# Restriction on the energy and luminosity of storage rings due to beamstrahlung

###### Abstract

The role of beamstrahlung in high-energy storage-ring colliders (SRCs) is examined. Particle loss due to the emission of single energetic beamstrahlung photons is shown to impose a fundamental limit on SRC luminosities at energies for head-on collisions and for crab-waist collisions. With beamstrahlung taken into account, we explore the viability of SRCs in the –500 range, which is of interest in the precision study of the Higgs boson. At , SRCs are found to be competitive with linear colliders; however, at –500, the attainable SRC luminosity would be a factor 15–25 smaller than desired.

###### pacs:

29.20^{†}

^{†}preprint: BINP 2012-5

The ATLAS and CMS experiments at the LHC recently reported higgsATLAS ; higgsCMS an excess of events at , which may be evidence for the long-sought Higgs boson. The precision study of the Higgs boson’s properties would require the construction of an energy- and luminosity-frontier collider Aarons .

The LEP collider at CERN is generally considered to have been the last energy-frontier storage-ring collider (SRC) due to synchrotron-radiation energy losses, which are proportional to . Linear colliders (LC) are free from this limitation and allow multi- energies to be reached. Two LC projects are in advanced stages of development: the ILC ILC and the –3000 CLIC CLIC .

Nevertheless, several proposals Zim ; Oide for a SRC for the study of the Higgs boson in have recently been put forward Sen . Lower cost and reliance on firmly established technologies are cited as these projects’ advantages over an LC. Moreover, it has been proposed that a SRC can provide superior luminosity, and that the “crab waist” collision scheme crab ; SuperB allows them to exceed the ILC and CLIC luminosities even at –500 . Parameters of the recently proposed SRCs are summarized in Table 1.

The present paper examines the role of beamstrahlung, i.e., synchrotron radiation in the field of the opposing beam, in high-energy SRCs. First discussed in Rees , beamstrahlung has been well-studied only in the LC case Chen . As we shall see, at energy-frontier SRCs beamstrahlung determines the beam lifetime through the emission of single photons in the tail of the beamstrahlung spectra, thus severely limiting the luminosity.

LEP | LEP3 | DLEP | STR1 | STR2 | STR3 | STR4 | STR5 | STR6 | |

cr-w | cr-w | cr-w | cr-w | ||||||

, | 209 | 240 | 240 | 240 | 240 | 240 | 400 | 400 | 500 |

Circumference, | 27 | 27 | 53 | 40 | 60 | 40 | 40 | 60 | 80 |

Beam current, mA | 4 | 7.2 | 14.4 | 14.5 | 23 | 14.7 | 1.5 | 2.7 | 1.55 |

Bunches/beam | 4 | 3 | 60 | 20 | 49 | 15 | 1 | 1.4 | 2.2 |

5.8 | 13.5 | 2.6 | 6 | 6 | 8.3 | 12.5 | 25. | 11.7 | |

, | 16 | 3 | 1.5 | 3 | 3 | 1.9 | 1.3 | 1.4 | 1.9 |

/, | 48/0.25 | 20/0.15 | 5/0.05 | 23.3/0.09 | 24.6/0.09 | 3/0.011 | 2/0.011 | 3.2/0.017 | 3.4/0.013 |

/, | 1500/50 | 150/1.2 | 200/2 | 80/2.5 | 80/2.5 | 26/0.25 | 20/0.2 | 30/0.32 | 34/0.26 |

/, | 270/3.5 | 54/0.42 | 32/0.32 | 43/0.47 | 44/0.47 | 8.8/0.05 | 6.3/0.047 | 9.8/0.074 | 10.7/0.06 |

SR power, MW | 22 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 |

Energy loss/turn, | 3.4 | 7 | 3.47 | 3.42 | 2.15 | 3.42 | 33.9 | 18.5 | 32.45 |

, | 0.013 | 1.3 | 1.6 | 1.7 | 2.7 | 17.6 | 4 | 7 | 2.2 |

, | 0.09 | 6.3 | 4.2 | 3.5 | 3.4 | 38 | 194 | 232 | 91 |

/electron | 0.09 | 1.1 | 0.37 | 0.61 | 0.6 | 4.2 | 8.7 | 11.3 | 4.8 |

lifetime([email protected]), s (Eq. 4) | 0.02 | 0.3 | 0.2 | 0.4 | 0.005 | 0.001 | 0.0005 | 0.005 | |

, | 0.013 | 0.2 | 0.4 | 0.5 | 0.8 | 0.46 | 0.02 | 0.03 | 0.024 |

At SRCs the particles that lose a certain fraction of their energy in a beam collision leave the beam; this fraction is typically around 0.01 (0.012 at LEP) and is known as the ring’s energy acceptance. Beamstrahlung was negligible at all previously built SRCs. Its importance considerably increases with energy. Table 1 lists the beamstrahlung characteristics of the newly proposed SRCs assuming a 1% energy acceptance: the critical photon energy for the maximum beam field , the average number of beamstrahlung photons per electron per beam crossing , and the beamstrahlung-driven beam lifetime. Please note that once beamstrahlung is taken into account, the beam lifetime drops to unacceptable values, from a fraction of a second to as low as a few revolution periods.

