Luminosity and beam-beam tune shifts with crossing angle and hourglass effects in e⁺-e^- colliders

Introducing a crossing angle reduces both the luminosity and beam-beam tune shifts if no optics changes are made. The lowered beam-beam tune shifts however allow the possibility of further reducing the beta functions at the IP to increase the luminosity while keeping the beam-beam tune shifts within allowed limits. Specifically, in an e⁺-e^- collider where $\beta_{y}^{*}\ll\beta_{x}^{*}$, a crossing angle in the horizontal plane allows a scheme where $\beta_{x}^{*}$ is reduced sufficiently to increase the luminosity beyond values without a crossing angle. This has been investigated in recent designs of colliders such as Super KEKB [1], FCC-ee [2] etc. We include both the crossing angle and the hourglass effects on the luminosity and the beam-beam tune shifts. Analytic expressions for the combined effects do not appear to be available in the literature; instead they are approximated as acting independently. The only assumption in our treatment below is that of symmetric interaction region optics for the electrons and positrons so that the bunch sizes in all three dimensions are the same in both beams. After developing exact general expressions, we consider several limiting cases and show that they reduce to known forms where applicable. The purpose of this paper is to find appropriate combinations of the crossing angle and $(\beta_{x}^{*},\beta_{y}^{*})$ which maximize the luminosity while restricting the beam-beam tune shifts to tolerable values.

We first apply the theory to the very preliminary design of a Higgs factory at Fermilab, called a site filler, and find parameters that increase the luminosity with a non-zero crossing angle. Next we apply the theory to the FCC-ee collider whose design is considerably more mature. We find that the luminosity in this design can also be increased with changes in the crossing angle and $\beta_{x}^{*}$.

The relativistically invariant luminosity per bunch and unit time is [3]

$${\cal L}=Kf_{rev}N_{+}N_{-}\int_{-\infty}^{infty}\int_{-\infty}^{infty}\int_{-\infty}^{infty}\int_{-\infty}^{infty}ds\;dt\;dx\;dy\;\rho_{-}(xy,s-ct)\rho_{+}(x,y,s-ct)$$

(2.1)

($N_{-},\rho_{-}$), ($N_{+},\rho_{+}$) are the (bunch intensities, three dimensional densities) of the electrons and positrons respectively and $K$ is a kinematic factor

$$K=\sqrt{({\bf v_{+}}-{\bf v_{-}})^{2}-\frac{(\bf v_{+}\times v_{-})^{2}}{c^{2}}}$$

(2.2)

We assume that the electrons move along the positive $s$ direction and the positrons in the opposite direction. When the beams cross in the horizontal plane at a full angle of $\theta_{C}$, the coordinates in the two beam frames are

	$\displaystyle x_{-}$	$\displaystyle=$	$\displaystyle C_{C}x-S_{C}s,\;\;\;s_{-}=C_{C}s+S_{C}x$
	$\displaystyle x_{+}$	$\displaystyle=$	$\displaystyle-C_{C}x-S_{C}s,\;\;\;s_{+}=-C_{C}s+S_{C}x$		(2.3)

where $(x,s)$ are the coordinates in the laboratory frame, $C_{C}=\cos(\theta_{C}/2),S_{C}=\sin(\theta_{C}/2)$. The transverse velocities are orders of magnitude smaller than the longitudinal velocity ($\simeq c$), so the only velocity components are the projections of the longitudinal velocity,

	$\displaystyle{\bf v_{-}}$	$\displaystyle\equiv$	$\displaystyle\frac{d}{dt}(x_{-}\hat{x}+0\hat{y}+s_{-}\hat{z})=(-S_{C}\hat{x}+0\hat{y}+C_{C}\hat{z})c$
	$\displaystyle{\bf v_{+}}$	$\displaystyle\equiv$	$\displaystyle\frac{d}{dt}(x_{+}\hat{x}+0\hat{y}+s_{+}\hat{z})=(-S_{C}\hat{x}+0\hat{y}-C_{C}\hat{z})c$
	$\displaystyle({\bf v_{-}}-{\bf v_{+}})^{2}$	$\displaystyle=$	$\displaystyle(0\hat{x}+0\hat{y}+2C_{C}\hat{z})^{2}c^{2}=4C_{C}^{2}c^{2}$
	$\displaystyle(\bf v_{+}\times v_{-})$	$\displaystyle=$	$\displaystyle(0\hat{x}+2S_{C}C_{C}\hat{y}+0\hat{z})c^{2}$

where $\hat{x},\hat{y},\hat{z}$ are unit vectors. Hence the kinematic factor is

$$K=c\sqrt{4C_{C}^{2}-4S_{C}^{2}C_{C}^{2}}=2cC_{C}^{2}$$

(2.4)

