Monochromatization of e + e − colliders with a large crossing angle
aa r X i v : . [ phy s i c s . acc - ph ] S e p Monochromatization of e + e − colliders with a large crossing angle V. I. Telnov ∗ Budker Institute of Nuclear Physics SB RAS, Novosibirsk, 630090, RussiaNovosibirsk State University, Novosibirsk, 630090, Russia (Dated: 31 August 2020)The relative center-of-mass energy spread σ W /W at e + e − colliders is O (10 − ), which is muchlarger than the widths of narrow resonances produced in the s -channel in e + e − collisions. Thisfact greatly lowers the resonance production rates of the J/ψ , ψ (2 S ), Υ (1 S ), Υ (2 S ) and Υ (3 S )mesons and makes it extremely difficult to observe resonance production of the Higgs boson. Thus,a significant reduction of the center-of-mass energy spread would open up great opportunities inthe search for new physics in rare decays of narrow resonances, the search for new narrow stateswith small Γ e + e − , the study of true muonium and tauonium, etc. The existing monochromatizationscheme is only suitable for head-on collisions, while e + e − colliders with crossing angles (the so-calledCrab Waist collision scheme) can provide significantly higher luminosity due to reduced collisioneffects. In this paper, we propose a new monochromatization method for colliders with a largecrossing angle. The contribution of the beam energy spread to the spread of the center-of-massenergy is canceled by introducing an appropriate energy–angle correlation at the interaction point; σ W /W ∼ (3–5) × − appears possible. Limitations of the proposed method are also considered. PACS numbers: 29.20
The point-like nature of the electron and a narrow en-ergy spread are important advantages of e + e − colliders.The energy spread occurs due to synchrotron radiation(SR) in the rings as well as beamstrahlung at the IP(important for Z and H factories). The energy spreaddue to SR depends mainly on the beam energy E andmagnetic radius of the ring R , and only weakly on thespecific design of the collider. For uniform rings withoutdamping wigglers σ E /E ≈ . × − E [GeV] / p R [m].The energy spreads for some of the existing and planned e + e − rings are given in Table I. TABLE I. Beam energy at circular e + e − collidersVEPP-2000 BEPC-II SuperKEKB FCC-ee E , GeV 1 ∼ πR , km 0.024 0.24 3 100 σ E /E, − ∼ . ∼ . One can see that the invariant mass spread σ W /W =(1 / √ σ E /E ∼ (0.35–0.5) × − . This spreadis much greater than the widths of the narrow e + e − res-onances J/ψ , ψ (2 S ), Υ (1 S ), Υ (2 S ), Υ (3 S ) and the Higgsboson, see Table II. The resonance width Γ is the fullwidth at half maximum, so one should compare Γ /m and2.36 σ W /W ≈ (0.8–1.2) × − . TABLE II. Width of some narrow e + e − resonances J/ψ ψ (2 S ) Υ (1 S ) Υ (2 S ) Υ (3 S ) H (125) m , GeV /c /m, − . σ W / Γ ∼ ∼ ∼ ∼ ∼ ∼ One of the promising directions for particle physicsis the study of rare and forbidden processes sensitive to new physics. Therefore,
J/ψ and Υ factories witha narrow invariant-mass spread would be good candi-dates for future experimental facilities. In the case ofa large continuum background, the signal-to-noise ratio S/ √ B ∝ ( L int /σ W ) / √L int = √L int /σ W ); therefore, theintegrated luminosity required to observe a rare decayof a known resonance (or to observe a narrow resonancewith a very small Γ e + e − ) L int ∝ (1 /σ W ) . A 100-fold im-provement in monochromaticity for Υ -mesons would beequivalent to a luminosity increase by a factor of 100 =10000! In the absence of a background, monochromatiza-tion lowers the observable branching limit proportionallyto σ W .The first consideration of energy monochromatizationin e + e − collisions dates back to mid-1970s [1]. In the pro-posed scheme (Fig. 1), beams collide head-on and havea horizontal or vertical energy dispersion at the inter-action point (IP), opposite in sign for the e + and e − beams. As a result, the particles collide with oppositeenergy deviations, E + ∆ E and E − ∆ E , and their in-variant mass W ≈ √ E E ≈ E − (∆ E ) /E is veryclose to 2 E . This monochromatization scheme was con- E+dEE E+dEE−dE E−dEE
