Unified dispersive approach to real and virtual photon-photon scattering into two pions
|
|
- θάνατος Παπαντωνίου
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Unified dispersive approach to real and virtual photon-photon scattering into two pions Bachir Moussallam Project started with Diogo Boito arxiv: Photon 2013 *** Paris, May 20-24
2 Introduction Goal: representations for amplitudes γγ (q 2 ) ππ or γ (q 2 ) γππ ( q 2 < 1 GeV 2 ) Method: combine nonperturbative QCD tools [Unitarity, analyticity, Chiral symmetry, soft photon theorems] Relevancefor muon g 2 HVP contribution from γππ channels[reduced model dependence] Future: 2π contribution in light-by-light amplitude γγ (q 2 2 ) ππ γ (q 2 3 )γ (q 2 ) ([Hoferichter et al.]) 4 Account for ππ re-scattering (exp. progress recently: [NA48/2],[DIRAC]...) 2/24
3 Experimental aspects q 2 >4m 2 π : e + e γπ 0 π 0 direct relation to γ γπ 0 π 0. Measurements performed (q < 1 GeV): SND:[Phys. Lett. B 537 (2002) 201] CMD-2:[Phys. Lett. B 580 (2004) 119] also KLOE[EPJ C49 (2007) 473] but q=m ϕ e + e γπ + π ISR and FSR amplitudesinterfere. q 1 p 1 k 1 q 2 (A) k 2 (B) p 2 must measure angular distributions 3/24
4 q 2 <0 : From (in principle) e + e e + e π 0 π 0, e + e π + π with single tagging 4/24
5 Theory: Unitarity key ingredient to FSI ππ scattering elastic: s< 1 GeV 2 Fermi-Watson theorem. Valid for γγ (q 2 ) ππ? Depends on q 2 q 2 4m 2 π : Im(γγ ππ) J =(γγ ππ) J (ππ ππ) J q 2 >4m 2 π : Im(γγ ππ) J =(γγ ππ) (ππ ππ) J J +(γ ππ)(γππ ππ) J [Creutz,Einhorn PR D1 (1970)2537.] 5/24
6 Chiral low energy expansion: Valid when q 2, s <<1 GeV 2 π 0 π 0 probes loops, NLO calculations [Bijnens, Cornet NP B296 (1988) 557] (q 2 =0) [Donoghue,Holstein,PR D48 (1993) 137] (q 2 =0) H n = 2(s m2 π ) Ḡ(s,q 2 ) ++ NLO F 2 π H c = s Ḡ(s,q 2 )+( l ++ NLO F 2 6 l 5 ) s q2 + H Born 48π 2 F 2 ++ π π with Ḡ(s,q 2 )= sḡ π (s) q 2 Ḡ π (q 2 ) s q 2 q 2 J π (s) J π (q2 ) s q 2 ImḠ π (z), J π (z) =0 when z>4m 2 π 6/24
7 Analyticity in QCD Combine unitarity with analyticity[omnès, NC 8 (1958) 316]. PW amplitudes analytic s with two cuts right-handcut:[4m 2 π, ] left-hand cut:[, 0] (usually!) Discontinuity on RHC: disc(γγ ππ)=(γγ ππ) s iε (ππ ππ) s+iε also when q 2 >4m 2. FSI problem solved π Amplitude from LHC from Muskhelishvili equation Appl. to γγ ππ[gourdin, Martin NC 17 (1960) 224] Matching w. ChPT[Morgan, Pennington PL B272 (1991) 134, Donoghue, Holstein PR D48 (1993) 137] 7/24
8 Phenomenology of LHC Pion pole (Born Amplitude) π + π + π + γ γ γ π π (a) (b) (c) π Resonance poles π 0 π + π ω, ρ 0 ρ + γ π 0 γ π γ π q 2 =0: form-factors 8/24