The normalized density of the electron and positron bunches are

	$\displaystyle\rho_{-}\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!\frac{1}{(2\pi)^{3/2}\sigma_{x-}\sigma_{y-}\sigma_{s-}}\exp\left[-\frac{(C_{C}x-S_{C}s)^{2}}{2\sigma_{x-}^{2}}-\frac{y^{2}}{2\sigma_{y-}^{2}}-\frac{(C_{C}s+S_{C}x-ct)^{2}}{2\sigma_{s-}^{2}}\right]$		(2.5)
	$\displaystyle\rho_{+}\!\!\!\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!\!\!\!\frac{1}{(2\pi)^{3/2}\sigma_{x+}\sigma_{y+}\sigma_{s+}}\exp\left[-\frac{(-C_{C}x-S_{C}s)^{2}}{2\sigma_{x+}^{2}}-\frac{y^{2}}{2\sigma_{y+}^{2}}-\frac{(S_{C}x-C_{C}s-ct)^{2}}{2\sigma_{s+}^{2}}\right]$		(2.6)

We assume that the bunches are symmetric at the IP so that the beam sizes in all 3 dimensions are matched, i.e.

$$\sigma_{x+}^{*}=\sigma_{x-}^{*},\;\;\;\sigma_{y+}^{*}=\sigma_{y-}^{*},\;\;\;\sigma_{s+}=\sigma_{s-}=\sigma_{s}$$

(2.7)

This assumption is true for equal high energy colliders that we consider but could be dropped to consider the more general case of asymmetric colliders. We ignore perturbative effects such as a non-zero dispersion or transverse offsets at the IP.

The beta functions depend only on the longitudinal coordinate $s$ in the lab frame, they do not depend on the coordinates in the beam frames. Thus

	$\displaystyle\sigma_{x-}(s)$	$\displaystyle=$	$\displaystyle\sigma_{x+}(s)=\sigma_{x}(s)=\sqrt{\epsilon_{x}(\beta_{x}^{*}+\frac{s^{2}}{\beta_{x}^{*}})}=\sigma_{x}^{*}\sqrt{1+\frac{s^{2}}{\beta_{x}^{*,2}}}$		(2.8)
	$\displaystyle\sigma_{y-}(s)$	$\displaystyle=$	$\displaystyle\sigma_{y+}(s)=\sigma_{y}(s)=\sigma_{y}^{*}\sqrt{1+\frac{s^{2}}{\beta_{y}^{*,2}}}$		(2.9)

Putting all the factors for the luminosity

	$\displaystyle{\cal L}$	$\displaystyle=$	$\displaystyle\frac{2cf_{rev}N_{+}N_{-}C_{C}^{2}}{(2\pi)^{3}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}ds\;dt\;dx\;dy\;\frac{1}{\sigma_{x}(s)^{2}\sigma_{y}(s)^{2}\sigma_{s}^{2}}\exp[-\frac{(C_{C}x-S_{C}s)^{2}}{2\sigma_{x}^{2}}]$
			$\displaystyle\exp[-\frac{y^{2}}{2\sigma_{y}^{2}}]\exp[-\frac{(C_{C}s+S_{C}x-ct)^{2}}{2\sigma_{s}^{2}}]\exp[-\frac{(-C_{C}x-S_{C}s)^{2}}{2\sigma_{x}^{2}}]\exp[-\frac{y^{2}}{2\sigma_{y}^{2}}]$
			$\displaystyle\exp[-\frac{(C_{C}s-S_{C}x+ct)^{2}}{2\sigma_{s}^{2}}]$

The integrations over $t,y,x$ are straightforward. The last integration over $s$ is

			$\displaystyle\int_{-\infty}^{\infty}\frac{ds}{\sigma_{s}}\;\frac{1}{\sqrt{(1+\frac{s^{2}}{\beta_{x}^{*2}})(1+\frac{s^{2}}{\beta_{x}^{*2}})}}\exp[-\frac{C_{C}^{2}s^{2}}{\sigma_{s}^{2}}-\frac{S_{C}^{2}s^{2}}{\sigma_{x}^{2}}]$
		$\displaystyle=$	$\displaystyle\int_{-\infty}^{\infty}du\;\exp[-C_{C}u^{2}(1+T_{C}^{2}\frac{\sigma_{s}^{2}/\sigma_{x}^{*2}}{1+u^{2}/(u_{x}^{2})})]\frac{1}{\sqrt{(1+\frac{u^{2}}{u_{x}^{2}})(1+\frac{u^{2}}{u_{y}^{2}})}}$

where we defined

$$u=\frac{s}{\sigma_{s}},\;\;\;u_{x}=\frac{\beta_{x}^{*}}{\sigma_{s}}\;\;\;u_{y}=\frac{\beta_{y}^{*}}{\sigma_{s}}\;\;\;T_{C}=\tan(\theta_{C}/2)$$