At the SRCs considered in Table 1, the beam lifetime due to the unavoidable radiative Bhabha scattering is 10 minutes or longer. One would therefore want the beam lifetime due to beamstrahlung to be at least 30 minutes. The simplest (but not optimum) way to suppress beamstrahlung is to decrease the number of particles per bunch with a simultaneous increase in the number of colliding bunches. As explained below, should be reduced to . Thus, beamstrahlung causes a great drop in luminosity, especially at crab-waist SRCs: compare the proposed and corrected (as suggested above) rows in Table 1.

To achieve a reasonable beam lifetime, one must make small the number of beamstrahlung photons with energies greater than the threshold energy that causes the electron to leave the beam. These photons belong to the high-energy tail of the beamstrahlung spectrum and have energies much greater than the critical energy. It will be shown below that the beam lifetime is determined by such single high-energy beamstrahlung photons, not by the energy spread due to the emission of multiple low-energy photons.

The critical energy for synchrotron radiation pdg

(1) |

where is the bending radius and . The spectrum of photons per unit length with energy well above the critical energy pdg

(2) |

where , . To evaluate the integral of this spectrum from the threshold energy to note that the minimum value of , the exponent decreases rapidly, and so one can integrate only the exponent and use the minimum value of outside the exponent. After integration and substitution of from Eq. 1, we obtain the number of photons emitted on the collision length with energy :

(3) |

where is the classical radius of the electron.

The regions of the beam where the field strength is the greatest contribute the most to the emission of the highest-energy photons. We need to find the critical energy for this field and the bunch size that yields an acceptable rate of beamstrahlung particle loss. The collision length for head-on and for crab-waist collisions. In the transverse direction, we can assume that the electron crosses the region with the strongest field with a 10% probability. The average number of beam collisions experienced by an electron before it leaves the beam can be estimated from , where is given by Eq. 3. Thus, and the beam lifetime due to beamstrahlung

(4) |

Assuming , , , and a ring circumference of 50, from Eqs. 3 and 4 we get

(5) |

The accuracy of this expression is quite good for any SRC because it depends on the values in front of the exponent in Eq. 4 only logarithmically.

Let us express the critical energy via the beam parameters. In beam collisions, the electrical and magnetic forces are equal in magnitude and act on the particles in the oncoming beam in the same direction. Thus, we can use the effective doubled magnetic field. The maximum effective field for flat Gaussian beams . The bending radius Substituting to Eq. 1, we find

(6) |

Combined with Eq. 5, this imposes a restriction on the beam parameters,

(7) |

This formula is the basis for the following discussion.

For Gaussian beams, the average number of beamstrahlung photons per electron for head-on collisions Chen , their average energy , and the average critical energy ; hence, . Above, we considered the maximum field, i.e., was equal to . Then, for the condition in Eq. 7 we obtain

(8) |

(9) |

For crab-waist collisions, is the same while the interaction length is shorter, instead of ; therefore, the number of photons is proportionally smaller.

So, when is large enough is determined by the rare photons with energies , a factor greater than .

The beam energy spread due to beamstrahlung can be estimated as follows. In the general case Wied ,

(10) |

In our case, the damping time (due to radiation in bending magnets) , , and Sands , which gives

(11) |

where is the energy loss per revolution and and are given by Eqs. 8 and 9. Taking the typical bunch length , GeV, and we get an estimate for the energy spread due to beamstrahlung (under the condition in Eq. 7) .

The beam energy spread due to synchrotron radiation (SR) in the bending magnets Wied :

(12) |

where is the partition number. For the projects in Table 1, due to SR varies between 0.17% and 0.24%. For , , one gets . For the given example, the beamstrahlung energy spread becomes larger than that due to SR in rings at .

The energy spread due to beamstrahlung contributes to the beam lifetime (if the lifetime is large enough) when the energy acceptance ; with [11] taken into account, this yields . For the typical , mm, we get , which is much larger than the realistic storage-ring energy acceptance 0.01–0.03. Therefore, the beam energy spread due to beamstrahlung never causes the beam lifetime; the lifetime is always determined by the emission of single photons.

In the “crab waist” collision scheme crab ; SuperB , the beams collide at an angle . The crab-waist scheme allows for higher luminosity when it is restricted only by the tune shift, characterized by the beam-beam strength parameter. One should work at a beam-beam strength parameter smaller than some threshold value, for high-energy SRCs Zim .

In head-on collisions, the vertical beam-beam strength parameter (further ”beam-beam parameter”) Wied

(13) |

In the crab-waist scheme crab ,

(14) |

The luminosity in head-on collisions

(15) |

in crab-waist collisions,

(16) |

In the crab-waist scheme, one can make , which enhances the luminosity by a factor of compared to head-on collisions. For example, at the proposed Italian SuperB factory SuperB this enhancement would be a factor of 20–30.

