Moment Analysis of Hadronic Vacuum Polarization Proposal for a lattice QCD evaluation of g µ 2

Transcription

1 Moen Analysis of Hadronic Vacuu Polarizaion Proposal for a laice QCD evaluaion of g Eduardo de Rafael Cenre de Physique Théorique CNRS-Luiny, Marseille 7h June 04 NAPLES Seinar

2 Moivaion a (E8 BNL) = (54) sa(33) sys 0 [0.54pp] Fuure Experiens: FNAL wih ±0.4 pp overall uncerainy JPARC wih siilar uncerainy bu very differen echnique Sandard Model Conribuions J.P. Miller, E. de Rafael, B.L. Robers, D. Söckinger, Annu. Rev. Par. Nucl. Phys. CONTRIBUTION RESULT IN 0 UNITS QED (lepons) ± 0.04 HVP(lo)[e + e ] 6 93±4 HVP(ho) 98.4 ± 0.7 HLxL 05±6 EW 53± Toal SM ± 49 Persisen 3.6σ discrepancy beween SM heory and Experien

3 HVP Conribuion o he Muon Anoaly X Hadrons Calculaion Seps (for non expers) g ν ( q q q q ν q g ν )Π( q ) q = ( q q q ν q g ν d 0 q ) IΠ() q. d gν = IΠ() 0 q. Muon anoaly fro assive phoon (ass ): (g )hadrons ahvp = α d dx 0 0 x ( x) x + ( x) IΠ(),

4 Euclidean Forulaion = α 0 d dx 0 x ( x) ( x) IΠ(), d Q Π(Q ) = 0 + Q x + HVP in Laice QCD In laice QCD useful o go Euclidean: = α 0 = α d dx ( x) 0 [ dx( x) 0 x x IΠ(), + x x ] x Π x. B.E. Laurup- de Rafael 69 = α 0 Euclidean Q = x Q x. Se ω = = x x, dω [ ( ( + ω) + ω ω ) ] ( 4 + ω ω d ) ω 4 dω Π ω Coen on Laice QCD Evaluaions

5 Coen on Laice QCD Evaluaions = α dω 0 ω 4 [ ( ( + ω) + ω ω ) ] ( 4 + ω ω d G(ω) = 4 [ ( + ω)( + ω ω 4 + ω) ) dω Π ω ] G Ω Ω So far, Laice QCD evaluaions need exrapolaions wih Models or Padé Approxians

6 Moen Analysis = α dω 0 ω 4 [ ( ( + ω) + ω ω ) ] ( 4 + ω ω d d dω Π ( ω ) = d + ω IΠ(). ) dω Π ω Mellin-Barnes Inegral Represenaion of ( + A) = i d dω Π ω d = ( α = c+i ds(a) s Γ(s)Γ( s) c i ( c+i ω ds i c i ) c+i i c i ds F(s) M(s) ) s Γ(s)Γ( s) IΠ() d s F(s) = Γ(3 s)γ( 3 + s)γ( + s) and M(s) = IΠ() Mellin Transfor of Specral Funcion Siilar Techniques used in QED by J.Ph. Aguilar, D. Greyna, de Rafael 08

7 Relevan Moens = α c+i i c i ds F(s) M(s) d s F(s) = Γ(3 s)γ( 3 + s)γ( + s) and M(s) = IΠ() Mellin Transfor of Specral Funcion The Mellin ransfor in QCD is finie for s < and singular a s = : d s M(s) = IΠ() Two ypes of Moens Noral Moens: Log weighed Moens: d M( n) = d M( n) = +n IΠ(), n = 0,,,... +n log IΠ(), n =,, 3,

8 Calculaion in ers of Moens = α c+i i c i ds F(s) M(s) d s F(s) = Γ(3 s)γ( 3 + s)γ( + s) and M(s) = IΠ() Singular expansion of F(s): Mellin Transfor of Specral Funcion F(s) 3 s (s + ) + 5 s + 6 (s + ) s + 8 (s + 3) s Expansion in Moen Approxians = { α M( ) + 6 M( ) M(0) + 5 M( ) + M( ) M( 3) + 8 M( 3) + O [ M( 4)] }

9 Phenoenological Toy Model Hadronic Specral Funcion 0 5 Π I GeV D. Bernecker and H.B. Meyer, ; L. Lelllouch, 4 The os recen experienal deerinaion fro e + e daa: = (6.93± 0.04) 0 8. This phenoenological paraerizaion: =