(2.10)

Thus the general expression for the luminosity per bunch is

$\displaystyle{\cal L}$

$\displaystyle=$

$\displaystyle\frac{f_{rev}N_{+}N_{-}C_{C}}{4\pi^{3/2}\sigma_{x}^{*}\sigma_{y}^{*}}\int_{-\infty}^{\infty}du\;\exp[-C_{C}^{2}u^{2}(1+T_{C}^{2}\frac{\sigma_{s}^{2}/\sigma_{x}^{*2}}{1+u^{2}/(u_{x}^{2})})]\frac{1}{\sqrt{(1+\frac{u^{2}}{u_{x}^{2}})(1+\frac{u^{2}}{u_{y}^{2}})}}$

and the general correction factor is

	$\displaystyle R_{L}\!\!\!$	$\displaystyle\equiv$	$\displaystyle\frac{\cal L}{\cal L_{0}}$
		$\displaystyle=$	$\displaystyle\!\!\!\frac{C_{C}}{\sqrt{\pi}}\int_{-\infty}^{\infty}du\;\exp[-\cos^{2}(\theta_{C}/2)u^{2}(1+\tan^{2}(\theta_{C}/2)\frac{\sigma_{s}^{2}/\sigma_{x}^{*2}}{1+u^{2}/(u_{x}^{2})})]\frac{1}{\sqrt{(1+\frac{u^{2}}{u_{x}^{2}})(1+\frac{u^{2}}{u_{y}^{2}})}}$

Limiting cases

1. 1.

   No crossing angle or hourglass

   Setting $C_{C}=1,T_{C}=0$ and $u_{x},u_{y}\to\infty$, we have

   $${\cal L}=\frac{f_{rev}N_{+}N_{-}}{4\pi^{3/2}\sigma_{x}^{*}\sigma_{y}^{*}}\int_{-\infty}^{\infty}du\;\exp[-u^{2}]=\frac{f_{rev}N_{+}N_{-}}{4\pi\sigma_{x}^{*}\sigma_{y}^{*}}$$

   (2.13)

   the standard expression for the nominal luminosity.

2. 2.

   Only the hourglass effect, no crossing angle

   $${\cal L}=\frac{f_{rev}N_{+}N_{-}}{4\pi^{3/2}\sigma_{x}^{*}\sigma_{y}^{*}}\int_{-\infty}^{\infty}du\;\exp[-u^{2}]\frac{1}{\sqrt{(1+\frac{u^{2}}{u_{x}^{2}})(1+\frac{u^{2}}{u_{y}^{2}})}}$$

   (2.14)

   Hence the luminosity correction factor is

   $$R_{L}\equiv\frac{\cal L}{\cal L_{0}}=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}du\;\exp[-u^{2}]\frac{1}{\sqrt{(1+\frac{u^{2}}{u_{x}^{2}})(1+\frac{u^{2}}{u_{y}^{2}})}}$$

   (2.15)

   which agrees with the expression Eq.(2.7) in [4].

3. 3.

   Only the crossing angle, no hourglass effect

   In the limit that $\beta_{x}^{*},\beta_{y}^{*}\gg\sigma_{z}^{*}$, there is very little variation in the transverse sizes across the bunch length. In this limit $(t_{x}^{2},t_{y}^{2}\gg 1$. The dominant contributions to the integral come from the regions close to $t=0$ because of the exponential factor. In this limit we can assume $(t/t_{x})^{2},(t/t_{y})^{2}\ll 1$ and the luminosity is

   	$\displaystyle{\cal L}$	$\displaystyle=$	$\displaystyle\frac{f_{rev}N_{+}N_{-}C_{C}}{4\pi^{3/2}\sigma_{x}^{*}\sigma_{y}^{*}}\int_{-\infty}^{\infty}du\;\exp[-C_{C}^{2}u^{2}(1+T_{C}^{2}\frac{\sigma_{s}^{2}}{\sigma_{x}^{*2}})]$		(2.16)
   		$\displaystyle=$	$\displaystyle\frac{f_{rev}N_{+}N_{-}C_{C}}{4\pi^{3/2}\sigma_{x}^{*}\sigma_{y}^{*}}\frac{\sqrt{\pi}}{C_{C}\sqrt{(1+T_{C}^{2}\frac{\sigma_{s}^{2}}{\sigma_{x}^{*2}})}}$

   Hence the luminosity correction factor is

   $$R_{L}=\frac{1}{\sqrt{(1+T_{C}^{2}\frac{\sigma_{s}^{2}}{\sigma_{x}^{*2}})}}$$

   (2.17)

   This is the standard correction factor for the crossing angle.