9 Born helicity amplitudes: H Born λλ (s,q 2,θ)=F v π (q2 ) H Born λλ (s,q 2,θ) H Born ++ (s,q2,θ)= 2(4m2 π q2 (1 σ 2 π cos2 θ)) (s q 2 )(1 σ 2 π cos2 θ) 2(s 4m 2 H Born + (s,q2 π,θ)= )sin2 θ (s q 2 )(1 σ 2 π cos2 θ) H Born +0 (s,q2,θ)= 2 2q 2 (s 4m 2 π )sinθcosθ s(s q 2 )(1 σ 2 π cos2 θ) Partial-waveJ=0 h Born 0,++ (s,q2 )= 1 s q 2 4m2 π σ π (s) log1+σ π(s) 1 σ π (s) 2q2 Singularities: LHC[,0], pole s=q 2 (soft photon), 9/24
10 Resonance exchange partial-waves: LHC in case q 2 >4m 2 π Im(s)/m 2 π Omnès applicable? 0 s s Re(s)/m 2 π 10/24
11 Yes a) Use Källen-Lehmann representation for propagators BW V (t)= 1 π 4m 2 π dt σ(t,m V,γ V ) (t t) b) Limiting prescriptions: q 2 =lim ε 0 q 2 +iε (generalized) LHC does not intersect RHC Im (s)/m 2 π Re (s)/m 2 π 11/24
12 Representation based on 2-subtracted DR H I ++ (s,q2,z)=f v π (q2 ) H I,Born ++ (s,q2,z)+ V=ρ,ω F Vπ(q 2 ) H I,V ++ (s,q2,z) +Ω (s q I 0 (s) 2 )b I (q 2 )+sf v s(j π (q2 ) I,π (s,q 2 ) J I,π (q 2,q 2 )) q 2 Ĵ I,π (q 2 ) s q 2 +s V=ρ,ω F Vπ(q 2 ) sj I,V (s,q 2 ) q 2 J I,V (q 2,q ) 2. Remarks: H I ++ isospin amplitude,hn ++, Hc linear combination ++ First line: tree diagrams Omnès function Ω I 0 (s)=exp 1 π 4m 2 π ds s (s s) δi 0 (s ) 12/24
13 (Continued) J I,π (s,q 2 ),J I,V (s,q 2 ): integrals of h I,Born 0,h I,V 0, phase-shifts J I,π (s,q 2 )= 1 π J I,V (s,q 2 )= 1 π 4m 2 π 4m 2 π (integrals well defined) also ds sinδ I 0 (s ) h I,π (s ) 2 (s s) Ω I 0 (s ) 0,++ (s,q 2 ) ds sinδ I 0 (s ) h I,V (s ) 2 (s s) Ω I 0 (s ) 0,++ (s,q 2 ) Ĵ I,π (q 2 )= JI,π (s,q 2 ) s Satisfies soft-photon theorem s=q 2 Extra polynomial in s: paramatrize higher energy portionsof cuts. Two unknownfunctions: b I (q 2 ) 13/24
14 Chiral symmetry constraints π 0 π 0 amplitudeat s=0 H n ++ (0,q2,z)= H n,v ++ (0,q2,z) q 2 b n (q 2 ) V Adlerzero b n (q 2 )=O(m 2 π ) Except if q 2 =0! NLO ChPT: H n ++ (0,q2,z) = 2m2 π Ḡπ NLO F 2 (q 2 ) J π (q 2 ) π q 2 = 96π 2 F 2 1+ q2 15m 2 + π π 14/24
15 Parametrization of subtraction functions: b n (q 2 )=b n (0) F(q 2 ) +β ρ (GS ρ (q 2 ) 1)+β ω (BW ω (q 2 ) 1) b c (q 2 )=b c (0) +β ρ (GS ρ (q 2 ) 1)+β ω (BW ω (q 2 ) 1) F(q 2 ) from chiral NLO b n (0), b c (0): comp. with chiral amplitude Note: relation to pion polarizabilities α π 0 β π 0= 2α 1 lim s=0 m π s Hn,V ++ (s,0,θ)+bn (0) α π + β π += 2α 1 lim s=0 s Hc,V ++ (s,0,θ)+bc (0) m π β ρ, β ω : experimental inputs 15/24
16 Comparison with NLO ChPT q 2 = 0.2 GeV 2 H++(s, q 2, z) t=m 2 π Re(H++) disp. Re(H++) chiral Im(H++) disp. Im(H++) chiral q 2 = 0.2 (GeV) s (GeV) 2 q 2 =0.2 GeV 2 H++(s, q 2, z) t=m 2 π Form factor=1 at NLO Re(H++) disp. Re(H++) chiral Im(H++) disp. Im(H++) chiral q 2 = 0.2 (GeV) s (GeV) 2 16/24
17 q 2 =0: comparison w. data[γγ π 0 π 0 ] 0(z < 0.8) (nb) σγγ π 0 π α β = 1.9 α β = 1.3 α β = 1.0 Crystal Ball Belle s (GeV) 2 Crystal Ball: [PR D41 (1990) 3324] Belle : [PR D78 (2008) ] KLOE-2 [expected] 17/24
