}}\nonumber \\{} & {} \quad \times \exp \left( \frac{-\varDelta _y^2}{2\left( (\sigma _x^2+\sigma _y^2)-(\sigma _x^2-\sigma _y^2)\cos (\theta )\right) }\right) . \end{aligned}$$
(24)

The tilt angle leads to non-factorizable transverse densities in the lab frame XYZ, but their resultant overlap integral is factorizable in terms of beam separation \(\varDelta _{x}\) and \(\varDelta _{y}\) with Gaussian form as in Eqs. (23) and (24). Thus, the horizontal and vertical convolved beam sizes \(\varSigma _{\theta ,x}^G\) and \(\varSigma _{\theta ,y}^G\) are not coupled, and they can be determined via two separate one-dimensional vdM scans as

$$\begin{aligned} \varSigma ^G_{\theta ,x}(\theta )= & {} \frac{1}{\sqrt{2\pi }} \frac{\int \varOmega _{\theta ,x}^G(\varDelta _x,\theta )\ d\varDelta _x}{\varOmega _{\theta ,x}^G(0,\theta )}\nonumber \\= & {} \sqrt{(\sigma _x^2+\sigma _y^2)+(\sigma _x^2-\sigma _y^2)\cos (\theta )}\;\nonumber \\ \varSigma ^G_{\theta ,y}(\theta )= & {} \frac{1}{\sqrt{2\pi }} \frac{\int \varOmega _{\theta ,y}^G(\varDelta _y,\theta \ )\ d\varDelta _y}{\varOmega _{\theta ,y}^G(0,\theta )}\nonumber \\= & {} \sqrt{(\sigma _x^2+\sigma _y^2)-(\sigma _x^2-\sigma _y^2)\cos (\theta )}. \end{aligned}$$
(25)

In principle, the tilt angle \(\theta \) leads to a reduction in the overlap integral, which can be considered a geometrical reduction factor due to the \(x-y\) coupling in the transverse particle densities. Figure 5 shows the maximum overlap integral \(\varOmega _\theta ^G\) at a tilt angle \(\theta \) (from 0 to \(90^o\)) normalized to that at zero tilt angle \(\varOmega _0^G\) for Gaussian bunches with different horizontal to vertical bunch dimension ratios. For round bunches, \(\frac{\sigma _x}{\sigma _y}=1\), the tilt angle does not produce any effect. As the \(\frac{\sigma _x}{\sigma _y}\) ratio increases, the reduction effect increases.

Fig. 5
figure 5

The ratio of maximum overlap integral at a tilt angle to that at zero tilt angle of Gaussian bunches \(\frac{\varOmega _\theta ^G}{\varOmega _0^G}\) at tilt angle \(\theta \) in the range 0 to \({90}^o\) for different horizontal to vertical bunch dimension ratios \(\frac{\sigma _x}{\sigma _y}= 0.9\), 1 and 1.2 for \(\sigma _y=100\) \(\upmu \)m at zero separation

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3.5.2 Non-Gaussian tails

Since the bunches with non-Gaussian tails can be represented by q-Gaussian distribution as mentioned earlier, let’s assume two q-Gaussian bunches \(\rho _{1,2}\) with equal bunch dimensions and tail densities in the horizontal and vertical directions (\(\sigma _{1x}^{qG}=\sigma _{2x}^{qG}=\sigma _x^{qG}\); \(\sigma _{1y}^{qG}=\sigma _{2y}^{qG}=\sigma _y^{qG}\); \(q_{1x}=q_{2x}=q_x;\) and \(q_{1y}=q_{2y}=q_y\)) at zero crossing angle, using similar frames of reference and coordinates as in Eq. (22). The particle densities in the lab frame are defined as:

Fig. 6
figure 6

The ratio of maximum overlap integral of q-Gaussian bunches at a tilt angle \(\theta \) to that at zero tilt angle \(\frac{\varOmega _\theta ^{qG}}{\varOmega _0^{qG}}\) with horizontal tail density \(q_x=0.9\) (a) and \(q_x=1.1\) (b) at tilt angle \(\theta \) in the range 0 to \({90}^o\) for different horizontal to vertical bunch dimension ratios \(\frac{\sigma _x^{qG}}{\sigma _y^{qG}}=0.9\), 1 and 1.2, where the vertical bunch dimension \(\sigma _y^{qG}=100\) \(\mu m\)