FIG. 1. Existing monochromatization scheme for head-on col-lisions [1]. sidered by many authors in 1980s–1990s [2–8] for use in c - τ and B -factory projects (i.e., in the energy range ofthe ψ and Υ resonances); however, none of the proposalswere implemented as they all resulted in an unaccept-able loss of luminosity. A new wave of interest in thistopic is connected with the FCC-ee and CEPC projects,where one of the processes of great interest is e + e − → H ,which has a tiny cross section and is observable only if σ W < ∼ Γ H [9, 10].The new generation of circular e + e − colliders(DAΦNE[11], SuperKEKB[13], c - τ [14], FCC-ee[15],CEPC[16]) rely on the so-called “crab-waist” collisionscheme [11, 12], where the beams collide at an angle θ c ≫ σ x /σ z . When the luminosity is limited by thetune shift, characterized by the beam-beam strength pa-rameter ξ y , the crab-waist collision scheme allows muchhigher luminosities to be reached compared to head-oncollisions. For the same beam currents, the maximum lu-minosity for head-on collisions L ∝ /σ z , where σ z is thebunch length, while for collisions at an angle L ∝ /β y ,where the vertical beta function β y ∼ σ x /θ c can be ∼ σ z ; as a result, the luminositycan be higher by the same factor. In existing designs, thecrossing angle θ c varies between 30 mrad (FCC-ee) and83 mrad (SuperKEKB). In what follows, we propose andexplore significant modifications to this collision schemeaimed at achieving monochromatization.First, let us consider the mass resolution in the un-modified collision scheme with a crossing angle, Fig 2.The invariant mass of the produced system θ θ E E
FIG. 2. Collisions with crossing angles. W = ( P + P ) = 2 m + 2( E E − ~p ~p ) ≈≈ E E (1 + cos( θ + θ )) . (1)Here, we neglect the terms of the relative order ( m/E ) and choose units so that c = 1. The contribution of thevertical angular spread is negligible in all practical cases.By differentiating this formula while assuming that theenergies and the angles are independent and setting θ = θ = θ c / E = E = E , we find the relative massspread (cid:16) σ W W (cid:17) = 12 (cid:16) σ E E (cid:17) + 12 sin θ c (1 + cos θ c ) σ θ , (2)where θ c is the beam crossing angle, σ E is the beam en-ergy spread, and σ θ is the beam angular spread at theIP, which is determined by the horizontal beam emit-tance ε x and beta function β ∗ x at the IP: σ θ = p ε x /β ∗ x .For head-on collisions, the second term vanishes, and themass resolution is determined solely by the beam energy d d θ θ E /2 /2 E + E θ E c c FIG. 3. Collisions with the energy-angle correlation. spread. In the aforementioned colliders with the crab-waist scheme, the contribution of beam energy spread isalso dominant.The presently proposed monochromatization methodis based on the fact that the invariant mass W dependson both the beam energies and their crossing angle. Thesecond term in Eq. 2 reflects the natural stochastic beamspread due to the horizontal beam emittance and cannotbe avoided; however, the first term can be suppressedvery significantly, as we shall demonstrate. For simplicityof further consideration, hereinafter we omit the secondterm (i.e., assume σ θ = 0) as it is independent of the firstand can be added back later. In the proposed method, weprovide the beams with an angular dispersion such thata beam particle arrives to the IP with a horizontal anglethat depends on its energy: the higher the energy, thelarger the angle, Fig. 3. We can choose such a dispersionthat when a particle coming from the left, with energy E + d E and angle θ = θ c / θ , collides with a particlecoming from the right, with the nominal (average) en-ergy E and angle θ c /