18 Fit of σ e + e γπ 0 π 0(q2 ) data β ρ β ω χ 2 /N dof ref. 0.14±0.12 ( 0.39±0.12) /27 SND (2002) 0.13±0.15 ( 0.31±0.15) /21 CMD-2 (2003) 0.05±0.09 ( 0.37±0.09) /50 Combined 18/24
19 Fit of σ e + e γπ 0 π 0(q2 ) data (cont.) Cross-section (nb) Cross-section (nb) Akhmetshin (2003) β ρ, β ω : fitted β ρ = β ω = Achasov (2002) β ρ, β ω : fitted β ρ = β ω = 0 q (GeV) q (GeV) 19/24
20 Differential cross-sections dσ ds (s,q2 ) (nb GeV 1) dσ e + e π 0 π 0 γ d s q = 0.75 q = m ρ q = m ω q = s (GeV) shape changeswhen q>(m ρ +m π ) no σ meson bump 20/24
21 Illustrative comp. w. other approaches Consider ρ 0 π 0 π 0 γ (GeV) Dispersive UChPT σ-model 10 4 B ρ π 0 π 0 γ d s s (GeV) Chiral Lagr.+V + unitarized pion loop [Palomar,Hirenzaki,Oset NP A707 (2002)161] Chiral Lagr.+V +pion loop +sigma meson[bramon et al. PL B517 (2001) 345] 21/24
22 Muon(g 2)/2: γππ contribution in HVP a [γππ] μ = 1 q 2 max 4π 3 dq 2 K μ q 2 σ e + e γ γππ(q 2 ) 4m 2 π cross-sections: σ c, σ n α 3 σ(q 2 )= 12(q 2 ) 3 q 2 4m 2 π ds(q 2 s)σ π (s) 1 dz 1 σ c : SeparateBorn H c λλ 2 = H Born λλ +Ĥ c λλ 2 σ c (q 2 )=σ Born (q 2 )+ ˆσ Born (q 2 )+ ˆσ c (q 2 ) H λλ (s,q 2,θ) 2 Define σ Born (e.g. add rad. corr. part σ(e + e π + π )) 22/24
23 σ Born corresponds to sqed σ Born (q 2 )= πα2 Numerical results: channel cross-section 3q 2 σ3 π (q2 ) F v π (q2 ) 2 α π η(q2 ) γπ + π σ Born γπ + π ˆσ Born (1.31±0.30) γπ + π ˆσ c (0.16±0.05) γπ 0 π 0 σ n (0.33±0.05) Remarks a Born comparableto Δa SM μ μ =± a μ [ˆσ Born ]>0 unlike[dubinskyetal. EPJC40 (2005)41] σ-meson approx: a [γσ] = [Narison (2003)], μ = [Ahmadov,Kuraev,Volkov (2010)] a μ 23/24
24 Conclusions Analyticity based treatment of FSI in γγ ππ extended to γγ (q 2 ) Main issue: left-hand cut [pion, resonances] becomes generalized one, but properly defined Good description of experimentaldata e + e γπ 0 π 0 (2 parameters) a μ : contributionsfrom γπ 0 π 0, γπ + π [q<0.95 GeV] Other applications: pion generalized polarizabilities, sigma meson (pole)-γ form factor Extensions possible [ q> 1 GeV ] (coupled-channel MO). Double virtual scattering? 24/24
Three coupled amplitudes for the πη, K K and πη channels without data
Three coupled amplitudes for the πη, K K and πη channels without data Robert Kamiński IFJ PAN, Kraków and Łukasz Bibrzycki Pedagogical University, Kraków HaSpect meeting, Kraków, V/VI 216 Present status
Διαβάστε περισσότεραDong Liu State Key Laboratory of Particle Detection and Electronics University of Science and Technology of China
Dong Liu State Key Laboratory of Particle Detection and Electronics University of Science and Technology of China ISSP, Erice, 7 Outline Introduction of BESIII experiment Motivation of the study Data sample
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραPredictions on the second-class currents