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$$\begin{aligned} \rho _1(x,y,\theta )= & {} QG\left( x\cos \left( \frac{\theta }{2} \right) + y\sin \left( \frac{\theta }{2}\right) ; q_x,\beta ^{qG}_x\right) \nonumber \\{} & {} \times QG\left( - x\sin \left( \frac{\theta }{2} \right) + y\cos \left( \frac{\theta }{2}\right) ;q_y,\beta _y^{qG}\right) ;\nonumber \\ \rho _2(x,y,\theta )= & {} QG\left( x\cos \left( \frac{\theta }{2} \right) - y\sin \left( \frac{\theta }{2}\right) ; q_x,\beta _x^{qG}\right) \nonumber \\{} & {} \times QG\left( - x\sin \left( \frac{\theta }{2} \right) - y\cos \left( \frac{\theta }{2}\right) ;q_y,\beta _y^{qG}\right) ,\nonumber \\ \end{aligned}$$
(26)

and the overlap integral \(\varOmega _\theta ^{qG}\) is given by

(27)

The solution of Eq. (27) is hard to find analytically; therefore, numerical integration techniques are used to estimate the impact of tilt angle on the overlap integral of q-Gaussian bunches. Similar to Gaussian, the existence of the tilt angle leads to a reduction in the overlap integral. The dependence of this reduction on the tilt angle at different horizontal tail densities \(q_x\) and bunch dimension ratios \(\frac{\sigma _x^{qG}}{\sigma _y^{qG}}\) is shown in Fig. 6, where the vertical tail density is assumed to be Gaussian, \(q_y=1\). Figure 6 shows that for the bunch dimension ratio \(\frac{\sigma _x}{\sigma _y}=1\), the reduction effect exists for both light- and heavy-tailed bunches, being larger for heavy-tailed bunches. For bunches with heavy-tailed horizontal density \(q_x>1\), if \(\frac{\sigma _x}{\sigma _y}>1\), the reduction effect is smaller than in the Gaussian case while for \(\frac{\sigma _x}{\sigma _y}<1\), the reduction effect is larger. For bunches with light-tailed horizontal density \(q_x<1\), if \(\frac{\sigma _x}{\sigma _y}>1\), the reduction effect is larger than in the Gaussian case while for \(\frac{\sigma _x}{\sigma _y}<1\), the reduction effect is smaller. It worth nothing that for q-Gaussian bunches, the tilt angle leads to a non-factorable overlap integral \(\varOmega _\theta ^{qG}\) in terms of beam separation \(\varDelta _{x}\) and \(\varDelta _{y}\), which leads to a non-factorization bias in vdM scan. The detailed study of this effect is abroad of the topic and will be published elsewhere.

4 Application

The previous section estimates how the non-Gaussian tails affect the overlap integral and convolved beam size, where the q-Gaussian bunches are used as a more realistic approximation to the actual bunches. In this section, the influence of non-Gaussian tails on quantities derived from van-der-Meer (vdM) scan is investigated. Namely, a “toy” vdM scan is modelled by q-Gaussian bunches, and the resultant scan data is fitted by Gaussian, double Gaussian and q-Gaussian fit models to estimate the precision of the Gaussian-based models when it is applied to bunches with non-Gaussian tails. After that, the procedure is tested using beam overlap width measurements performed in 2015 CMS vdM scan program.