4. 4.

   Flat bunch

   In this limit, $\sigma_{x}^{*}\gg\sigma_{y}^{*}$. Since the equilibrium emittances obey $\epsilon_{x}\gg\epsilon_{y}$, this is easily satisfied if $\beta_{x}^{*}\gg\beta_{y}^{*}$, which typically is the case. In this limit we drop all terms with $u_{x}$ from Eq.(LABEL:eq:_hg_cross)

   $\displaystyle{\cal L}$

   $\displaystyle=$

   $\displaystyle\frac{f_{rev}N_{+}N_{-}C_{C}}{4\pi^{3/2}\sigma_{x}^{*}\sigma_{y}^{*}}\int_{-\infty}^{\infty}du\;\exp[-C_{C}^{2}u^{2}(1+T_{C}^{2}\frac{\sigma_{s}^{2}}{\sigma_{x}^{*2}})]\frac{1}{\sqrt{(1+\frac{u^{2}}{u_{y}^{2}})}}$

   (2.18)

   The luminosity correction factor is

   	$\displaystyle R_{flat}$	$\displaystyle=$	$\displaystyle\frac{C_{C}}{\sqrt{\pi}}\int_{-\infty}^{\infty}du\;\exp[-C_{C}^{2}u^{2}(1+T_{C}^{2}\frac{\sigma_{s}^{2}}{\sigma_{x}^{*2}})]\frac{1}{\sqrt{(1+\frac{u^{2}}{u_{y}^{2}})}}$		(2.19)
   		$\displaystyle=$	$\displaystyle\frac{C_{C}}{\sqrt{\pi}}u_{Y}\exp[\frac{1}{2}b^{2}u_{y}^{2}]K_{0}(\frac{1}{2}b^{2}u_{y}^{2})$		(2.20)

   where

   $$b^{2}=C_{C}^{2}[1+T_{C}^{2}\frac{\sigma_{z}^{2}}{\sigma_{x}^{*2}}]\geq 0$$

   (2.21)

   and $K_{0}$ is a Bessel function. There is a similar expression Eq. (2) in [5].

   In the absence of a crossing angle so that $C_{C}=1,T_{C}=0$, we have $b=1$ and Eq, 2.20 is the same as Eq. (2.12) in Furman.

The beam-beam potential for electrons interacting with a positron bunch which has a longitudinally Gaussian density is

	$\displaystyle U(x,y)$	$\displaystyle=$	$\displaystyle\frac{N_{+}r_{e}}{\gamma_{e}}\int\frac{\exp[-s^{2}/(2\sigma_{s}^{2})]}{\sqrt{2\pi}\sigma_{s}}\;ds\int_{0}^{\infty}dq\;\frac{1}{(2\sigma_{x}(s)^{2}+q)^{1/2}(2\sigma_{y}(s)^{2}+q)^{1/2}}$		(3.1)
			$\displaystyle\left\{1-\exp[-\frac{x^{2}}{2\sigma_{x}(s)^{2}+q}-\frac{y^{2}}{2\sigma_{y}(s)^{2}+q}]\right\}$

where the parameters are those of the positron bunch. The potential for electrons interacting with positrons at a full crossing angle $\theta_{C}$ can be found by replacing

$$x\to C_{C}x-S_{C}s,\;\;\;s\to C_{C}s+S_{C}x$$

(3.2)

The amplitude dependent beam-beam tune shifts can be obtained from the second derivatives of the potential as

$$\Delta\nu_{x}(x,y)=-\frac{\beta_{x}^{*}N_{+}r_{e}}{4\pi\gamma_{e}}\frac{\partial^{2}U}{\partial x^{2}},\;\;\;\Delta\nu_{y}(x,y)=-\frac{\beta_{y}^{*}N_{+}r_{e}}{4\pi\gamma_{e}}\frac{\partial^{2}U}{\partial y^{2}}$$

