. Update on Mellin-Barnes Approximants to HVP . . . Eduardo de Rafael Aix-Marseille Universit´ e, Univ. Toulon, CNRS, CPT, Marseille, France 21st June 2018 Second Workshop of the Muon g − 2 Theory Initiative MAINZ June 2018 Talk Based on: E.de R. Phys. Rev. (2017), J. Charles, D. Greynat, E.de R. Phys.Rev. (2018): ArXiv:1712.02202v3. Work in progress with J´ erˆ ome Charles and David Greynat.
. HVP Contribution to the Muon Anomaly . Hadronic Spectral Function Representation . . ∫ 1 ∫ ∞ x 2 ( 1 − x ) = α dt 1 a HVP dx π Im Π( t ) µ x 2 + t π t µ ( 1 − x ) 4 m 2 0 m 2 π σ ( t ) [ e + e − → ( γ ) → Hadrons ] = 4 π 2 α 1 Im Π( t ) . . . t π . Euclidean Hadronic Self-Energy Representation B.E. Lautrup-E. de Rafael ’69 , EdeR ’94 , T. Blum ’03 . . ∫ 1 x 2 ∫ ∞ 1 − x m 2 = α dt 1 µ a HVP dx ( 1 − x ) π Im Π( t ) , µ π t x 2 t + 1 − x m 2 4 m 2 0 µ π � �� � Dispersion Relation ∫ 1 ( ) x 2 = − α Q 2 ≡ 1 − x m 2 dx ( 1 − x ) Π . µ π 0 � �� � Accessible via LQCD . . . EdeR Mellin-Barnes Approximants to HVP
. Mellin- Barnes Representation EdeR’14 ∫ 1 ∫ ∞ x 2 ( 1 − x ) = α dt 1 a HVP π Im Π( t ) dx µ x 2 + t π t µ ( 1 − x ) 4 m 2 0 m 2 π ∫ 1 ∫ ∞ m 2 = α dt 1 1 µ dx x 2 π Im Π( t ) , π t t x2 1 − x m 2 4 m 2 0 µ π 1 + t ( ) − s x2 cs + i ∞ 1 − x m2 ∫ µ 1 1 Inserting = ds Γ( s )Γ( 1 − s ) and integrating over x x2 2 π i t 1 − x m2 cs − i ∞ µ 1 + t Mellin-Barnes Representation c s + i ∞ ( ) − s ) m 2 ∫ m 2 ( α 1 µ µ a HVP = F ( s ) M ( s ) , c s ≡ Re ( s ) ∈ ] 0 , 1 [ ds µ π t 0 2 π i t 0 c s − i ∞ t 0 = 4 m 2 F ( s ) = − Γ( 3 − 2 s ) Γ( − 3 + s ) Γ( 1 + s ) , π ; ( t ) s − 1 1 ∫ ∞ dt M ( s ) = π Im Π( t ) t t 0 t 0 � �� � Mellin Transform of the Spectral Function EdeR Mellin-Barnes Approximants to HVP
. Properties of the Mellin Transform of the Spectral Function ( t ∫ ∞ ) s − 1 1 dt 1 t ≥ t 0 = 4 m 2 M ( s ) = π Im Π( t ) , π Im Π( t ) ≥ 0 π ± . for t t 0 t 0 . Complete Monotonicity . . The Positivity of 1 π Im Π( t ) implies that M ( s ) and all its derivatives are Monotonically Increasing functions for −∞ < s < 1 , with extension to the full complex s-plane by Analytic Continuation. . . . . Spectral Function Moments: M ( s = 0 , − 1 , − 2 , · · · ) . . ∞ ( t 0 ( ) ) 1 + n 1 ∫ = ( − 1 ) n + 1 ∂ n + 1 dt ( n + 1 )! ( t 0 ) n + 1 ( ∂ Q 2 ) n + 1 Π( Q 2 ) π Im Π( t ) , n = 0 , 1 , 2 , · · · t t Q 2 = 0 t 0 � �� � � �� � LQCD and / or Dedicated Experiment Experiment . . . . The Leading Moment is an upper bound to a HVP ( J.S. Bell-EdeR ’69 ) µ . . ( α ) 1 ∫ ∞ m 2 m 2 ( ) ≤ α 1 dt t 0 1 ∂ µ µ a HVP ∂ Q 2 Π( Q 2 ) π Im Π( t ) = − t 0 µ π 3 t 0 t t π 3 t 0 4 m 2 Q 2 = 0 π � �� � � �� � LQCD M ( 0 ) . . . EdeR Mellin-Barnes Approximants to HVP
