Single kick approximations for beam-beam deflections

In early beam-beam simulations the dependence longitudinal position has been ignored in calculating beam-beam force exerted on a particle. The kick calculated as if the particle is located at the center the bunch and collides with all of the opposing bunch the interaction point (IP). As a result of this simplifie approach, the strengths of kicks are the same for a par in the middle of the bunch and the tail of the bunch as lo as those two particles have the same transverse coord Clearly, this approach has only limited validity. For today’s high luminosity storage ring, where b can be very small at the IP, this simplified approach must replaced by a method which includes longitudinal effe [1]. Such longitudinal effects can be divided into tw parts: the transverse kick should be influenced by longitudinal position of the particle and there should a longitudinal kick appearing as an energy change. First, we will review a synchro-beam map developed Hirataet al. In this introductory section we emphasize t physics of beam-beam interaction rather than the pu mathematical manipulation of formulas using Hamilton formulation. Then, we will apply the map to infinitesimal stron slices of a longitudinally Gaussian distributed stro beam. With proper approximation, we will make drama simplification of the formulas. As an application of the general theory developed h we will further customize the method to round Gauss beams. The integrations over infinitesimal slices carried out with approximations and a point interact formula is obtained. Finally, to test the accuracy of our result, a simulat has been carried out. Comparisons with discrete s methods using the synchro-beam map suggested by H et al. are made with various levels of approximatio developed in this paper.


I. INTRODUCTION
In early beam-beam simulations the dependence on longitudinal position has been ignored in calculating the beam-beam force exerted on a particle. The kick is calculated as if the particle is located at the center of the bunch and collides with all of the opposing bunch at the interaction point (IP). As a result of this simplified approach, the strengths of kicks are the same for a particle in the middle of the bunch and the tail of the bunch as long as those two particles have the same transverse coordinate. Clearly, this approach has only limited validity.
For today's high luminosity storage ring, where b can be very small at the IP, this simplified approach must be replaced by a method which includes longitudinal effects [1]. Such longitudinal effects can be divided into two parts: the transverse kick should be influenced by the longitudinal position of the particle and there should be a longitudinal kick appearing as an energy change.
First, we will review a synchro-beam map developed by Hirata et al. In this introductory section we emphasize the physics of beam-beam interaction rather than the purely mathematical manipulation of formulas using Hamiltonian formulation.
Then, we will apply the map to infinitesimal strong slices of a longitudinally Gaussian distributed strong beam. With proper approximation, we will make dramatic simplification of the formulas.
As an application of the general theory developed here, we will further customize the method to round Gaussian beams. The integrations over infinitesimal slices are carried out with approximations and a point interaction formula is obtained.
Finally, to test the accuracy of our result, a simulation has been carried out. Comparisons with discrete slice methods using the synchro-beam map suggested by Hirata et al. are made with various levels of approximations developed in this paper.

A. Synchro-beam map by Hirata et al.
Here, we will review the method called synchro-beam map, developed by Hirata et al. [2]. Before we go any farther, we should state clearly that throughout this paper we use weak-strong formalism. Thus, the opposing strong beam is assumed to be unaffected by the interaction.
Suppose that there is a test particle located at ͑x, p x , y, p y , z, e͒, where p x dx͞ds, p y dy͞ds, e p͞p 0 , and a strong slice is located at z ‫ء‬ when the centers of both bunches are at IP. As can be seen from Fig. 1, in general, the test particle and the slice do not collide at this time. In this particular case, they will collide some time later. After dt ͑z 2 z ‫ء‬ ͒͞2c (seen in Fig. 2), they will collide at a collision point (CP). Ignoring synchrotron oscillations of the particle temporarily 1 (i.e., the particle moves at uniform speed c in the longitudinal direction), the transverse coordinates of the test particle at the CP will be Note that in the synchro-beam interaction, the size (or shape) of the strong bunch also changes according to the Twiss parameters at IP. It is easy to verify that the transformation in Eq. (1) is induced by where a new notation : D : x is a Poisson bracket ͓D, x͔.
Since this transformation is generated by continuous Lie transformation, it is guaranteed to be canonical. We FIG. 1. A test particle and a strong slice under consideration when the center of the bunch and its opposing bunch are at IP. In this figure, suppose that the bunch containing the test particle advances to the right.
shall obtain the new longitudinal coordinate under this transformation, The Poisson bracket ͓D, z͔ is easily calculated as " Thus, the second and higher orders vanish.
In the same manner, The second-order term is Hirata et al. proposed a synchro-beam mapping for a particle-slice interaction, which is generated by where n ‫ء‬ is the number of particles in the slice, Q ‫ء‬ represents a set of auxiliary parameters characterizing the transverse distribution function of the strong beam, and U is the electric scalar potential of the strong slice. In particular, for bi-Gaussian, Q ‫ء‬ is just s x and s y . To reflect the change in the strong beam due to translation of CP, Q ‫ء‬ is a function of the displacement S. Then, using this map, we can apply the beam-beam interaction based on the coordinate at the CP, It is obvious that the spatial coordinates X, Y , and Z remain unchanged under the transformation due to the commutations to the operator U͓X, Y; Q ‫ء‬ ͑ ͑ ͑S͑Z, z ‫ء‬ ͒͒ ͒ ͔͒. However, the momenta are changed.
Let us define where, following Hirata et al., we have defined It is clear that the second or higher order of : U : does not contribute since the f's and g are functions of X, Y, and Z only.
Since U depends on Z implicitly, we need to take special care to obtain g: For the last equality, we have used Eq. (2).
After we apply the kicks, we transfer the test particle back to the original z. We summarize the final result of synchro-beam interaction by a single slice below: The same calculation is performed slice after slice to obtain the total effect due to the strong bunch.