2, they produce the same invariantmass as two colliding particles that both have the nom-inal energies and angles, E and θ c /
2. From Eq. 1, weobtain the required condition:( E + d E ) E (1 + cos( θ c + d θ )) = E (1 + cos θ c ) . (3)In the linear approximation, this gives the required an-gular dispersion (the same for both particles)d θ i = 1 + cos θ c sin θ c d E i E . (4)In what follows, the exact dispersion relation (Eq. 3) willbe called “nonlinear”, while Eq. 4 will be referred to as“linear” dispersion. Taking the first derivative of W (Eq. 1) and substituting the linear dispersion from Eq. 4,one can confirm that the resulting variation of the invari-ant mass is indeed zero:d W = 2 E ((1 + cos θ c ) d E − E sin θ c d θ )++ 2 E ((1 + cos θ c ) d E − E sin θ c d θ ) = 0 . (5)Note that the proposed monochromatization methodworks even for unequal beam energies. For simplicity,in the rest of the paper we will usually assume E = E .If W were a product of two functions that depend, re-spectively, only on the parameters of the left or the rightcolliding particles, then the contribution of the beam en-ergy spreads σ E i to σ W could have been completely ze-roed using the nonlinear dispersion (Eq. 3). However, thecontributions of the two beams are not factorized due tothe cos( θ + θ ) term in Eq. 1. Therefore, a second-ordercontribution from the energy spread remains; below, wewill find it for both the linear and nonlinear dispersions.Since the first derivative of W is zero (Eq. 5), we mustuse the quadratic term of the Taylor series:d W = (1 / W ( E , E , θ , θ ) , (6)where W is given by Eq. 1 and d W = (cid:18) d E ∂∂E + d E ∂∂E + d θ ∂∂θ + d θ ∂∂θ (cid:19) W . (7) Then, in the resulting expression we replace d θ and d θ by d E and d E using Eq. 4. As a result, we get an ex-pression in the form d W/W = a ((d E ) + (d E ) )) + b (d E d E ). In the case of ideal nonlinear dispersion(Eq. 3), only the unfactorized term cos( θ + θ ) con-tributes to d W ; therefore, a = 0, and d W is describedby the term b (d E d E ), which depends on both beams.In the case of linear dispersion (Eq. 4), both terms con-tribute and, in addition to fluctuations, there is also asmall shift of the mean invariant mass, ∆ W .In the case of Gaussian beam energy distributions withr.m.s. spread σ E , we have (d E ) = σ E , σ (d E ) = √ σ E ,d E d E = 0, σ (d E d E ) = σ E , the mass spreads fromthe two beams must be summed quadratically. In addi-tion, the fluctuations of the first and the second termsare independent and must be summed quadratically.Finally, for ideal nonlinear dispersion (Eq. 3), the massspread due to the beam energy spread (cid:16) σ W W (cid:17) E = σ E E θ c sin θ c , ∆ WW = 0 . (8)For linear dispersion (Eq. 4) (cid:16) σ W W (cid:17) E = σ E E "(cid:18)
1+ 1 + cos θ c sin θ c (cid:19) + (cid:18) θ c sin θ c (cid:19) / , (9) ∆ WW = σ E E (cid:18) θ c sin θ c (cid:19) . (10)The total invariant mass spread is the sum of the resid-ual contribution of the energy spread (Eq. 8 or Eq. 9) andthe second term of Eq. 2, which is due to the angularspread: (cid:16) σ W W (cid:17) = (cid:16) σ W W (cid:17) E + 12 sin θ c (1 + cos θ c ) σ θ (11) ! " ! "$ ! "% &’() * !+, !+- !+$ !+. !" $$$ /01 ) (cid:1) ) /56/1 ) "% ,/01 (cid:1) /56/1 "9 /0:’(+;’&<+49/=2324 (cid:1) /56/1 "9 /0:’(+;’&<+4-/01 (cid:1) /56/1 "9 /0(+:’(+;’&<+4 FIG. 4. Monochromaticity of collisions vs collision angle.