4.1 Toy van-der-Meer scan

The “toy” van-der-Meer scan was performed by calculating the overlap integral of two q-Gaussian bunches with equal bunch dimension \(\sigma _u^{qG}\) and tail density q, where the overlap integral is calculated by the analytical formulas of \(\varOmega _u^{qG}\), Eqs. (15) and (17), for different separations \(\varDelta _u\) from 0 to 6 \(\sigma _u^{qG}\), using 60 points of \(\left( \varDelta _u, \varOmega _u^{qG}(\varDelta _u;q)\right) \), and the convolved beam size of these bunches \(\varSigma _u^{qG}\) is calculated by the analytical formula (25). For the vdM scan, the standard fit model for application is Gaussian \(f^{G}\), Eq. (28), but since the simple Gaussian does not adequately fit the scan data, especially the tails, a double Gaussian fit \(f^{DG}\), Eq. (29), with two different widths is widely used in RHIC [11] and LHC [14], where the Gaussian with smaller width \(\sigma _1\) fits the core, and the Gaussian with the larger width \(\sigma _{2}\) fits the tails, and \(\varepsilon \) is the fraction of Gaussian with the smaller width. The convolved beam size \(\varSigma _u^{DG}\) \((u=x,y)\) is defined as in Eq. (30).

$$\begin{aligned} f^G(\varDelta _u)= & {} \frac{1}{\sqrt{2\pi }\varSigma _u^G}\exp \left( -\frac{\varDelta _u^2}{2\varSigma _u^{G^2}}\right) . \end{aligned}$$
(28)
$$\begin{aligned} f^{DG} (\varDelta _u )= & {} \frac{1}{\sqrt{2\pi }} \bigg [\frac{\varepsilon }{\sigma _{1u}}\exp \left( -\frac{\varDelta _u^2}{2\sigma _{1u}^{2}}\right) \nonumber \\{} & {} +\frac{1-\varepsilon }{\sigma _{2u}}exp\left( -\frac{\varDelta _u^2}{2\sigma _{2u}^{2}}\right) \bigg ]. \end{aligned}$$
(29)
$$\begin{aligned} \varSigma _u^{DG}= & {} \frac{\sigma _{1u}\sigma _{2u}}{\varepsilon _u\sigma _{2u}+(1-\varepsilon _u)\sigma _{1u}}. \end{aligned}$$
(30)

Based on the q-Gaussian bunches assumption and the derived analytical formulae of the overlap integral \(\varOmega _u^{qG}\), a vdM scan fit model is proposed to account for the tail populations. The model is based on the ability of the q-Gaussian distribution to describe various tails, ranging from finite light tails for \(q<1\) to heavy tails for \(q>1\). The proposed model \(f_u^{qG}\) is defined in terms of q and \(\varSigma _u^{qG}\) as:

$$\begin{aligned} f_u^{qG}(\varDelta _u;q)=\frac{1}{C_u^{qG}\sqrt{5-3q}\varSigma _u^{qG}}e_q\left( - \frac{\varDelta _u^2}{(5-3q)\varSigma _u^{qG^2}}\right) , \nonumber \\ \end{aligned}$$
(31)

where \(e_q\) and \(C^{qG}\) are defined by Eqs. (7) and (8).

The toy scan is conducted for light-tailed bunches with “\(q=0.8\)” and heavy-tailed bunches with “\(q=1.2\)”, with equal bunch dimensions \(\sigma _u^{qG}=100\) \(\upmu \)m, The resultant toy scans data are then fitted by models (28), (29), and (31) using least-squares minimization by the trust region reflective method from Non-Linear Least-Squares Minimization and Curve-Fitting Python Package “lmfit”. Since the overlap integral of light-tailed bunches has underpopulated tails, the double Gaussian model (29) was applied only for heavy-tailed bunches, as the concept of its application does not coincide with the light-tailed bunches.

The fitting of vdM toy scan data is shown in Fig. 7. For light-tailed beams “\(q=0.8\)”, Fig. 7a, the Gaussian model failed to fit the data at small and large separations, where it overestimates the data core and tails; on the other hand, the q-Gaussian model fits the data well. For heavy-tailed beams at “\(q=1.2\)”, Fig. 7b, the Gaussian model underestimates the core and the tails of the data; on the other hand, the double Gaussian and q-Gaussian models provide a good description of the data. Compared to Gaussian and double Gaussian fits, the q-Gaussian fit provides the best description of the data, especially for the tails. The fitting parameters are summarized in Table 2. The goodness of fit statistics are based on the root mean square error (RMSE) and \(R^2\) statistics and the deviations from the predicted values of \(\varSigma _u^{qG}\) and the maximum \(\varOmega _u^{qG}\) (at zero separation) are summarized in Table 3.

