Asymptotic expansions and Mellin-Barnes representation David Greynat - PowerPoint PPT Presentation

Asymptotic expansions and Mellin-Barnes representation David Greynat I.F.A.E. Universitat Autonoma de Barcelona 29 September 2009 in collaboration with Jean-Philippe AGUILAR, Samuel FRIOT and Eduardo de RAFAEL Conference on ”Approximation and extrapolation of convergent and divergent sequences and series”

Introduction Mellin Transform A physical example MDMT Asymptotic analysis Scheme Context: Phenomenology in Quantum Field theory Mellin-Barnes representation may be of use for at least two important questions in the phenomenology of QFT: Perturbative calculations in QFT imply to evaluate numerous Feynman 1 diagrams often with several masses and momenta. How can we evaluate them analytically ? Perturbative expressions are series in powers of the coupling constants. 2 How can we have non-perturbative ( i.e. exponentially suppressed) informations ? D. Greynat Asymptotic expansions and Mellin-Barnes representation

Introduction Mellin Transform A physical example MDMT Asymptotic analysis Scheme Why using Mellin-Barnes representation? The first observation is that the Mellin transform has the following scale property M [ f ( ax )] ( s ) = a − s M [ f ( x )] ( s ) Renormalization Group solutions lead to consider asymptotic behaviours of the diagrams as a α ln β a . The Mellin transformation kernel is the most pertinent to obtain this type of asymptotic behaviour. The Mellin-Barnes representation allows expansion in several parameters and it also gives an explicit formula for the remainder that permits a control on perturbative and (in some cases) non-perturbative expansions. D. Greynat Asymptotic expansions and Mellin-Barnes representation

Introduction Mellin Transform A physical example MDMT Asymptotic analysis Scheme Mellin-Barnes representation and Feynman diagrams calculation S. Friot, D. Greynat and E. de Rafael, Phys. Lett. B 628 , 73 (2005) J.-Ph. Aguilar, D. Greynat and E. de Rafael, Phys. Rev. D 77 , 093010 (2008) D. Greynat Asymptotic expansions and Mellin-Barnes representation

Converse Mapping theorem Introduction Mellin Transform A physical example MDMT Asymptotic analysis Definitions Scheme One dimensional Mellin Transform The Mellin transform of a function f and its inverse transform are defined as c + i ∞ ˆ ∞ ds ˆ M [ f ( x )]( s ) . dx x s − 1 f ( x ) 2 i π x − s M [ f ( x )]( s ) f ( x ) = = ← → 0 c − i ∞ If and only if c . = Re s ∈ ] α, β [ Fundamental strip written � α, β � It corresponds to the behaviours f ( x ) x → 0 + O ( x − α ) f ( x ) x → + ∞ O ( x − β ) = & = π ( 1 + x ) − 1 ← → � 0 , 1 � sin π s Γ( ν − s )Γ( s ) ( 1 + x ) − ν � 0 , Re ν � ← → Γ( ν ) π ln ( 1 + x ) ← → �− 1 , 0 � s sin π s D. Greynat Asymptotic expansions and Mellin-Barnes representation

Converse Mapping theorem Introduction Mellin Transform A physical example MDMT Asymptotic analysis Definitions Scheme Idea The singularities in the complex Mellin’s plan govern completely the asymptotic behaviour of the associated function We need to define the singular expansion From the Laurent series of a function ϕ in p ( s − p ) n + · · · + A − 1 A − n ϕ ( s ) = s − p + A 0 + A 1 ( s − p ) + · · · one can build the formal series by summing all over the poles of ϕ of the singular part: � � ( s − p ) n + · · · + A − 1 A − n � s − p p this is the singular expansion of ϕ and it is written as � ( s − p ) n + · · · + A − 1 A − n � � ϕ ( s ) ≍ s − p p D. Greynat Asymptotic expansions and Mellin-Barnes representation

Converse Mapping theorem Introduction Mellin Transform A physical example MDMT Asymptotic analysis Definitions Scheme Converse Mapping Theorem If f satisfies the condition to have a Mellin transform in the fundamental strip � α, β � � | s | − η � and M [ f ] ( s ) = O for η > 1. Converse Mapping Theorem c p , n ( − 1 ) n c n , p ( n − 1 )! x − p ln n − 1 x � � M [ f ( x )] right ( s ) ≍ f ( x ) ↔ ∼ ( s − p ) n x → + ∞ p >β, n p >β, n ( − 1 ) n − 1 d n , p d p , n x p ln n − 1 x � � M [ f ( x )] left ( s ) ≍ f ( x ) ∼ ↔ ( s + p ) n ( n − 1 )! x → 0 p <α, n p <α, n D. Greynat Asymptotic expansions and Mellin-Barnes representation

Introduction Mellin Transform A physical example MDMT Asymptotic analysis Description Scheme Resummation A physical and practical example: g − 2 Lautrup and de Rafael (1964) Friot, Greynat and de Rafael (2005) � α � 2 ˆ 1 dx � x 2 � ℓ a µ = x ( 1 − x )Π ( ℓ ) 1 − x m 2 − R µ π 0 � α � 2 ˆ 1 ˆ 1 dx y ( 1 − y ) x ( 1 − x )( 2 − x ) dy = m 2 π 1 − x 1 + 0 0 ℓ m 2 x 2 y ( 1 − y ) µ c + i ∞ ds 1 ˆ π 2 i π X − s Inverse Mellin Representation: 1 + X = � 0 , 1 � sin π s c − i ∞ � m 2 c + i ∞ � − s � α ds � 2 ˆ ˆ 1 ˆ 1 π a µ = dx x 2 s − 1 ( 1 − x ) 1 − s ( 2 − x ) dy y 1 + s ( 1 − y ) 1 + s ℓ 2 i π m 2 sin π s π 0 0 µ c − i ∞ � m 2 c + i ∞ � − s � � α ds 1 − s � 2 � 2 ˆ π ℓ = 2 i π m 2 sin π s ( 2 + s )( 1 + 2 s )( 3 + 2 s ) π µ c − i ∞ D. Greynat Asymptotic expansions and Mellin-Barnes representation