These formulas have been verified by direct simulation.The dependence of the invariant mass spread on the colli-sion angle θ c is shown in Fig. 4. It can be seen that lineardispersion works almost as well as the best-case nonlineardispersion. Now, let us discuss some of the limitations ofthis monochromatization method.First, the required dispersion (Eq. 4) is unacceptablylarge at small collision angles. The large horizontal angu-lar spread at the IP requires very strong final quadrupolemagnets that are located in the places with high disper-sion, which leads to strong synchrotron radiation and, asa consequence, to an additional energy spread and dete-rioration of the horizontal emittance ε x . For the energyspread σ E /E = 0 . × − , the r.m.s. angular spread dueto the dispersion is σ θ,d = 1 . × − for sin θ c = 0 . . × − at sin θ c = 1. There are fewer prob-lems at large collision angles; however, W decreases as W ( θ c ) /W (0) = cos ( θ c / θ c = 0 . , . , . ,
1, respectively. Therefore,we take sin θ c = 0 . σ θ = p ε x /β ∗ x ,where β ∗ x is the value of the horizontal beta function atthe IP. From Fig. 4, we see that in order for the contri-butions of the residual energy spread and the horizontalangular spread (only due to emittance) to be equal, thelatter should be σ θ < ∼ − for sin θ c = 0 .
5. The horizon-tal emittance at the 7 (4) GeV KEK Super B factory is ε x =4.8 (3.3) × − m. Specialized synchrotron-radiationsources have (and are planning) ε x < − m at energies E = 3–6 GeV. The maximum value of β ∗ x is limited bythe distance between the IP and the final quadrupole,which is about 1 m. So, σ θ = p − / ∼ − canbe thought of as a feasible ultimate target that achieves σ W /W ≈ . × − , which is about 150 to 200 timesbetter than that at current and past e + e − storage rings.Let us consider further limitations. Below, we considerthe following effects: • the increase of ε x and σ E /E due to radiation inthe final focus quadrupoles; • the increase of ε x and σ E /E in the magnetic fieldof the detector; • the increase of ε x and σ E /E in the part of the ringwhere dispersion is created; • the beam-beam attraction, which influences thecollision angle.The difference in the horizontal positions of the parti-cles with energies E + d E and E is d x = D x d E/E ,where D x is the horizontal dispersion function. Syn-chrotron radiation in regions with D x = 0 leads to anincrease of the emittance [17]:∆ ε x = A Z H ( s ) | ρ | d s, A = 5548 √ r e γ α . (12)where α = e / ¯ hc ≈ / r e = e /mc is the classicalelectron radius, γ = E/mc . The dispersion invariant H ( s ) = β x D ′ x + 2 α x D x D ′ x + γ x D x , where β x , α x , γ x areoptical functions (Twist parameters), α x = − β x / , γ x =(1 + α x ) /β x .Synchrotron radiation in quadrupoles and the disper-sion sections (in addition to the ring magnets) also leadsto an increase in the energy spread:∆ (cid:16) σ E E (cid:17) = A Z d s | ρ | . (13)To understand the importance of these effects, let usestimate the equilibrium energy spread and horizontalemittance when they are caused only by these effects buttheir damping is provided by synchrotron radiation in therings with damping times τ E = 3 R / γ cr e , τ x = 2 τ E .For one interaction point, they are∆ (cid:16) σ E E (cid:17) = τ E A T rev Z d s | ρ | ≈ . r e γ R Z d s | ρ | (14)∆ ε x = τ x A T rev Z H ( s ) d s | ρ | ≈ . r e γ R Z H ( s ) d s | ρ | (15)Let us estimate the radiation effects in the finalquadrupole of length l q located at a distance F from theIP. The particles enter the quadrupole from the IP at an-gles given by Eq. 4 and exit parallel to its axis, therefore ρ ∼ θ c l q sin θ c d EE ; h d s | ρ | i ≈ . l q (cid:18) θ c sin θ c (cid:19) (cid:16) σ E E (cid:17) , (16) where the coefficient 1.7 is obtained numerically for F = l q after averaging over the energies (although these cal-culations are estimates, I keep the numerical coefficientswhere possible).Similarly, for F = l q we obtain H ( s ) d s | ρ | ≈ β x (cid:18) θ c sin θ c (cid:19) (cid:16) σ E E (cid:17) , (17) where β x is the horizontal beta function on the outsideof the quadrupole. Substituting to Eqs. 16, 17 and mul-tiplying by 2 to account for the quadrupoles from bothsides of the IP, we obtain a reasonable estimate of theadditional energy spread and emittance due to radiationin the final quadrupoles:∆ (cid:16) σ E E (cid:17) ≈ r e γ RF (cid:18) θ c sin θ c (cid:19) (cid:16) σ E E (cid:17) ∝ E √ R ; (18)∆ ε x = 170 r e γ Rβ x (cid:18) θ c sin θ c (cid:19) (cid:16) σ E E (cid:17) ∝ E √ R . (19)For example: for E = 5 GeV, R = 500 m, sin θ c = 0 . σ E /E = 0 . × − , F = β x = 1 m, we get ∆( σ E /E ) =3 . × − , ∆ ε x = 2 . × − m. Thus, the effect onthe energy spread is negligible, and the increase of thehorizontal emittance is two times smaller than the Su-perKEKB emittance.For FCC-ee with R ≈
15 km and E = 50 GeV, the en-ergy spread will still be acceptable but the emittance willbe 4 orders of magnitude greater than its design value,3 × − m. Unfortunately, this means that this methodof monochromatization does not work at high energies( E > ∼ z = F is the same as in the lastquadrupole. So, to evaluate the effects in the detector wecan compare the field in the quadrupole and the effectivetransverse field of the solenoid B s, ⊥ = B s sin ( θ c / E = E can befound from l q /ρ ≈ ( σ E /E )(1 + cos θ c ) / sin θ c , which gives B q ∼ . E [ GeV] /l q [m] T. For E = 5 GeV and l q = 1 m, B q ∼ .
03 T. The effective detector field for B s = 1 Tand sin θ c = 0 . B s, ⊥ ∼ .
25 T, which is 8 times greaterthan that in the quadrupoles. This is a very serious prob-lem because, as shown above, ∆ ε x due to radiation in thequadrupoles is already close to the acceptable limit for E = 5 GeV. Possible solutions: a) compensate the fieldof the detector solenoid on the axis with an antisolenoid(they are used in any case in storage rings for the re-moval of x - y coupling), leaving only a short free space( ∼ ±
10 cm) for detecting the produced particles, and atthe same time reduce B s . In this approach, the solenoidand quadrupole contributions would be comparable at B s ∼ . D x = 1 . F (1 + cos θ c ) / sin θ c at theentrance to the final focus quadrupoles (the factor 1.5is due to the difference in dispersions at the quadrupoleentrance and exit for F = l q ). In order for β ∗ x at theIP to be as large as possible ( ∼ F ), the β function atthe entrance to the quadrupole should also be β ∼ F (because F ≈ β ∗ x β ). The length of the dispersion sys-tem is determined by the increase of emittance due toradiation. We consider the scheme shown in Fig. 5 withexplanatory notes. FIG. 5. A scheme of the chromatic section used for estimates.
For this scheme, R H ( s ) d s / | ρ | ≈
20 ( D x ) /L β ,where L = 1 . l is the total length. Substituting intoEqs. 16, 17 and multiplying by 2 (both sides of the IP),we obtain the additional emittance due to radiation inthe dispersion sections (for β = F )∆ ε x ∼ r e γ RF L (cid:18) θ c sin θ c (cid:19) . (20)For E = 5 GeV, R = 500 m, sin θ c = 0 . F = 1 m,we get ∆ ε x ∼ − m at L = 230 m. It would be 2 timesshorter for F = 0 . θ c = 0 . Υ mesons. For the above example(2 E = 10 GeV, R = 500 m, sin θ c = 0 .
5) with the hori-zontal emittance limited by Eq. 19 one can have an angu-lar spread σ θ x ∼ p ε x /β ⋆x ∼ p × − / ∼ . × − ,which corresponds to the mass width σ W /W ∼ . × − . In this case the monochromaticity is determinedby the angular spread, see Fig. 4. Similar considerationsfor sin θ c = 0 . .
7) give σ W /W ∼ . . × − . Thesevalues of the monochromaticity (for sin θ c = 0 . − .