. Mellin-Barnes Approximants Ramanujan’s Master Theorem (-G.H. Hardy’s proof-) ( ) ( ) s − 1 ∫ ∞ Q 2 Q 2 d t 0 t 0 0 {( ( ) 2 } ) M ( 0 ) − Q 2 Q 2 − t 0 Q 2 Π( Q 2 ) ≡ M ( − 1 ) + M ( − 2 ) + · · · = Γ( s )Γ( 1 − s ) M ( s ) t 0 t 0 Q 2 → 0 Convergence of Discrete Moments M ( − n ) to the Full Mellin Transform M ( s ) ( − n ⇒ s ) . Marichev’s Class of Mellin Transforms Superpositions of Standard Products of gamma functions of the type: Γ( a i − s )Γ( c j + s ) ∑ ∏ M ( s ) = λ n , a i , b k , c j , d l constants λ n Γ( b k − s )Γ( d l + s ) , n i , j , k , l Practically all functions in Mathematical Physics have Mellin transforms of this type. We propose to consider Mellin-Approximants to M HVP ( s ) of this type, restricted by QCD-properties to the subclass: N Γ( a k − s ) ∑ ∏ M N ( s ) = λ n Γ( b k − s ) n k = 1 with λ n , a i , b k , c j , d l constrained by Monotonicity, and fixed by Matching to Input Moments. EdeR Mellin-Barnes Approximants to HVP
. QED Vacuum Polarization Test J. Mignaco-E. Remiddi ’69 ( α ) 3 { 673 108 − 41 81 π 2 − 4 9 π 2 log ( 2 ) − 4 9 π 2 log 2 ( 2 ) + 4 7 9 log 4 ( 2 ) − 270 π 4 a VP = µ π ( α [ ]} ) 3 + 13 18 ζ ( 3 ) + 32 4 , 1 = 0 . 0528707 · · · 3 PolyLog 2 π ( α ) 3 Results from Mellin Approximants M N ( s ) in units of π Input Moments Numerical result Accuracy M ( 0 ) 0 . 0500007 5 % M ( 0 ) , M ( − 1 ) 0 . 0531447 0 . 5 % M ( 0 ) , M ( − 1 ) , M ( − 2 ) 0 . 0528678 0 . 004 % M ( 0 ) , M ( − 1 ) , M ( − 2 ) , M ( − 3 ) 0 . 0528711 0 . 00075 % M ( 0 ) , M ( − 1 ) , M ( − 2 ) , M ( − 3 ) , M ( − 4 ) 0 . 0528706 0 . 00018 % Convergence of Mellin-Approximants tested numerically up to N = 9 EdeR Mellin-Barnes Approximants to HVP
. QED (fourth order) Fast Convergence for a VP µ � � a VP µ ( N ) − a VP � � µ ( exact ) Logarithmic Plot of � versus number of input moments � � a VP µ ( exact ) � Convergence speed for the muon anomaly 1 0.001 10 - 6 10 - 9 0 2 4 6 8 nb of input moments EdeR Mellin-Barnes Approximants to HVP
from e + e − → Hadrons QCD Test with Experimental Moments kindly provided to us by Alex Keshavarzi and Thomas Teubner . a HVP ( exp . ) = ( 6 . 933 ± 0 . 025 ) × 10 − 8 µ A. Keshavarzi, D. Nomura, T. Teubner, arXiv:1802.02995v1 [hep-ph] M ( s ) Moments and Errors in 10 − 3 units Moment Experimental Value Relative Error M ( 0 ) 0 . 7176 ± 0 . 0026 0 . 36 % M ( − 1 ) 0 . 11644 ± 0 . 00063 0 . 54 % M ( − 2 ) 0 . 03041 ± 0 . 00029 0 . 95 % M ( − 3 ) 0 . 01195 ± 0 . 00017 1 . 4 % M ( − 4 ) 0 . 00625 ± 0 . 00011 1 . 8 % M ( − 5 ) 0 . 003859 ± 0 . 000078 2 . 0 % · · · · · · · · · Results from Mellin Approximants in 10 − 8 units a HVP µ Input Moments Type of Approximant Central Value Stat. Uncert. s = 0 N = ( 1 ) 6 . 991 0 . 023 s = 0 , − 1 N = ( 2 ) 6 . 970 0 . 024 s = 0 , − 1 , − 2 N = ( 2 ) + ( 1 ) 6 . 957 0 . 025 s = 0 , − 1 , − 2 , − 3 N = ( 2 ) + ( 1 ) + ( 1 ) 6 . 932 0 . 025 EdeR Mellin-Barnes Approximants to HVP
. Results for a HVP with Errors µ EdeR Mellin-Barnes Approximants to HVP