(3.3)

The beam-beam tune shift parameters are the values of the tune shifts at the origin, i.e.

$$\xi_{x}=\Delta\nu_{x}(0,0),\;\;\;\xi_{y}=\Delta\nu_{y}(0,0)$$

(3.4)

Substituting the rotated forms in Eq.(3.2) into the potential, taking the derivatives, evaluating the terms at the origin and leads to

	$\displaystyle\xi_{x}$	$\displaystyle=$	$\displaystyle\frac{\beta_{x}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\int_{0}^{\infty}\frac{ds}{\sqrt{2\pi}\sigma_{s}}\int_{0}^{\infty}dq\;2\frac{\exp[-\frac{s^{2}C_{C}^{2}}{(2\sigma_{s}^{2})}-\frac{(s^{2}S_{C}^{2})}{(q+2\sigma_{x}^{2}(s))}]}{\sqrt{(q+2\sigma_{x}^{2}(s))^{3}(q+2\sigma_{y}^{2}(s))}}$		(3.5)
			$\displaystyle\times C_{C}^{2}\left\{1+2s^{2}S_{C}^{2}[\frac{1}{\sigma_{s}^{2}}-\frac{1}{q+2\sigma_{x}^{2}(s)}]\right\}$
			$\displaystyle-\frac{\exp[-\frac{s^{2}C_{C}^{2}}{2\sigma_{s}^{2}}]\left(1-\exp[-\frac{s^{2}S_{C}^{2}}{q+2\sigma_{x}^{2}(s)}]\right)}{\sigma_{s}^{2}\sqrt{(q+2\sigma_{x}^{2}(s))(q+2\sigma_{y}^{2}(s))}}S_{C}^{2}\left[1-\frac{s^{2}C_{C}^{2}}{\sigma_{s}^{2}}\right]$
	$\displaystyle\xi_{y}$	$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\int_{-\infty}^{\infty}\frac{ds}{\sqrt{2\pi}\sigma_{s}}\int_{0}^{\infty}dq\frac{2\exp\left(-\frac{t^{2}S_{c}^{2}\sigma_{s}^{2}}{q+2\sigma_{x}^{2}}-\frac{t^{2}C_{C}^{2}}{2}\right)}{\left(q+2\sigma_{y}^{2}\right)\sqrt{\left(q+2\sigma_{x}^{2}\right)\left(q+2\sigma_{y}^{2}\right)}}$		(3.6)

The different integrations over $s$ cannot all be done analytically, so the 2D integrations have to be done numerically. Transform to dimensionless variables $(u,t)$ and define other dimensionless variables

	$\displaystyle t$	$\displaystyle=$	$\displaystyle\frac{s}{\sigma_{s}},\;\;\;t_{x}=\frac{\beta_{x}*}{\sigma_{s}},\;\;\;t_{y}=\frac{\beta_{y}*}{\sigma_{s}}$		(3.7)
	$\displaystyle u$	$\displaystyle=$	$\displaystyle\frac{2\sigma_{x}^{*,2}}{q+2\sigma_{x}^{*,2}}\;\;\;\Rightarrow q=2\sigma_{x}^{*,2}(\frac{1}{u}-1);\;\;\;0\leq u\leq 1$		(3.8)
		$\displaystyle\Rightarrow$	$\displaystyle\sigma_{x}(t)=\sigma_{x}^{*}\sqrt{1+\frac{s^{2}}{\beta_{x}^{*,2}}}=\sigma_{x}^{*}\sqrt{1+\frac{t^{2}}{t_{x}^{2}}}$		(3.9)
		$\displaystyle\Rightarrow$	$\displaystyle\sigma_{y}(t)=\sigma_{y}^{*}\sqrt{1+\frac{s^{2}}{\beta_{y}^{*,2}}}=\sigma_{y}^{*}\sqrt{1+\frac{t^{2}}{t_{y}^{2}}}$		(3.10)
	$\displaystyle r_{yx}$	$\displaystyle=$	$\displaystyle\frac{\sigma_{y}^{*2}}{\sigma_{x}^{*2}},\;\;\;r_{sx}=\frac{\sigma_{s}^{2}}{2\sigma_{x}^{*2}}$		(3.11)

The Jacobian of the transformation is $J(q,s;u,t)=\frac{2\sigma_{x}^{*,2}\sigma_{s}}{u^{2}}$. Carrying out the transformation leads to the equations for the general case