Fig. 7
figure 7

The fitting of the overlap integral of two q-Gaussian bunches with equal bunch size \(\sigma ^{qG}_{u}=100\ \mu m\) and tail density q obtained at separation \(\varDelta _u\) in the range from 0 to 6 \(\sigma _u^{qG}\) during the “toy” vdM scan. Two cases were considered: light-tailed bunches “\(q=0.8\)” (a) and heavy-tailed bunches “\(q=1.2\)” (b), where the following fit models were applied: Gaussian (28); double Gaussian (29); and q-Gaussian (31)

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Table 2 The estimated fitting parameters for the “toy” vdM scan data of two q-Gaussian bunches with equal size \(\sigma _{u}^{qG}=100\ \mu m\) and tail density q. Two cases were considered: “light-tailed beams” \(q=0.8\) and “heavy-tailed beams” \(q=1.2\). The following fit models were applied: Gaussian (28); double Gaussian (29); and q-Gaussian (31)
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Table 3 A comparison of goodness of fit statistics based on RMSE and \(R^2\) for different fit models applied to the “toy” vdM scan data. The deviation of the predicted convolved beam size \(\varSigma _u^{fit\ model}\) and overlap integral \(\varOmega _u^{fit\ model}\) from their respective analytical values \(\varSigma _u^{Analytical}\) and the maximum \(\varOmega _u^{Analytical}\) (at zero separation) is also presented. The fit models are Gaussian (28); double Gaussian (29); and q-Gaussian (31)
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Based on the previous results, the q-Gaussian fit model (\(f^{qG}\)), Eq. (31), provides the best fit for the scan data and predicts the convolved beam size and the overlap integral with high precision with deviations less than \(\pm 5\%\) and \(\pm 0.025\%\), respectively. Double Gaussian fit model (\(f^{DG}\)), Eq. (29), provides a good fit for the data, and predicts overlap integral with a deviation less than \(0.04\%\), but it can only be applied for infinite heavy-tailed bunches, since for bounded light-tailed bunches, it gives the same results as Gaussian, and it fails to provide any better predictions than single Gaussian. The high deviation of the Gaussian-based models compared to the q-Gaussian model is because the Gaussian-based models do not account for the relation between the convolved beam size and the tail density, even for \(f^{DG}\), which was only successful for the heavy-tailed bunches.

To investigate the effect of the non-Gaussian tails on the accuracy of prediction of different fit models, the toy vdM scan was performed with tail density q in the range 0.8 to 1.2. Figure 8 shows the deviation of the predicted convolved beam size \(\varSigma _u^{fit\ model}\) and overlap integral \(\varOmega _u^{fit\ model}\) obtained by the previous fit models from their respective analytical values calculated by Eqs. (15) and (17), respectively. For light-tailed bunches \(f^{G}\) and \(f^{DG}\) overestimate the convolved beam size and the overlap integral, while for heavy-tailed beams \(f^{G}\), \(f^{DG}\) and \(f^{qG}\) underestimate them. When q tends to 1, all models have predictions close to the analytical values. The goodness of fit statistics based on RMSE and adjusted \(R^2\) at various tail densities q is shown in Fig. 9.

Fig. 8
figure 8

The deviation of the convolved beam size \(\varSigma _u^{fit\ model}\) (a) and the overlap integral \(\varOmega _u^{fit\ model}\) (b), determined by the different fit models, from their respective analytical values \(\varSigma _u^{Analytical}\) and \(\varOmega _u^{Analytical}\), calculated by Eqs. (18) and (12), (15) and (17), for “toy” vdM scan of two equal-size q-Gaussian bunches with tail density q in the range 0.8 to 1.2

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Fig. 9
figure 9

Goodness of fit analysis based on RMSE (a) and Adj. \(R^2\) (b) for different fit models of toy vdM scan of two equal-size q-Gaussian bunches with tail density q in the range 0.8–1.2

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Based on Figs. 8 and 9, the q-Gaussian fit model, Eq. (31), can predict the convolved beam size and the overlap integral with higher accuracy than Gaussian and double Gaussian models; it also provides a better description of the scan data according to the goodness of fit statistics. It can be applied for both light- and heavy-tailed bunches. Compared to the analytical solution in Eqs. (15) and (17), the q-Gaussian fit model represents a good and straightforward approximation.