B. Synchro-beam map applied to infinitesimal slices
Here, the synchro-beam map is applied to a discrete number of slices. Suppose there are L discrete slices of opposing bunch. Thus, we apply Eq. (11) L times in the order opposite to the physical order; x new exp͑2n ‫1ء‬ : U 1 :͒ exp͑2n ‫2ء‬ : U 2 :͒ · · · exp͑2n ‫ء‬L : U L :͒x where x represents ͑x, y, z, p x , p y , e͒.
The second-order term, such as : U k :: This is a small dimensionless value, even after multiplication by the total number of particles in a strong bunch N. As a good approximation, we will ignore the second or higher terms. Thus, an incoming particle is influenced by strong slices directly and independently. Then, the expression simplifies to The slices considered in the previous subsection are made infinitesimal and integrated to obtain the effects given by the entire opposing bunch.
Assuming that the opposing beam is Gaussian distributed in longitudinal direction, we can convert the summation into an integration as Using this expression, Eq. (18) can be converted to This simple-looking equation is the core of this entire paper, and all following sections are devoted to instances of its evaluation. However, this equation cannot be evaluated easily. Instead, we transform x into X first and apply the beam-beam interaction, Then, we transform X new back to the IP to obtain x new .

II. APPLICATION TO ROUND GAUSSIAN BEAM A. Introduction
Now, we assume that a bunch and its opposing strong bunch are round and have transverse Gaussian distributions. Then, the "potential" of the beam can be expressed [3] as where S 11 ͑S͒ is the adjusted beam size at the corresponding CP. Ignoring the effects of beam-beam interaction, the region around IP is treated as drift space. Thus, where a 0 , b 0 , and g 0 are Twiss parameters at the IP.
To shorten the writing, we drop the subscript 0 in further discussion. And, to avoid the confusion between b͑s͒ and b 0 , we explicitly address b͑s͒. Then, it is a good approximation to take S 11 ͑s͒ e 0 b͑s͒ e 0 ͑b 2 2as 1 gs 2 ͒ .
To obtain f or g we use Eq. (13), To integrate this, we change the variable u to Then, In the same manner, Incorporating this into Eq. (21), Since the Poisson bracket Thus, transforming back to the original coordinate, we obtain where we defined As for the transverse spatial coordinate r, as expected. Transforming back to the original coordinate, In terms of the original coordinates, where we defined (37) As for z, Thus, Note that this is always true in synchro-beam map. In other words, the longitudinal coordinate is invariant under the synchro-beam map. Finally, where we defined Now, going back to the original coordinate system, This implies

B. Detailed computation
Inserting r 1 p r S in place of R in Eq. (32) and in place of f in Eq. (27), we obtain an expression Now, changing the variable of the integral from z ‫ء‬ to S, according to Eq. (2), The integration cannot be carried out easily, mainly due to the S in the denominator of the exponential. However, if some conditions are met, we can truncate the high power of S. For our purpose, we will keep only the terms up to the second order of S. The validity of this truncation will be investigated numerically in a later section. To the second order, the integral can be carried out easily, Note that the factor e z 2 ͞2s 2 z , which is in all the integrals above, will cancel e 2͑z 2 ͞2s 2 z ͒ outside the original integral. To calculate Eq. (48), we need to expand the integrand in S, keep up to the second order, and integrate it. To make the writing short and concise, we define We expand the exponential into an infinite series When r ¿ jzp r j, with the help of the binomial expansion, we obtain ͑r 1 Sp r ͒ n r n 1 nr n21 Sp r 1 Now, we expand the sum over m up to S 2 , and take the power k of this. The result is Combining these two results and keeping only up to the second order in s, Before integrating, we should perform summation first. The generic form of the sum can be written as For Now, resetting the starting point of the index k to 0, X k1 ͑21͒ k kj k k!͑2e 0 b͒ k 2 j 2e 0 bX k0 (62) With these ingredients, we can sum f f over k. After a little algebra, we obtain Now, one might ask if some of the denominators will cause singularity at r 0. This is not the case. Those singular terms cancel each other and only the positive power of r remains after the expansion of the exponentials. In fact, when r 0,   Now, let us move to the next column marked first order. Now, the quantities are substantially smaller; however, the quantity minimizes at the 1-slice method. This implies that the first-order method is not a bad approximation to synchro-beam map and the first-order method resembles the 1-slice method most. The latter further implies that we should make better approximation to include slice effects. Hence, the first-order approximation is still insufficient to describe the beam-beam interaction, in general.
Finally, we look at the last column, where the quantities are taken with respect to the second-order method. The quantity has been reduced further and now minimizes at the 5-slice method. This implies that the approximation using up to the second order includes not only the benefit of synchro-beam map but also the benefit of slice effects. The latter implication is backed more strongly by the fact that the quantities remain small even with the 7-and 9-slice methods.
Those who are discontented with the accuracy of the 5-slice method are encouraged to derive third-or even higher-order corrections to our calculation. Since all the necessary techniques to derive such corrections are shown here, it would be just tedious algebra.
Since a is 0 at IP for many cases, one can simplify the formula derived here considerably and the reduction would further facilitate the derivation of higher-order correction, if necessary.
This new method has the advantage of being a point interaction or a single kick. It is not necessary to propagate back and forth between IP and CPs to apply beam-beam interaction.
ACKNOWLEDGMENT I thank my thesis advisor Richard Talman for numerous suggestions and corrections.