7) are70–110 times better than at the existing e + e − storagerings. According to Eq. 19 the horizontal emittance limitdue to radiation in final quadrupoles varies as E / √ R and it becomes insignificant for low energies. As noted,the existing and planned synchrotron radiation sourceswith E =3–6 GeV have emittances of ε x < − m,then the angular spread σ θ x ∼ p − / − and σ W /W ∼ . × − is in reach, that corresponds to a150-fold improvement in monochromaticity. In the en-ergy region of J/ Ψ and below, the intra-beam scatteringis important, and realistic numbers can be obtained onlyafter careful consideration. With decreasing energy, theemittance ε x and σ θ x increase, therefore the optimumcrossing angle will decrease, see Fig. 4.At high-luminosity e + e − factories, the number of par-ticles per bunch is large, N ≈ (5–10) × . The ques-tion arises: how does the collision angle change due tothe attraction of the beams? Simple estimates indicatethat this effect can be problematic. However, a detailedexamination unexpectedly shows that beam attractiondoes not affect the invariant mass of the colliding par-ticles. Indeed, let us consider relativistic particles withthe energy E that are attracted by an opposing relativis-tic beam that creates an electric field E and a magneticfield B ≈ E . At distance d s , the particle receives energyd E = e E sin θ c d s ≈ eB sin θ c d s and an additional angled θ ≈ ( e E cos θ c + eB ) d s/E ≈ eB (1 + cos θ c ) d s/E . Sub-stituting d E and d θ for this and similar opposite particlein Eq. 1, we show that d W = 0!A few words about the possible loss of luminositydue to monochromatization. The only difference com-pared to the SuperKEKB design would be a larger cross-ing angle, 500 mrad instead of 90 mrad. The luminosity L ∝ N ( N f ) / ( σ z σ y θ c ). For the same beams, an increaseof the crossing angle by a factor of 6 would means a lossof luminosity by the same factor. However, the collisioneffects would become weaker, and one can partially com-pensate for the loss by increasing N and decreasing σ z .Other limitations are present, and so the resulting lumi-nosity would be lower, perhaps be a factor of two or three.Such a decrease is acceptable because monochromatiza-tion would significantly increase the effective luminosity, ∝ /σ W when studying rare decays in presence of a largecontinuum background.In conclusion, a new method of monochromatization of e + e − collisions is being proposed, which works at largecrossing angles ( θ c > ∼ E < ∼
10 GeV.There are other problems, such as the influence of thedetector field, which can be overcome but require non-standard solutions. The achievable invariant-mass spreadis σ W /W ∼ (3–5) × − , which is about 100 times bet-ter than at the past and existing e + e − storage rings.There are two main directions in particle physics: higherenergies—or very high luminosities at relatively low en-ergies. Monochromatization is a very natural next stepin the development of the next generation of luminosity-frontier colliders. It can increase by several orders ofmagnitude the effective luminosity in the study of raredecays or looking for narrow states with a small Γ e + e − .The full potential of this method can be realized at thevery narrow Υ ( nS ) resonances as well as at lower ener-gies, where a lot of interesting physics is also present.This paper is just the beginning; the next step towardrealistic projects requires the efforts of accelerator de-signers.This work was supported by Russian Ministry of Ed-ucation and Science and RFBR-DFG Grant No 20-52-12056. ∗ [email protected][1] A. Renieri, “Possibility of Achieving Very High-EnergyResolution in electron-Positron Storage Rings,” Labora-tori Nazionali di Frascati Report No. LNF-75/6(R).[2] I. Y. Protopopov, A. N. Skrinsky and A. A. Zholents,“Energy monochromatization of particle interaction instorage rings,” IYF-79-06.[3] A. A. Avdienko et al. , “The Project of Modernizationof the VEPP-4 Storage Ring for Monochromatic Experi-ments in the Energy Range of J/ψ and Υ Mesons,” Proc.12th Intern. Conf. High Energy Accelerators, Fermilab, 1983, p. 186.[4] K. Wille and A. W. Chao, “Investigation of a Monochro-mator Scheme for SPEAR,” SLAC/AP-32 (1984).[5] M. Jowett, “Feasibility of a Monochromator Scheme inLEP,” CERN LEP Note 544, September (1985).[6] Yu. I. Alexahin, A. Dubrovin, A. A. Zholents, “Pro-posal on a Tau-Charm Factory with Monochromatiza-tion,” Proc. 2nd European Particle Accelerator Confer-ence, Nice, France, 12–16 June 1990, p. 398.[7] A. Zholents, “Polarized
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