4.2 Influence of the non-Gaussian tails at the vdM scan at LHC

The beam overlap \(\varOmega \) is not a measurable quantity. The real measurable is the response rate R mentioned above. The response R is proportional to \(\varOmega \) because R characterizes the flux of the collision product, whereas \(\varOmega \) characterizes the intensity of collisions. Therefore, the measured rates R in the vdM scan form a scan curve with a shape similar to that of \(\varOmega \). In this subsection, we apply the models developed before to response rates R.

The Gaussian, double Gaussian, and q-Gaussian fit models are applied to the actual vdM scan dataset, from CMS at the LHC run 2 published in [10], to assess the proposed q-Gaussian fit model in comparison with fit models used in [10], and to investigate its statistical significance in describing the vdM scan data of actual beams and its ability to predict the overlap integral with higher precision. The previous fit models are rewritten in the general form as:

$$\begin{aligned} f^G(\varDelta _u)= & {} A\exp \left( -\frac{(\varDelta _u-\mu )^2}{2\varSigma _u^{G^2}}\right) +Const; \end{aligned}$$
(32)
$$\begin{aligned} f^{DG} (\varDelta _u )= & {} A \Bigg [\varepsilon \exp \left( -\frac{(\varDelta _u-\mu )^2}{2\sigma _{1u}^{2}}\right) \nonumber \\{} & {} +(1-\varepsilon )\exp \left( -\frac{(\varDelta _u-\mu )^2}{2\sigma _{2u}^{2}}\right) \Bigg ]+Const; \end{aligned}$$
(33)
$$\begin{aligned} f_u^{qG}(\varDelta _u;q)= & {} A e_q\left( - \frac{(\varDelta _u-\mu )^2}{(5-3q)\varSigma _u^{qG^2}}\right) +Const, \end{aligned}$$
(34)

where Const is added to account for the background, and A represents the amplitude of the normalized rates. The fitting of the normalized rates and their fitting residuals are presented in Fig. 10, with the horizontal X scan in Fig. 10a and the vertical Y scan in Fig. 10b, while the resultant fitting parameters are summarized in Table 4.

Fig. 10
figure 10

Fits (32), (33) and (34) applied to dataset [10] where the normalized rates are represented as a function of the beam separation in the horizontal X (a) and vertical Y (b) directions. The resultant fitting parameters are summarized in Table 4

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It worth noticing that for the horizontal scan, the q-Gaussian fit predicts a scan curve with slightly underpopulated tails with a value of \(q=0.9968\). On the other hand, the double Gaussian fit exhibits a strong dependence on the initial values of the fitting parameters (\(\sigma _1\), \(\sigma _2\), \(\varepsilon \)), resulting in infinite sets of fitting parameters where the double Gaussian is equivalent to a single Gaussian. This implies that the double Gaussian fit cannot further enhance precision for scans with underpopulated tails. For the vertical scan, the q-Gaussian fitting predicts slightly overpopulated tails of the scan curve with \(q=1.0157\). The statistical analysis of the fit models and the predicted \(\varSigma _u^{fit\ model}\) and \(\varOmega _u^{fit\ model}\) summarized in Table 4, where the RMSE, Adj. \(R^2\) and \(\chi ^2/dof\) are used for the goodness of fit analysis.