. Beta-Function Approximants to HVP ( Particular type of Approximants) Beta-Function Approximants to the Mellin Transform of the Spectral Function N M N ( s ) = α 5 λ n Γ( b n − n ) Γ( n − s ) ∑ , λ 1 = 1 , b n ≥ n + 1 . π 3 Γ( b n − s ) n = 1 � �� � Beta ( n − s , b n − n ) They have simple Hadronic Self-Energy Approximants: ( 1 � N ) Q 2 � − Q 2 Π N ( Q 2 ) = − α 5 λ n Γ( b n − n ) n ∑ � Γ( n ) 2 F 1 � b n π 3 t 0 Γ( b n ) t 0 n = 1 � �� � Gauss Hypergeometric Function and Equivalent simple Spectral Functions: N ( 4 m 2 ) n − 1 ( ) b n − n − 1 1 − 4 m 2 1 π Im Π N ( t ) = α 5 ∑ π π θ ( t − 4 m 2 λ n π ) , π 3 t t n = 1 with the matching solutions for λ n and b n ≥ n + 1 constrained by the positivity of 1 π Im Π N ( t ) . EdeR Mellin-Barnes Approximants to HVP
. Example: Results of three superpositions Using the central values of the first five moments from experiment: ( N = 3 ) = 6 . 9335 × 10 − 8 . a HVP µ Shape of the “Equivalent” Spectral Function in α π units: 1.5 1.5 1.0 Im Π t ) 1.0 Im Π t ) 1 0.5 1 0.5 0.0 0.0 0 20 40 60 80 100 2 4 6 8 10 t / t 0 t / t 0 EdeR Mellin-Barnes Approximants to HVP
. Shape of the “Equivalent” Spectral Function Number of Input Moments =9 QCD spectral function approximant , N = 9 2.0 1.5 1.0 0.5 0.0 0 20 40 60 80 100 t / t 0 EdeR Mellin-Barnes Approximants to HVP
. Convergence Test � � a HVP ( N ) − a HVP � � ( exp . ) µ µ Log Plot of � versus N � � a HVP ( exp . ) � µ Convergence speed for the muon anomaly 0.005 0.001 5. × 10 - 4 1. × 10 - 4 5. × 10 - 5 1. × 10 - 5 0 2 4 6 8 nb of input moments EdeR Mellin-Barnes Approximants to HVP
. Conclusions We claim that, from an Accurate LQCD Determination , of the first few moments, one could reach an evaluation of a HVP with competitive µ precision -or even higher- than the present experimental determinations. Accurate determination of the First Moment is an excellent Test ( α ) 1 ∫ ∞ m 2 ( ) ≤ α 1 dt 1 ∂ µ a HVP − m 2 ∂ Q 2 Π( Q 2 ) π Im Π( t ) = µ µ π 3 t t π 3 4 m 2 Q 2 = 0 π � �� � � �� � LQCD and / or DEDICATED EXPERIMENT M ( 0 ) from experiment The fact that the Π( Q 2 ) Beta-Function Approximants are simple superpositions of simple Gauss-Hypergeometric-Functions offers the possibility of using LQCD information on values of Π( Q 2 ) at fixed Q 2 -values , or an Alternative Input to Moments. EdeR Mellin-Barnes Approximants to HVP
. Breit-Wigner plus Theta-like Spectral Function ( α ) 1 Γ M N c ∑ f 2 V M 2 q 2 π Im Π( t ) = θ ( t − t 0 ) + 3 θ ( t − t pQCD ) ( t − M 2 ) 2 + Γ 2 M 2 f π f � �� � ⇒ πδ ( t − M 2 ) for Γ → 0 Shape of this Spectral Function (M = M ρ , Γ = Γ ρ , t 0 = 4 m 2 π , f 2 V = 0 . 51 ). 2.5 2.0 Im Π x ) 1.5 1.0 1 0.5 0.0 0 2 4 6 8 10 t x = M 2 EdeR Mellin-Barnes Approximants to HVP
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