Predictions on the second-class currents τ η ( ) ν τ decays Sergi Gonzàlez-Solís Institut de Física d Altes Energies Universitat Autònoma de Barcelona in collaboration with Rafel Escribano and Pablo Roig,
Διαβάστε περισσότεραExpIntegralE. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
ExpIntegralE Notations Traditional name Exponential integral E Traditional notation E Mathematica StandardForm notation ExpIntegralE, Primary definition 06.34.0.000.0 E t t t ; Re 0 Specific values Specialied
Διαβάστε περισσότεραHadronic Tau Decays at BaBar
Hadronic Tau Decays at BaBar Swagato Banerjee Joint Meeting of Pacific Region Particle Physics Communities (DPF006+JPS006 Honolulu, Hawaii 9 October - 3 November 006 (Page: 1 Hadronic τ decays Only lepton
Διαβάστε περισσότερα상대론적고에너지중이온충돌에서 제트입자와관련된제동복사 박가영 인하대학교 윤진희교수님, 권민정교수님
상대론적고에너지중이온충돌에서 제트입자와관련된제동복사 박가영 인하대학교 윤진희교수님, 권민정교수님 Motivation Bremsstrahlung is a major rocess losing energies while jet articles get through the medium. BUT it should be quite different from low energy
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραDiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραLow Frequency Plasma Conductivity in the Average-Atom Approximation
Low Frequency Plasma Conductivity in the Average-Atom Approximation Walter Johnson & Michael Kuchiev Physical Review E 78, 026401 (2008) 1. Review of Average-Atom Linear Response Theory 2. Demonstration
Διαβάστε περισσότεραRevisiting the S-matrix approach to the open superstring low energy eective lagrangian
Revisiting the S-matrix approach to the open superstring low energy eective lagrangian IX Simposio Latino Americano de Altas Energías Memorial da América Latina, São Paulo. Diciembre de 2012. L. A. Barreiro
Διαβάστε περισσότεραAdS black disk model for small-x DIS
AdS black disk model for small-x DIS Miguel S. Costa Faculdade de Ciências da Universidade do Porto 0911.0043 [hep-th], 1001.1157 [hep-ph] Work with. Cornalba and J. Penedones Rencontres de Moriond, March
Διαβάστε περισσότεραProblem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Διαβάστε περισσότεραBessel functions. ν + 1 ; 1 = 0 for k = 0, 1, 2,..., n 1. Γ( n + k + 1) = ( 1) n J n (z). Γ(n + k + 1) k!
Bessel functions The Bessel function J ν (z of the first kind of order ν is defined by J ν (z ( (z/ν ν Γ(ν + F ν + ; z 4 ( k k ( Γ(ν + k + k! For ν this is a solution of the Bessel differential equation
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραEPS-HEP 2015 DOUBLE-SCATTERING MECHANISM. Antoni Szczurek 1,2 Mariola Kłusek-Gawenda 1
EPS-HEP 5 DOUBLE-SCTTERING MECHNISM OF PRODUCTION OF TWO MESONS IN ULTRPERIPHERL, ULTRRELTIISTIC HEY ION COLLISIONS ntoni Szczurek, Mariola Kłusek-Gawenda Institute of Nuclear Physics PN Kraków University
Διαβάστε περισσότεραForced Pendulum Numerical approach
Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.