	$\displaystyle\xi_{x}$	$\displaystyle=$	$\displaystyle\frac{\beta_{x}^{*}N_{p}r_{e}}{2\pi\gamma_{e}}\left\{\frac{C_{C}^{2}}{(\sigma_{x}^{*,2})}\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi}}\;\exp[-\frac{C_{C}^{2}t^{2}}{2}]\int_{0}^{1}du\;\exp[-\frac{(r_{sx}S_{C}^{2}ut^{2})}{1+u(t/t_{x})^{2}}]\right.$		(3.12)
			$\displaystyle\times\left(1+2S_{C}^{2}t^{2}[1-r_{sx}\frac{u}{1+u(t/t_{x})^{2}}]\right)\left[\frac{1}{(1+u(t/t_{x})^{2})^{3}(1+[r_{yx}(1+(t/t_{y})^{2})-1]u)}\right]^{1/2}$
			$\displaystyle-\frac{S_{C}^{2}}{\sigma_{s}^{2}}\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi}}\;\exp[-\frac{C_{C}^{2}t^{2}}{2}]\left[1-C_{C}^{2}t^{2}\right]\int_{0}^{1}\frac{du}{u}\;\left(1-\exp[-\frac{r_{sx}S_{C}^{2}ut^{2}}{1+u(t/t_{y})^{2}}]\right)$
			$\displaystyle\left.\times\left[\frac{1}{(1+u(t/t_{x})^{2})(1+[r_{yx}(1+(t/t_{y})^{2})-1]u)}\right]^{1/2}\right\}$
	$\displaystyle\xi_{y}$	$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{p}r_{e}}{\pi\gamma_{e}}\frac{1}{(2\sigma_{x}^{*,2})}\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi}}\;\exp[-\frac{C_{C}^{2}t^{2}}{2}]\int_{0}^{1}du\;\exp\left[-\frac{r_{sx}S_{C}^{2}ut^{2}}{(1+u(t/t_{x})^{2})}\right]$		(3.13)
			$\displaystyle\times\left[\frac{1}{(1+u(t/t_{x})^{2})(1+[r_{yx}(1+(t/t_{y})^{2})-1]u)^{3}}\right]^{1/2}$

These expressions involve a finite range of integration (from $0\to 1$) over $u$ compared to the infinite range over $q$ in Eq.(3.5) and (3.6). In those equations, the convergence rate is poor and evaluate slowly while the double integrals evaluate quickly in the set Eq.(3.12) and (3.13) above. We note that the second set of terms in Eq.(3.12) proportional to $\frac{S_{C}^{2}}{\sigma_{s}^{2}}$ are typically very small in the collider applications considered below and can be ignored.

Limiting cases

1. 1.

   No crossing angle or hourglass effect

   In this case $S_{C}=0,C_{C}=1$, so

   	$\displaystyle\xi_{x}$	$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}t_{x}\frac{2}{(2\sigma_{x}^{*,2})}\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi}}\exp[-\frac{t_{x}^{2}t^{2}}{2}]\int_{0}^{1}du\;\left[\frac{1}{(1+(r_{yx}-1)u)}\right]^{1/2}$
   	$\displaystyle\xi_{y}$	$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{+}r_{e}}{\pi\gamma_{e}}\frac{t_{x}}{(2\sigma_{x}^{*,2})}\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi}}\exp[-\frac{t_{x}^{2}t^{2}}{2}]\int_{0}^{1}du\left[\frac{1}{(1+(r_{yx}-1)u)^{3}}\right]^{1/2}$

   Using Eqs.(A.1) and (A.2) in the Appendix, we find

   	$\displaystyle\xi_{x}$	$\displaystyle=$	$\displaystyle\frac{\beta_{x}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\frac{1}{\sigma_{x}^{*}(\sigma_{y}^{*}+\sigma_{x}^{*})}$		(3.14)
   	$\displaystyle\xi_{y}$	$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\frac{1}{\sigma_{y}^{*}(\sigma_{y}^{*}+\sigma_{x}^{*})}$		(3.15)

   These are the standard expressions for the tune shifts without a crossing angle or hourglass effects.

2. 2.

