On accuracy of central mass energy determination for - PowerPoint PPT Presentation

On accuracy of central mass energy determination for FCCee_z_202_nosol_13.seq A. Bogomyagkov Budker Institute of Nuclear Physics Novosibirsk FCC-ee polarization workshop October 2017 A. Bogomyagkov (BINP) FCC-ee c.m. energy 1 / 22

Introduction: different energies Π Circumference: Design energy: E 0 magnets fields � E ( s ) ds Average energy: � E � = Π Measured energy: E meas = f ( W ) function of spin tune Invariant mass: M (central mass energy) A. Bogomyagkov (BINP) FCC-ee c.m. energy 2 / 22

Introduction: spin precession frequency Ω 0 is revolution frequency. W is spin precession frequency. Gyromagnetic ratio: q = q 0 + q ′ = mc + q ′ . e 1 + q ′ W = 1 � � q 0 � � � B ⊥ � � γ + q ′ B ⊥ ( θ ) d θ = Ω 0 · 2 π q 0 � B ⊥ /γ � 1 + � γ � q ′ � � ≈ Ω 0 · , q 0 q ′ = g − 2 = 1 . 1596521859 · 10 − 3 ± 3 . 8 · 10 − 12 . q 0 2 � W � E [ MeV ] = 440 . 64843 ( 3 ) − 1 . Ω 0 A. Bogomyagkov (BINP) FCC-ee c.m. energy 3 / 22

Spin distribution width: synchrotron oscillations Synchrotron oscillations: δ = ∆ E / E 0 = a · cos( ω syn t ) . a 2 a 2 � � W = Ω 0 1 + ν 0 − α 0 ν 0 + Ω 0 ( ν 0 ( 1 − α 0 ) − α 0 ) sin( ω syn t ) + α 0 Ω 0 ν 0 2 cos( 2 ω syn t ) 2 FCCee_z_202_nosol_13 Spin precession frequency distribution shifts and becomes wider by a 2 α 0 ν 0 � � = − α 0 ν 0 σ 2 � W − Ω 0 ( 1 + ν 0 ) � 2 δ = − 2 · 10 − 12 = − Ω 0 ( 1 + ν 0 ) 1 + ν 0 1 + ν 0 ∆ E = − 2 · 10 − 14 E 0 A. Bogomyagkov (BINP) FCC-ee c.m. energy 4 / 22

Energy dependent momentum compaction Momentum compaction: α = α 0 + α 1 δ α 1 ¨ δ = − ω 2 syn δ − ω 2 δ 2 Synchrotron oscillations: syn α 0 α 0 σ 2 , � δ � = − α 1 δ 2 � = σ 2 � Average and RMS: � 1 − α 0 σ 2 − α 1 α 0 σ 2 � q ′ Average W : � W � δ = γ 0 Ω 0 q 0 � α 0 σ 2 � 1 − α 1 Average energy: � E � = E 0 � α 0 σ 2 − α 0 σ 2 � 1 − α 1 Measured energy: E meas = E 0 A. Bogomyagkov (BINP) FCC-ee c.m. energy 5 / 22

Energy dependent momentum compaction FCCee_z_202_nosol_13 E 0 = 45 . 6 GeV, α 0 = 1 . 5 · 10 − 5 , α 1 = − 9 . 8 · 10 − 6 , σ = 3 . 8 · 10 − 4 � E � − E meas = α 0 σ 2 = 2 · 10 − 12 E 0 � E � − E 0 = − α 1 σ 2 = 1 · 10 − 7 E 0 α 0 A. Bogomyagkov (BINP) FCC-ee c.m. energy 6 / 22

Longitudinal field compensation Detector field is B 0 = 2 T. Deviation of compensating field is ∆ B c = 0 . 1 T. Length of compensating solenoid is L c = 0 . 75 m. B ρ = 152 . 105 T · m, E 0 = 45 . 6 GeV, ν = 103 . 484. FCCee_z_202_nosol_13 � 2 ∆ ν = ϕ 2 8 π cot( πν ) ≈ 1 � ∆ B c 2 B 0 L c ≈ 2 × 10 − 9 . 8 π cot( πν ) B 0 B ρ ∆ E = ∆ ν · 440 . 65 ≈ 2 × 10 − 11 . E 0 E 0 A. Bogomyagkov (BINP) FCC-ee c.m. energy 7 / 22