Διαβάστε περισσότεραmeasured by ALICE in pp, p-pb and Pb-Pb collisions at the LHC
Σ(85) Ξ(5) Production of and measured by ALICE in pp, ppb and PbPb collisions at the LHC for the ALICE Collaboration Pusan National University, OREA Quark Matter 7 in Chicago 7.. QM7 * suppressed, no suppression
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραNucleon resonances extracted from Bonn-Gatchina coupled channel analysis
Nucleon resonances extracted from Bonn-Gatchina coupled channel analysis NSTAR Nucleon resonances extracted from Bonn-Gatchina coupled channel analysis Petersburg Nuclear Physics Institute A. Sarantsev
Διαβάστε περισσότερα6.4 Superposition of Linear Plane Progressive Waves
.0 - Marine Hydrodynamics, Spring 005 Lecture.0 - Marine Hydrodynamics Lecture 6.4 Superposition of Linear Plane Progressive Waves. Oblique Plane Waves z v k k k z v k = ( k, k z ) θ (Looking up the y-ais
Διαβάστε περισσότεραTheory predictions for the muon (g 2): Status and Perspectives
Theory predictions for the muon (g 2): Status and Perspectives Matthias Steinhauser Mass 2018, Odense, May 29 June 1, 2018 TTP KARLSRUHE KIT University of the State of Baden-Wuerttemberg and National Laboratory
Διαβάστε περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραLarge β 0 corrections to the energy levels and wave function at N 3 LO
Large β corrections to the energy levels and wave function at N LO Matthias Steinhauser, University of Karlsruhe LCWS5, March 5 [In collaboration with A. Penin and V. Smirnov] M. Steinhauser, LCWS5, March
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ Ä664
ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ 2017.. 48.. 5.. 653Ä664 ˆ Œ ˆ ˆ e + e K + K nπ (n =1, 2, 3) Š Œ ŠŒ -3 Š - ˆ Œ Š -2000 ƒ.. μéμ Î 1,2, μé ³ ±μ²² μ Í ŠŒ -3: A.. ß ±μ 1,2,. Œ. ʲÓÎ ±μ 1,2,.. ̳ ÉÏ 1,2,.. μ 1,.. ÏÉμ μ 1,.
Διαβάστε περισσότεραL. F avart. CLAS12 Workshop Genova th of Feb CLAS12 workshop Feb L.Favart p.1/28
L. F avart I.I.H.E. Université Libre de Bruxelles H Collaboration HERA at DESY CLAS Workshop Genova - 4-8 th of Feb. 9 CLAS workshop Feb. 9 - L.Favart p./8 e p Integrated luminosity 96- + 3-7 (high energy)
Διαβάστε περισσότεραLifting Entry (continued)
ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραGeneral 2 2 PT -Symmetric Matrices and Jordan Blocks 1
General 2 2 PT -Symmetric Matrices and Jordan Blocks 1 Qing-hai Wang National University of Singapore Quantum Physics with Non-Hermitian Operators Max-Planck-Institut für Physik komplexer Systeme Dresden,
Διαβάστε περισσότεραis like multiplying by the conversion factor of. Dividing by 2π gives you the
Chapter Graphs of Trigonometric Functions Answer Ke. Radian Measure Answers. π. π. π. π. 7π. π 7. 70 8. 9. 0 0. 0. 00. 80. Multipling b π π is like multipling b the conversion factor of. Dividing b 0 gives
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραHartree-Fock Theory. Solving electronic structure problem on computers
Hartree-Foc Theory Solving electronic structure problem on computers Hartree product of non-interacting electrons mean field molecular orbitals expectations values one and two electron operators Pauli
Διαβάστε περισσότεραOn the Galois Group of Linear Difference-Differential Equations
On the Galois Group of Linear Difference-Differential Equations Ruyong Feng KLMM, Chinese Academy of Sciences, China Ruyong Feng (KLMM, CAS) Galois Group 1 / 19 Contents 1 Basic Notations and Concepts
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραD Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότεραNotations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation. Mathematica StandardForm notation
KelvinKei Notations Traditional name Kelvin function of the second kind Traditional notation kei Mathematica StandardForm notation KelvinKei Primary definition 03.5.0.000.0 kei kei 0 Specific values Values
Διαβάστε περισσότεραSolutions to the Schrodinger equation atomic orbitals. Ψ 1 s Ψ 2 s Ψ 2 px Ψ 2 py Ψ 2 pz
Solutions to the Schrodinger equation atomic orbitals Ψ 1 s Ψ 2 s Ψ 2 px Ψ 2 py Ψ 2 pz ybridization Valence Bond Approach to bonding sp 3 (Ψ 2 s + Ψ 2 px + Ψ 2 py + Ψ 2 pz) sp 2 (Ψ 2 s + Ψ 2 px + Ψ 2 py)
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραSPECIAL FUNCTIONS and POLYNOMIALS
SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 8.195
Διαβάστε περισσότεραLight Hadrons and New Enhancements in J/ψ Decays at BESII
Light Hadrons and New Enhancements in J/ψ Decays at BESII Guofa XU Representing BES Collaboration Institute of High Energy Physics Chinese Academy of Sciences Beijing, China xugf@ihep.ac.cn Outline Introduction
Διαβάστε περισσότεραGraded Refractive-Index