10 Mellin Transfor in he Phenoenological Toy Model d s M(s) = IΠ() Mellin Transfor of Specral Funcion α M pqcd(s) s > N c 3 3 s s s The Mellin Transfor of IΠ() is a Monoonously Decreasing Funcion

11 Mellin Transfor and Derivaive of Mellin Transfor in Toy Model d M(s) = s IΠ(), M(s) = d ds M(s) = d s log IΠ() 0 5 s s s s Boh Mellin Transfors of IΠ() are decreasing sooh Funcions

12 The Moen Approxians in he Phenoenological Toy Model Expansion of in Moen Approxians = { α 5 M(0) + 3 M( ) + M( ) M( ) + 6 M( ) M( 3) + 8 M( 3) + O [ M( 4)] } Nuerical Values of he Moen Approxians (Toy Model) α 3 M(0) = (6%) [ ] α 5 M(0) + 3 M( ) + M( ) = (4%) [ α 5 M(0) + 3 M( ) + M( ) + 97 ] 0 M( ) + 6 M( ) = (%) Fourh Approxiaion is already wihin 0.4% of oy odel resul (oy odel) = (e+ e ) = (6.93 ± 0.04) 0 8 (0.6%)

13 The Moen Approxians in Laice QCD The Leading Moen is a rigorous upper bound ( α ) d < 3 ( ( α ) IΠ() = 3 ) d ) dq Π(Q Q =0 J.S. Bell-de Rafael 69 Overesiaes by less han 0% (no bad for a rigorous bound) ( The slope of Π Q ) a he origin (second er) can be evaluaed in laice QCD I is difficul o iagine ha, unless laice QCD does beer han phenoenology in his siple case, i will ever reach a copeiive accuracy of he full deerinaion of. M( n) Moens correspond o successive derivaives of Π(Q ) a Q = 0 d M( n) = n=0,,... +n ( )n+ n+ IΠ() = (n + )! ( )n+ ) ( Q ) n+π(q Q =0 These derivaives can be deerined in Laice QCD

14 The Log Weighed Moens in Laice QCD Firs do a change of scale: d M( n) = n log ( M( n) = log d M( n) + IΠ() ) n log IΠ() Consruc he Euclidean Inegral Σ( ) Σ( ) ( dq Q ) Π(Q ) d = Q } {{ } Laice QCD + d M(0) log( log + ) IΠ() IΠ() Noice he relaion: M( ) = log M( ) + Σ( ) La. QCD M(0) + M( ) +

15 The Log Weighed Moens in Laice QCD (coninuaion) Consruc Nex he Euclidean Inegral Σ( ) (wih one ore Q Σ( ) + ( dq Q ) 3 Π(Q ) d = Q } {{ } Laice QCD ) M(0) ( M( ) + 3 log d 3 log( + power) IΠ() ) IΠ() The relaion o he waned M( ) is hen: M( ) = log M( ) Σ( ) + La. QCD M(0) M( )+ M( 3)+

16 The Log Weighed Moens in Laice QCD (conclusion) Inegral Moens o be evaluaed in laice QCD Σ( n) ( ) n+ dq Π(Q ) Q Q n =,, 3... Conrary o he evaluaion of, he oens Σ( ), Σ( ),... are no weighed by a heavily peaked kernel a sall Q. The hreshold of inegraion is a a raher large value Q = insead of zero. The deerinaion of hese inegral oens in laice QCD and heir coparison wih he corresponding phenoenological expressions in ers of he hadronic specral funcion given above, can provide valuable furher ess.

17 Successive Aproxiaions o (phen. odel) = Quaniies o be evaluaed in laice QCD M(0) ; Σ( ), M( ) ; Σ( ), M( ) and M( 3) 0.00 Final Resuls (nubers are hose fro he Phenoenological Toy Model) s Approxiaion (sae as Bell-de Rafael): α M(0) = nd Approxiaion {( α 3 ) M(0) + = 7.65(34) log M( ) + Σ( ) + M( ) 3rd Approxiaion α M(0) + log 6 M( ) ( log + M( ) + Σ( ) 6Σ( ) + 6 ) } M( 3) = 7.07(6) 0 8. }

18 Conclusion My conclusion is ha he oen analysis approach described above ay gradually lead o an accurae deerinaion of providing a he sae ie any ess of laice QCD evaluaions o be confroned wih phenoenological deerinaions using experienal daa.,