Using Eqs. 13 and 14 and the restriction in Eq. 7, we find the minimum beam energy when beamstrahlung becomes important. For head-on collisions,

(17) |

for crab-waist collisions,

(18) |

In the crab-waist scheme, beamstrahlung becomes important at much lower energies because . This can be understood from Eq. 14: smaller corresponds to denser beams, leading to a higher beamstrahlung rate.

Examples: a) SuperB SuperB : crab waist, , , , . Then, , i.e., beamstrahlung is not important. b) The STR3 project (Table 1): crab crossing, , , , . Then, , a factor 7 lower than ; thus, beamstrahlung is very important. c) For projects STR1 and STR2: head-on, , , , ; , beamstrahlung is important.

We have shown that beamstrahlung restricts the maximum value of and becomes important at energies for storage rings with head-on collisions; when the crab-waist scheme is employed, this changes to the more strict . All newly proposed projects listed in Table 1 are affected as they have .

Now, let us find the luminosity when it is restricted both by beam-beam strength parameter and beamstrahlung. For head-on collisions,

(19) |

and . This can be rewritten as

(20) |

Thus, in the beamstrahlung-dominated regime the luminosity is proportional to the bunch length, and its maximum value is determined by the beam-beam strength parameter. Together, these equations give

(21) |

(22) |

Similarly, for the crab-waist collision scheme,

(23) |

and . Substituting, we obtain

(24) |

The corresponding solutions are

(25) |

(26) |

We have obtained a very important result: in the beamstrahlung-dominated regime, the luminosities attainable in crab-waist and head-on collisions are practically the same. The gain from using the crab-waist scheme is only a factor of , contrary to the low-energy case, where the gain may be greater than one order of magnitude. For this reason, from this point on we will consider only the case of head-on collisions.

From the above considerations, one can find the ratio of the luminosities with and without taking beamstrahlung into account: it is equal to for head-on collisions and for crab-waist collisions and scales as for .

LEP | LEP3 | DLEP | STR1 | STR2 | STR3 | STR4 | STR5 | STR6 | |

cr-w | cr-w | cr-w | cr-w | ||||||

, | 209 | 240 | 240 | 240 | 240 | 240 | 400 | 400 | 500 |

Circumference, | 27 | 27 | 53 | 40 | 60 | 40 | 40 | 60 | 80 |

Bunches/beam | 70 | 24 | 53 | 240 | 36 | 45 | 31 | ||

33 | 5.9 | 2.35 | 3.9 | 4. | 0.4 | 0.34 | 0.6 | 0.65 | |

, | 8.1 | 8.1 | 5.7 | 6.9 | 6.9 | 3.4 | 6.7 | 7.8 | 9.6 |

, | 1.4 | 1.1 | 0.53 | 0.78 | 0.78 | 0.19 | 0.27 | 0.36 | 0.35 |

, | 0.47 | 0.31 | 0.89 | 0.55 | 0.83 | 1.1 | 0.12 | 0.16 | 0.087 |

In practical units,

(27) |

For example, for , , and the vertical emittances from Table 1 ( to 0.15), we get to 6.4.

According to Eq. 21, the maximum luminosity at high-energy SRCs with beamstrahlung taken into account

(28) |

where is the hourglass loss factor, is the collision rate, the average ring radius, and the number of bunches in the beam.

The energy loss by one electron in a circular orbit of radius pdg , then the power radiated by the two beams in the ring

(29) |

Substituting from Eq. 29 to Eq. 28, we obtain

(30) |

or, in practical units,

(31) |

Once the vertical emittance is given as an input parameter, we find the luminosity and the optimum bunch length by applying Eq. 27. Beamstrahlung and the beam-beam strength parameter determine only the combination ; additional technical arguments are needed to find and separately. When they are fixed, the optimal number of bunches is found from the total SR power, Eq. 29.

In Table 2, we present the luminosities and beam parameters for the rings listed in Table 1 after beamstrahlung is taken into account. The following assumptions are made: SR power MW, , , , ; the values of , and are taken from Table 1.

Comparing Tables 1 and 2, one can see that at taking beamstrahlung into account lowers the luminosities at storage-ring colliders with crab-waist collisions by a factor of 15. Nevertheless, these luminosities are comparable to those at the ILC, – at ILCinterim . However, at the ILC can achieve –, which is a factor 15–25 greater than the luminosities achievable at storage rings.

In conclusion, we have shown that the beamstrahlung phenomenon must be properly taken into account in the design and optimization of high-energy storage rings colliders (SRC). We have demonstrated that beamstrahlung suppresses the luminosities as at energies for head-on collisions and for crab-waist collisions. Beamstrahlung makes the luminosities attainable in head-on and crab-waist collisions approximately equal above these threshold energies. At –, beamstrahlung lowers the luminosity of crab-waist rings by a factor of 15–40. Some increase in SRC luminosities can be achieved at rings with larger radius, larger energy acceptance, and smaller beam vertical emittance.

We also conclude that the luminosities attainable at storage rings (at one interaction point) and linear colliders are comparable at . However, at storage-ring luminosities are smaller by a factor of 15–25. Linear colliders remain the most promising instrument for energies .

This work was supported by Russian Ministry of Education and Science.

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