Graded Refractive-Index Common Devices Methodologies for Graded Refractive Index Methodologies: Ray Optics WKB Multilayer Modelling Solution requires: some knowledge of index profile n 2 x Ray Optics for
Διαβάστε περισσότεραDifferential equations
Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential
Διαβάστε περισσότερα( ) Sine wave travelling to the right side
SOUND WAVES (1) Sound wave: Varia2on of density of air Change in density at posi2on x and 2me t: Δρ(x,t) = Δρ m sin kx ωt (2) Sound wave: Varia2on of pressure Bulk modulus B is defined as: B = V dp dv
Διαβάστε περισσότεραBaryon Studies. Dongliang Zhang (University of Michigan) Hadron2015, Jefferson Lab September 13-18, on behalf of ATLAS Collaboration
Λ b Baryon Studies Dongliang Zhang (University of Michigan) on behalf of Collaboration Hadron215, Jefferson Lab September 13-18, 215 Introduction Λ b reconstruction Lifetime measurement Helicity study
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραGeodesic Equations for the Wormhole Metric
Geodesic Equations for the Wormhole Metric Dr R Herman Physics & Physical Oceanography, UNCW February 14, 2018 The Wormhole Metric Morris and Thorne wormhole metric: [M S Morris, K S Thorne, Wormholes
Διαβάστε περισσότεραME 365: SYSTEMS, MEASUREMENTS, AND CONTROL (SMAC) I
ME 365: SYSTEMS, MEASUREMENTS, AND CONTROL SMAC) I Dynamicresponseof 2 nd ordersystem Prof.SongZhangMEG088) Solutions to ODEs Forann@thorderLTIsystem a n yn) + a n 1 y n 1) ++ a 1 "y + a 0 y = b m u m)
Διαβάστε περισσότεραOn the summability of divergent power series solutions for certain first-order linear PDEs Masaki HIBINO (Meijo University)
On the summability of divergent power series solutions for certain first-order linear PDEs Masaki HIBINO (Meijo University) 1 1 Introduction (E) {1+x 2 +β(x,y)}y u x (x,y)+{x+b(x,y)}y2 u y (x,y) +u(x,y)=f(x,y)
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραLecture 26: Circular domains
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains
Διαβάστε περισσότεραwave energy Superposition of linear plane progressive waves Marine Hydrodynamics Lecture Oblique Plane Waves:
3.0 Marine Hydrodynamics, Fall 004 Lecture 0 Copyriht c 004 MIT - Department of Ocean Enineerin, All rihts reserved. 3.0 - Marine Hydrodynamics Lecture 0 Free-surface waves: wave enery linear superposition,
Διαβάστε περισσότεραPg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is
Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =
Διαβάστε περισσότερα1. 3. ([12], Matsumura[13], Kikuchi[10] ) [12], [13], [10] ( [12], [13], [10]
3. 3 2 2) [2] ) ) Newton[4] Colton-Kress[2] ) ) OK) [5] [] ) [2] Matsumura[3] Kikuchi[] ) [2] [3] [] 2 ) 3 2 P P )+ P + ) V + + P H + ) [2] [3] [] P V P ) ) V H ) P V ) ) ) 2 C) 25473) 2 3 Dermenian-Guillot[3]
Διαβάστε περισσότεραg-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King
Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i
Διαβάστε περισσότεραProblem 7.19 Ignoring reflection at the air soil boundary, if the amplitude of a 3-GHz incident wave is 10 V/m at the surface of a wet soil medium, at what depth will it be down to 1 mv/m? Wet soil is
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότερα4.- Littlest Higgs Model with T-parity. 5.- hhh at one loop in LHM with T-parity
1.- Introduction. 2.- Higgs physics 3.- hhh at one loop level in SM 4.- Littlest Higgs Model with T-parity 5.- hhh at one loop in LHM with T-parity 7.- Conclusions Higgs Decay modes at LHC Direct measurement
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραA summation formula ramified with hypergeometric function and involving recurrence relation
South Asian Journal of Mathematics 017, Vol. 7 ( 1): 1 4 www.sajm-online.com ISSN 51-151 RESEARCH ARTICLE A summation formula ramified with hypergeometric function and involving recurrence relation Salahuddin
Διαβάστε περισσότεραAppendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee
Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραEE101: Resonance in RLC circuits
EE11: Resonance in RLC circuits M. B. Patil mbatil@ee.iitb.ac.in www.ee.iitb.ac.in/~sequel Deartment of Electrical Engineering Indian Institute of Technology Bombay I V R V L V C I = I m = R + jωl + 1/jωC
Διαβάστε περισσότεραThe kinetic and potential energies as T = 1 2. (m i η2 i k(η i+1 η i ) 2 ). (3) The Hooke s law F = Y ξ, (6) with a discrete analog
Lecture 12: Introduction to Analytical Mechanics of Continuous Systems Lagrangian Density for Continuous Systems The kinetic and potential energies as T = 1 2 i η2 i (1 and V = 1 2 i+1 η i 2, i (2 where
Διαβάστε περισσότεραBayesian statistics. DS GA 1002 Probability and Statistics for Data Science.