   Only a crossing angle

   We use the expressions Eq.(3.12) and (3.13), let $t_{x},t_{y}\to\infty$ and do the integration over $t$ first. Define shorthand variables

   $$a^{2}=r_{sx}S_{C}^{2},\;\;\;c^{2}=\frac{1}{2}C_{C}^{2}$$

   (3.16)

   We use the integration results in Eqs. (A.5) - (A.7) to obtain

   	$\displaystyle\xi_{x}$	$\displaystyle=$	$\displaystyle\frac{\beta_{x}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\frac{1}{4\sqrt{2}}\int_{0}^{1}du\;\left[\frac{1}{(1+[r_{yx}-1]u)}\right]^{1/2}\left\{\frac{2C_{C}^{2}}{(2\sigma_{x}^{*,2})}\frac{2c^{2}+2t_{x}^{2}S_{C}^{2}}{(c^{2}+a^{2}u)^{3/2}}\right.$		(3.17)
   			$\displaystyle\left.+\frac{S_{C}^{2}}{\sigma_{s}^{2}}\frac{1}{u}\frac{2a^{2}u}{(c^{2}+a^{2}u)^{3/2}}\right\}$
   		$\displaystyle=$	$\displaystyle\frac{\beta_{x}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\frac{1}{C_{C}^{3}}\frac{1}{[\sigma_{y}^{*}+\sqrt{\sigma_{x}^{*2}+T_{C}^{2}\sigma_{s}^{2}}]\sqrt{\sigma_{x}^{*2}+T_{C}^{2}\sigma_{s}^{2}}}$
   	$\displaystyle\xi_{y}$	$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\frac{1}{C_{C}}\int_{0}^{1)}du\frac{1}{\sqrt{(2(\sigma_{y}^{2}-\sigma_{x}^{2})u+1)^{3}[1+2(T_{C}^{2}\sigma_{s}^{2})u/(2\sigma_{x}^{2})]}}$		(3.18)
   		$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\frac{1}{C_{C}}\frac{1}{\sigma_{y}(\sigma_{y}+\sqrt{\sigma_{x}^{2}+T_{C}^{2}\sigma_{s}^{2}})}$

   These expressions can be obtained from the equations (3.14), (3.15) by replacing the transverse beam size $\sigma_{x}$ in the crossing plane by the effective beam size $\sqrt{\sigma_{x}^{2}+T_{C}^{2}\sigma_{s}^{2}})$ and including the factors $1/C_{C}^{3}$ in $\xi_{x}$, $1/C_{C}$ in $\xi_{y}$ which are $\sim 1$ for typical crossing angles.

3. 3.

   Only the hourglass, no crossing angle

   Setting $C_{C}=1,S_{C}=0,=1$ in the general forms Eq.(3.12), and (3.13)

   	$\displaystyle\xi_{x}\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!\frac{\beta_{x}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\left\{\frac{2}{(2\sigma_{x}^{*,2})}\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi}}\exp[-\frac{t^{2}}{2}]\right.$
   			$\displaystyle\left.\times\int_{0}^{1}du\;\left[\frac{1}{(1+ut^{2}/t_{x}^{2})^{3}(1+[r_{yx}(1+t^{2}/t_{y}^{2})-1]u)}\right]^{1/2}\right\}$
   	$\displaystyle\xi_{y}$	$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{p}r_{e}}{\pi\gamma_{e}}\frac{1}{(2\sigma_{x}^{*,2})}\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi}}\;\exp[-\frac{t^{2}}{2}]$
   			$\displaystyle\times\int_{0}^{1}du\;\left[\frac{1}{(1+u(t/t_{x})^{2})(1+[r_{yx}(1+(t/t_{y})^{2})-1]u)^{3}}\right]^{1/2}$

   Integrating over $u$ yields the expressions

   	$\displaystyle\xi_{x}$	$\displaystyle=$	$\displaystyle\frac{\beta_{x}^{*}N_{+}r_{e}}{\pi\gamma_{e}\sigma_{x}^{*,2}}\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi}}\exp[-t^{2}/2]\frac{1}{\sqrt{(1+\frac{t^{2}}{t_{x}^{2}})}\left[\sqrt{1+\frac{t^{2}}{t_{x}^{2}}}+\frac{\sigma_{y}^{*}}{\sigma_{x}^{*}}\sqrt{1+\frac{t^{2}}{t_{y}^{2}}}\right]}$		(3.19)
   	$\displaystyle\xi_{Y}$	$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{+}r_{e}}{2\pi\gamma_{e}\sigma_{y}^{*,2}}\int\frac{dt}{\sqrt{2\pi}}\exp[-t^{2}/2]\frac{1}{\sqrt{(1+\frac{t^{2}}{t_{y}^{2}})}\left[\sqrt{1+\frac{t^{2}}{t_{y}^{2}}}+\frac{\sigma_{x}^{*}}{\sigma_{y}^{*}}\sqrt{1+\frac{t^{2}}{t_{x}^{2}}}\right]}$		(3.20)

   These agree with Eq.(3.4) (evaluated at $s=0$) in[4].