Spin distribution width: horizontal betatron oscillations Ya.S. Derbenev, et al., “Accurate calibration of the beam energy in a storage ring based on measurement of spin precession frequency of polarized particles”, Part. Accel. 10 (1980) 177-180 FCCee_z_202_nosol_13 ∂ 2 B y Sextupole fields introduce additional B ⊥ ∝ x 2 , K 2 = 1 ∂ x 2 . B ρ Spin precession frequency distribution shifts and becomes wider by ∆ ν = − 1 � � � ε x β x ( s ) + η x ( s ) 2 σ 2 K 2 ( s ) ds . δ ν 2 π ∆ ν = ∆ E = − 2 . 5 · 10 − 7 . ν E 0 A. Bogomyagkov (BINP) FCC-ee c.m. energy 8 / 22

Vertical magnetic fields: horizontal correctors � ∆ B y One corrector with deflection χ : ∆ E = − χη x α Π , χ = B ρ ds . E 0 √ � ∆ E � = 2 2 sin( πν x ) � η x � RMS of energy shift: σ � β x � σ x E 0 α Π σ x is RMS of horizontal orbit variation. FCCee_z_202_nosol_13 � ∆ E � = − 1 . 2 · 10 − 3 [ m − 1 ] · σ x [ m ] , σ E 0 � � = 10 − 6 demands stability of the horizontal orbit between calibrations ∆ E σ E 0 σ x = 0 . 8 mm. A. Bogomyagkov (BINP) FCC-ee c.m. energy 9 / 22

Vertical magnetic fields: quadrupoles ∂ B y Shifted quadrupole: ∆ E = − χη x χ = K 1 L · ∆ x , K 1 = 1 α Π , ∂ x . E 0 B ρ FCCee_z_202_nosol_13 ∆ E = 10 − 6 demands stability of quadrupoles position between calibrations (10 min) E 0 Quadrupole ∆ x , m 2 · 10 − 4 QC7.1: 7 . 6 · 10 − 5 QY2.1: 1 . 6 · 10 − 4 QFG2.4: 1 . 4 · 10 − 4 QF4.1: 3 . 5 · 10 − 5 QG6.1: √ 720 = 5 · 10 − 6 QF4: ∆ x / A. Bogomyagkov (BINP) FCC-ee c.m. energy 10 / 22

Central mass energy: β chromaticity Invariant mass: M 2 = ( E 1 + E 2 ) 2 cos 2 ( θ ) + O ( m 2 e ) + O ( σ 2 α ) + O ( σ 2 E ) . Beta function chromaticity at IP: β x , y = β 0 x , y + β 1 x , y δ , σ 2 x , y = ε x , y β x , y . Particles with energy deviation have higher collision rate. h0 h1 × 3 × 3 10 10 h1 h1 h0 h0 1e+008 1e+008 Entries Entries Entries 1e+008 Entries 1e+008 − − 700 700 Mean Mean 5.926e 5.926e 008 008 − − − − Mean Mean 1.085e 1.085e 006 006 0.0005332 0.0005332 RMS RMS 0.0005332 0.0005332 RMS RMS 600 600 β d 500 500 1 y =15 β δ d 400 400 0y 300 300 200 200 100 100 0 0 − − − − − − − − 0.004 0.003 0.002 0.001 0 0.001 0.002 0.003 0.004 0.004 0.003 0.002 0.001 0 0.001 0.002 0.003 0.004 E1+E2 E1+E2 -2 -2 E0 E0 A. Bogomyagkov (BINP) FCC-ee c.m. energy 11 / 22

Central mass energy: β chromaticity FCCee_z_202_nosol_13 d β x d β y 1 1 ∆ M ∆ M , keV β x d δ β y d δ E 0 − 1 . 1 · 10 − 6 ± 5 · 10 − 8 0 15 − 49 ± 2 . 4 − 5 . 7 · 10 − 7 ± 5 · 10 − 8 200 0 − 26 ± 2 . 4 − 1 . 6 · 10 − 6 ± 5 · 10 − 8 200 15 − 75 ± 2 . 4 1 d β y Need to measure and adjust d δ . β 0 y A. Bogomyagkov (BINP) FCC-ee c.m. energy 12 / 22