To determine the best fit of the scan, the criterion is the fit with the lowest RMSE has Adj. \(R^2\) and \(\chi ^2/dof\) closest to 1. For the horizontal scan, the q-Gaussian fit provides the lowest RMSE, while the Gaussian fit has the Adj. \(R^2\) and \(\chi ^2/dof\) closest to 1. The q-Gaussian represents the best fit as the Gaussian model is a special case of the q-Gaussian where one of its parameters is fixed \((q=1)\). It is worth mentioning that such statistics will be observed for any scan with \(q \approx 1\). For the vertical scan, the double Gaussian has the lowest RMSE and the highest \(\chi ^2/dof\), which shows that double Gaussian could predict the data, but it could not explain the variance well; the q-Gaussian has the Adj. \(R^2\) and \(\chi ^2/dof\) closest to 1, and it has RMSE very close to the double Gaussian, which shows that the q-Gaussian can be considered the best fit since it can predict the data with high precision and can explain the variance as well. The overlap integral \(\varOmega \) obtained by the q-Gaussian model deviates from the CMS scan models by \(-0.027\%\) and \(0.077\%\) for the horizontal and vertical scans, respectively. The fitting results generally show that the q-Gaussian fit model represents a promising base for beam overlap modeling for vdM scans with underpopulated and overpopulated tails.

5 Conclusion

In high-energy colliders, the particles in the bunches undergo various effects that lead to slight deviations in their distributions from the exact Gaussian distribution. Hence, a more realistic approximation, such as q-Gaussian, is used to describe the beam profiles more efficiently. In this work, the overlap integral of q-Gaussian beams is investigated. It was shown that the tail density q controls the dependence of the overlap integral \(\varOmega \) on the beam separation \(\varDelta \): Beams with heavy tails “\(q>1\)” have larger beam overlap at small and large beam separations, while Beams with light tails “\(q<1\)” have larger beam overlap at intermediate beam separation. This dependency tends to that of Gaussian at q tends to 1. The width of the van-der-Meer (vdM) scan curve of q-Gaussian beams is also controlled by the tail density q. For bunches with the same bunch dimensions, the deviation of convolved beam size \(\varSigma \) of q-Gaussian beams from that of Gaussian beams is up to ± 4% for tail density q in the range 0.8–1.2.

Table 4 The estimated fitting parameters of the dataset [10] using Gaussian (32), double Gaussian (33) and q-Gaussian (34) fit models
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Table 5 Statistical analysis and the predicted convolved beam size \(\varSigma _u^{fit\ model}\) and overlap integral \(\varOmega _u^{fit\ model}\) from the fitting of dataset [10] by Gaussian (32), double Gaussian (33) and q-Gaussian (34) fitting models
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For q-Gaussian bunches, the tilt angle results in non-factorizable transverse densities and overlap integral. The resultant geometrical reduction due to the tilt angle depends on the horizontal and vertical tail densities, \(q_x\) and \(q_y\), respectively, and the ratio between the horizontal and vertical bunch dimensions \(\sigma _x/\sigma _y\). The detailed study of the resulting non-factorization bias is outside of the topic of this article and will be published elsewhere (Table 5).

A vdM scan fit model based on q-Gaussian is proposed and applied to a toy vdM scan of two similar q-Gaussian beams. The fitting results show that unlike the double Gaussian fit, which can be used only for separation scans with overpopulated tails, the proposed model can be applied for scans with underpopulated and overpopulated tails. It describes the scan data well, particularly the tails.

Applying the q-Gaussian fit model to the CMS vdM scan dataset shows that the horizontal scan has underpopulated tails, whereas the double Gaussian fit model tends to a single Gaussian, and it fails to improve the fit accuracy. On the other hand, the q-Gaussian fit model represents the best fit for horizontal and vertical scans. The visible cross-section \(\sigma ^{vis}\) obtained by the q-Gaussian model deviates from the CMS scan models by \(0.027\%\) and \(-0.077\%\) for the horizontal and vertical scans, respectively.

In conclusion, the effect of the tails on the overlap integral is getting more critical since the collider machines are constantly upgraded, and the requirements for luminosity precision have become stronger, demanding a more accurate account for bunch shape; hence, this shape should be taken into account accurately. The results of this study suggest that models based on q-Gaussian provide a strong foundation for vdM scan data analysis capable of accounting for non-Gaussian tails of colliding bunches with higher accuracy. Further research on actual vdM scan datasets using such models is recommended.