4. 4.

   Flat beams
   Here we consider the limit $t_{x}\to\infty$ in the general expressions in Eqs (3.12), (3.13). Interchanging the integrations and we drop the second set of terms $\propto S_{C}^{2}$ in $\xi_{x}$ that are negligibly small

   	$\displaystyle\xi_{x}$	$\displaystyle=$	$\displaystyle\frac{\beta_{x}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\frac{2C_{C}^{2}}{(2\sigma_{x}^{*,2})}\int_{0}^{1}du\;\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi}}\exp[-t^{2}(\frac{1}{2}C_{C}^{2}+r_{sx}S_{C}^{2}u)]$		(3.21)
   			$\displaystyle\times\left(1+2S_{C}^{2}t^{2}[1-r_{sx}u]\right)\left[\frac{1}{(1+(r_{yx}-1)u+r_{yx}ut^{2}/t_{y}^{2})}\right]^{1/2}$
   	$\displaystyle\xi_{y}$	$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{+}r_{e}}{\pi\gamma_{e}}\frac{1}{(2\sigma_{x}^{*,2})}\int_{0}^{1}\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi}}\exp[-t^{2}(\frac{1}{2}C_{C}^{2}+r_{sx}S_{C}^{2}u)]$		(3.22)
   			$\displaystyle\times\left[\frac{1}{(1+(r_{yx}-1)u+r_{yx}ut^{2}/t_{y}^{2})}\right]^{3/2}$

   The integrations over $t$ can be done and expressed in terms of confluent hypergeometric function $U$ and the Bessel function $K_{0}$. Using integration results in the Appendix and identifying the coefficients in the equations (7) and (A.9)

   	$\displaystyle a^{2}$	$\displaystyle=$	$\displaystyle\frac{1}{2}C_{C}^{2}+r_{sx}S_{C}^{2}u,\;\;({\rm Note}\;\;u\geq 0),\;\;\;b=2S_{C}^{2}[1-r_{sx}u]$
   	$\displaystyle c$	$\displaystyle=$	$\displaystyle 1+(r_{yx}-1)u,\;\;\;d=r_{yx}u/t_{y}^{2}$

   leads to the expressions for the tune shifts in the flat beam limit as

   	$\displaystyle\xi_{x}$	$\displaystyle=$	$\displaystyle\frac{\beta_{x}^{*}N_{+}r_{e}}{2\pi\gamma_{e}}\frac{2C_{C}^{2}}{2\sigma_{x}^{*,2}\sqrt{2\pi}}\int_{0}^{1}du\;\frac{1}{4(\frac{1}{2}C_{C}^{2}+r_{sx}S_{C}^{2}u)\sqrt{r_{yx}u/t_{y}^{2}}}$
   			$\displaystyle\left\{(C_{C}^{2}+2r_{sx}S_{C}^{2}u)\exp[{\rm arg}]K_{0}({\rm arg})+\sqrt{\pi}2S_{C}^{2}[1-r_{sx}u]U(\frac{1}{2},0,2\;{\rm arg}))\right\}$
   	$\displaystyle\xi_{y}$	$\displaystyle=$	$\displaystyle\frac{\beta_{y}^{*}N_{+}r_{e}}{\pi\gamma_{e}}\frac{1}{2\sqrt{2}\sigma_{x}^{*,2}}\int_{0}^{1}du\;\frac{1}{2(1+(r_{yx}-1)u)\sqrt{r_{yx}u/t_{y}^{2}}}U(1/2,0,2\;{\rm arg})$
   	$\displaystyle{\rm arg}$	$\displaystyle=$	$\displaystyle\frac{(\frac{1}{2}C_{C}^{2}+r_{sx}S_{C}^{2}u)(1+(r_{yx}-1)u)}{2r_{yx}u/t_{y}^{2}}]$		(3.25)

   This in principle, leaves the integration over $u$ to be evaluated numerically. While the integrations over the hypergeometric function $U$ in both $\xi_{x},\xi_{y}$ converge rapidly for typical parameter values, the integration over the Bessel function $K_{o}$ in $\xi_{x}$ is poorly convergent. We found it more convenient to use the double integration in Eq.(3.21) to evaluate $\xi_{x}$ and the single integration in Eq.(LABEL:eq:_xiy_fl_2) to evaluate $\xi_{y}$. In the following sections, we have used the exact expressions for the luminosity and the beam-beam tune shifts for both the Fermi site filler and the FCC-ee but in both cases, the flat beam expressions are very good approximations.