Energy dependence on azimuth: full tapering Two diametrically opposite RF cavities, U 0 — energy loss per revolution, E ( 0 ) — after RF cavity. Full tapering — magnets fields are adjusted to keep design curvature, quadrupole strength etc. U 2 dE E ( 0 ) k ≈ 3 E ( 0 ) + 3 U 0 ds ∝ E 4 , E ( 0 ) 2 + O ( U 3 0 E ( s ) = , 0 ) 1 Π Π ( 1 + k · s ) 3 U 2 Average energy: � E � ≈ E ( 0 ) − U 0 0 4 − 12 E ( 0 ) . Energy at the IP: E ( IP ) = E ( 0 ) − U 0 4 . U 2 The difference: � E � − E ( IP ) ≈ − 1 E ( 0 ) 2 = 5 · 10 − 8 , for E 0 = 45 . 6 GeV (Z). 0 E ( 0 ) 12 U 2 The difference: � E � − E ( IP ) ≈ − 1 E ( 0 ) 2 = 2 · 10 − 7 , for E 0 = 80 . 5 GeV (WW). 0 E ( 0 ) 12 A. Bogomyagkov (BINP) FCC-ee c.m. energy 13 / 22

Energy dependence on azimuth: partial tapering Partial tapering ( ∆ K 0 ) — fields of magnets groups are adjusted to keep approximately design curvature ( K 0 ). Equations of motion (canonical variables) σ ′ = − K 0 x ,    e 2 γ 4 � � � � − eV 0 2 π δ ( s − s 0 ) − 2 p t ′ = p 0 c K 2 sin φ s + 0 σ .  p 0 c λ RF σ 3  Solution: p t ( s ) = p 0 t − f ( s ) . � Π σ = 0 = − K 0 ( s ) x ( s ) ds = − p 0 t α Π + Π � ( K 0 f + ∆ K 0 ) η � s . 0 p 0 t = 1 α � ( K 0 f + ∆ K 0 ) η � s . A. Bogomyagkov (BINP) FCC-ee c.m. energy 14 / 22

Energy dependence on azimuth: partial tapering For simple (symmetrical) cases we do need to know function f ( s ) , just at certain points. Two RF cavities and symmetrical arcs  � p t � = p 0 t − � f � = p 0 t − U 0 = � E � − E 0 ,   4 E 0 E 0  = E IP − E 0 p t ( IP ) = p 0 t − f ( IP ) = p 0 t − U 0  ,   4 E 0 E 0 � E � = E 0 + E 0 p 0 t − U 0  4 ,   E IP = E 0 + E 0 p 0 t − U 0  4 .  There is no difference between � E � and E IP in the first order. Numerical calculations are needed for not symmetrical arcs, magnet misalignments. A. Bogomyagkov (BINP) FCC-ee c.m. energy 15 / 22

Collective field of the own bunch Electron in the field of own bunch will have potential energy �� 10 − 7 N p e 2 [ Gs ] � � σ x + σ y U [ eV ] = √ γ e + ln( 2 ) − 2 ln e [ C ] , r 2 πσ z [ cm ] γ e = 0 . 577 Euler constant, N p = 4 · 10 10 — bunch population, r ip = 15 mm and r arc = 20 mm — vacuum chamber radius at IP and in the arcs, σ x , IP = 6 . 2 · 10 − 6 m, σ y , IP = 3 . 1 · 10 − 8 m, , σ x , arc = 1 . 9 · 10 − 4 m, σ y , arc = 1 . 2 · 10 − 5 m. U ip = 192 keV 45 . 6 GeV = 4 . 2 · 10 − 6 , E 0 U arc = 120 keV 45 . 6 GeV = 2 . 6 · 10 − 6 . E 0 A. Bogomyagkov (BINP) FCC-ee c.m. energy 16 / 22

Collective field of the opposite bunch Potential energy at the center of the bunch { x , y , s , z = s − ct } = { 0 , 0 , 0 , 0 } � − ( x + s · 2 θ ) 2 y + q − γ 2 ( s + ct ) 2 � y 2 � ∞ exp − U ( x , y , s , ct ) = − γ N p r e mc 2 2 σ 2 2 σ 2 2 γ 2 σ 2 x + q s + q √ π dq , � � � 2 σ 2 2 σ 2 2 γ 2 σ 2 0 x + q y + q s + q U ( 0 , 0 , 0 , 0 ) = − 0 . 4 MeV 45 . 6 GeV = − 9 . 3 · 10 − 6 . E 0 A. Bogomyagkov (BINP) FCC-ee c.m. energy 17 / 22