Bayesian statistics DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Frequentist vs Bayesian statistics In frequentist
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραMATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Διαβάστε περισσότεραProblem 3.1 Vector A starts at point (1, 1, 3) and ends at point (2, 1,0). Find a unit vector in the direction of A. Solution: A = 1+9 = 3.
Problem 3.1 Vector A starts at point (1, 1, 3) and ends at point (, 1,0). Find a unit vector in the direction of A. Solution: A = ˆx( 1)+ŷ( 1 ( 1))+ẑ(0 ( 3)) = ˆx+ẑ3, A = 1+9 = 3.16, â = A A = ˆx+ẑ3 3.16
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραWhat happens when two or more waves overlap in a certain region of space at the same time?
Wave Superposition What happens when two or more waves overlap in a certain region of space at the same time? To find the resulting wave according to the principle of superposition we should sum the fields
Διαβάστε περισσότεραMock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =
Mock Eam 7 Mock Eam 7 Section A. Reference: HKDSE Math M 0 Q (a) ( + k) n nn ( )( k) + nk ( ) + + nn ( ) k + nk + + + A nk... () nn ( ) k... () From (), k...() n Substituting () into (), nn ( ) n 76n 76n
Διαβάστε περισσότερα([28] Bao-Feng Feng (UTP-TX), ( ), [20], [16], [24]. 1 ([3], [17]) p t = 1 2 κ2 T + κ s N -259-
5,..,. [8]..,,.,.., Bao-Feng Feng UTP-TX,, UTP-TX,,. [0], [6], [4].. ps ps, t. t ps, 0 = ps. s 970 [0] []. [3], [7] p t = κ T + κ s N -59- , κs, t κ t + 3 κ κ s + κ sss = 0. T s, t, Ns, t., - mkdv. mkdv.
Διαβάστε περισσότεραSection 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
Διαβάστε περισσότεραParametrized Surfaces
Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραP AND P. P : actual probability. P : risk neutral probability. Realtionship: mutual absolute continuity P P. For example:
(B t, S (t) t P AND P,..., S (p) t ): securities P : actual probability P : risk neutral probability Realtionship: mutual absolute continuity P P For example: P : ds t = µ t S t dt + σ t S t dw t P : ds
Διαβάστε περισσότεραDETERMINATION OF DYNAMIC CHARACTERISTICS OF A 2DOF SYSTEM. by Zoran VARGA, Ms.C.E.
DETERMINATION OF DYNAMIC CHARACTERISTICS OF A 2DOF SYSTEM by Zoran VARGA, Ms.C.E. Euro-Apex B.V. 1990-2012 All Rights Reserved. The 2 DOF System Symbols m 1 =3m [kg] m 2 =8m m=10 [kg] l=2 [m] E=210000
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραw o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Διαβάστε περισσότεραThe Standard Model. Antonio Pich. IFIC, CSIC Univ. Valencia
http://arxiv.org/pd/0705.464 The Standard Mode Antonio Pich IFIC, CSIC Univ. Vaencia Gauge Invariance: QED, QCD Eectroweak Uniication: SU() Symmetry Breaking: Higgs Mechanism Eectroweak Phenomenoogy Favour
Διαβάστε περισσότερα