Introduction Mellin Transform A physical example MDMT Asymptotic analysis Description Scheme Resummation c + i ∞ � − s � � � α � 2 � 2 ds m 2 1 − s ˆ π a µ = ℓ 2 i π m 2 sin π s ( 2 + s )( 1 + 2 s )( 3 + 2 s ) π µ c − i ∞ • For m ℓ = m τ then m 2 ℓ ≫ 1: Right side of the fundamental strip m 2 µ � 2 1 − s � π sin π s ( 2 + s )( 1 + 2 s )( 3 + 2 s ) � � � � � � − 1 1 − 1 1 9 1 s − 1 + ( s − 2 ) 2 + s − 2 + · · · ≍ − 45 140 19600 then   � 2 � 2 � � � � � 2 m 2 m 2 m 2 m 2 � α  1 1 9 a µ = µ µ µ µ + + + · · · ln  m 2 m 2 m 2 m 2 45 140 19600 π ℓ ℓ ℓ ℓ • For m ℓ = m e then m 2 ℓ ≪ 1: Left side of the fundamental strip m 2 µ � 1 � 2 � 1 − s � s + π 2 ( 2 + s )( 1 + 2 s )( 3 + 2 s ) ≍ 1 1 − 25 1 π s 2 + + · · · sin π s s + 1 6 36 4 2 then  � 2  � � � � α � 2 m 2 m 2 + π 2 m ℓ  1 − 25 9 µ µ a µ = 6 ln 36 + · · ·  m 2 m 2 m µ 19600 4 π ℓ ℓ D. Greynat Asymptotic expansions and Mellin-Barnes representation

Introduction Mellin Transform A physical example MDMT Asymptotic analysis Description Scheme Resummation Exact solution Flajolet et al. (1994) Friot et Grunberg JHEP 0709 :002 (2007) Those two expansions of the anomaly are the exact representation for m 2 m 2 r = µ ≫ 1 and r = ℓ ℓ µ ≪ 1 because in the rest of the two asymptotic series m 2 m 2 ⌊ T ⌋ c np r n ln p r + R ( T ) � a µ = n , p there are no exponential corrections � � ± T + iT � � ds 1 − s 2 i π r − s � � 2 ˆ π � � | R ( T ) | = � � sin π s ( 2 + s )( 1 + 2 s )( 3 + 2 s ) � � � � ± T − iT � 1 ± T � � r ± T 2 T π 2 � � � � ( 2 ± T )( 1 ± 2 T )( 3 ± 2 T ) � � � r ± T � R ( T ) T →∞ o = Therefore the resummation of all the contributions from each poles is convergent and give the exact function. D. Greynat Asymptotic expansions and Mellin-Barnes representation

Introduction Mellin Transform A physical example MDMT Asymptotic analysis Description Scheme Resummation Resummation It is easy to perform resummations with the Mellin representation, in our example on right-side � 2 � � 1 − s ∞ � 2 + n + 1 1 1 − 5 1 1 � π ( 2 + s )( 1 + 2 s )( 3 + 2 s ) ≍ sin π s 2 + n 2 + n ( s − p ) 2 1 3 4 4 p = 1 � � ∞ ( 2 + n ) 2 + 1 1 2 + n � 2 − 5 1 1 1 � + � 1 � 3 s − p 2 + n � 2 4 4 p = 1 m 2 And using the Converse Mapping Theorem ( with r = µ ≫ 1) we have the ℓ m 2 convergent expression by identification of the series as ”usual” functions � 1 � 1 � 2 � � α � � r , 2 , 3 r , 2 , 5 ln r − ln r − 1 4 − r + Φ − 5 Φ + 1 � √ � a µ = √ r ArcCoth r 2 2 4 r 4 r 2 6 π � 1 � � �� + 3 2 r ln ( r ) − 5 � √ � 1 − 1 ln r − r 2 ln 2 r 3 / 2 ArcCoth r ln r + r 2 Li 2 r r D. Greynat Asymptotic expansions and Mellin-Barnes representation

Introduction Mellin Transform A physical example MDMT Asymptotic analysis Description Scheme Resummation m 2 Using the Converse Mapping Theorem ( with r = µ ≪ 1) we have the ℓ m 2 expression � 2 � � � � α 36 − π 2 4 r 1 / 2 + 3 r − 5 π 2 9 + π 2 � � − 25 44 r 2 + 5 r , 2 , 3 a µ = 4 r 3 / 2 + 4 r 3 Φ 3 2 π � � − 1 r , 2 , 5 − 1 6 ln r + 3 2 r ln r + 1 � √ � √ 4 r 3 Φ r ArcTanh r 2 2 � − 5 ln r − r 2 ln ( 1 − r ) ln r + 1 2 r 2 ln 2 r − r 2 Li 2 ( r ) � √ � 2 r 3 / 2 ArcTanh r D. Greynat Asymptotic expansions and Mellin-Barnes representation