Asymptotic expansions and Mellin-Barnes representation. David Greynat. I.F.A.E.
. Universitat Autonoma de Barcelona. 29 September 2009 in collaboration with.
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: 1
Perturbative calculations in QFT imply to evaluate numerous Feynman diagrams often with several masses and momenta. How can we evaluate them analytically ?
2
Perturbative expressions are series in powers of the coupling constants. 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 aα lnβ 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,
D. Greynat
Phys. Rev. D 77, 093010 (2008)
Asymptotic expansions and Mellin-Barnes representation
Introduction Mellin Transform A physical example MDMT Asymptotic analysisDefinitions Scheme Converse Mapping theorem
One dimensional Mellin Transform The Mellin transform of a function f and its inverse transform are defined as . M[f (x)](s) =
ˆ
∞
dx x s−1 f (x)
0
If and only if . c = Re s ∈]α, β[
←→
f (x) =
c+i∞ ˆ
ds −s x M[f (x)](s) 2iπ
c−i∞
written
hα, βi
Fundamental strip
It corresponds to the behaviours f (x) = O(x −α )
&
x →0+
(1 + x)−1
←→
(1 + x)−ν
←→
ln(1 + x)
←→ D. Greynat
f (x)
π sin πs Γ(ν − s)Γ(s) Γ(ν) π s sin πs
=
x →+∞
O(x −β )
h0, 1i h0, Re νi h−1, 0i
Asymptotic expansions and Mellin-Barnes representation
Introduction Mellin Transform A physical example MDMT Asymptotic analysisDefinitions Scheme Converse Mapping theorem
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 ϕ(s) =
A−n A−1 +··· + + A0 + A1 (s − p) + · · · (s − p)n s−p
one can build the formal series by summing all over the poles of ϕ of the singular part: X A−n A−1 + · · · + (s − p)n s−p p this is the singular expansion of ϕ and it is written as ϕ(s)
X p
A−1 A−n + ··· + (s − p)n s−p
D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Introduction Mellin Transform A physical example MDMT Asymptotic analysisDefinitions Scheme Converse Mapping theorem
Converse Mapping Theorem
If f satisfies the condition to have a Mellin transform in the fundamental strip hα, βi and M [f ] (s) = O |s|−η for η > 1.
Converse Mapping Theorem M [f (x)]right (s)
M [f (x)]left (s)
X
cp,n (s − p)n
↔
X
dp,n (s + p)n
↔
p>β,n
pβ,n ∼
X (−1)n−1 dn,p p n−1 x ln x x →0 (n − 1)! p 0 We will focus on the vacuum-vacuum transitions 1 Z (0) = √ 2π
ˆ
D. Greynat
+∞
1
dφ e − 2! φ
2
λ φ4 − 4!
−∞
Asymptotic expansions and Mellin-Barnes representation
Introduction Mellin Transform A physical example MDMT Asymptotic analysis0-dim Scheme example Perturbative expansion Numerical analysis
Perturbative expansion Taking λ small, one has the perturbative expansion of Z (0) k ∞ k 1 1 X (−1) Γ 2 + 2k 35 2 385 3 λ 1 Z (0) ∼ √ λ − λ +O(λ4 ) = 1− λ+ λ→0 k! 6 8 384 3072 π k =0 Yet
(−1)k Γ 1 + 2k λ k 2 ∀λ, −−−→ ∞ k! 6 k →∞
then this is a divergent series. Let us define SnPert. and Rn
k k n−1 ∞ k k 1 1 1 X (−1) Γ 2 + 2k λ λ 1 X (−1) Γ 2 + 2k +√ Z (0) ∼ √ λ→0 k! 6 k! 6 π k =0 π k =n {z } {z } | | . . =Rn =S Pert. n−1
D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Introduction Mellin Transform A physical example MDMT Asymptotic analysis0-dim Scheme example Perturbative expansion Numerical analysis
Counting diagrams One of the interests of the 0-dimensional theory is that SNPert. counts the number of Feynman diagrams to a given order N in dimension 4. Z (0) = 1 − λ→0
1 35 2 385 3 λ+ λ − λ + O(λ4 ) 8 384 3072
Order λ Z (0) =
(−1) λ 3 1! 4!
+ O(λ2 ) = −
1 λ 8
+ O(λ2 )
Order λ2 24 384
λ2
+
3 384
D. Greynat
λ2
+
8 384
λ2
Asymptotic expansions and Mellin-Barnes representation
Introduction Mellin Transform A physical example MDMT Asymptotic analysis0-dim Scheme example Perturbative expansion Numerical analysis
Numerical analysis 1
From now, for all numerical calculations, we fix λ = 13 . Mathematica gives with 8 decimal precision Z (0)|λ= 1 ≈ 0.96556048...
3
5
7
9
11
1
13 1
0.98
0.98 0.96556048
Pert
Sn-1 0.96
0.96
0.94
0.94
3
0.92
0.92 1
3
5
7
9
11
n
Choosing a resummation procedure for the perturbative series leads to the property that |Rn | < |un | and |Rn | < |un−1 |. One then has η−1
Z (0) =
X
uk +
k =0
here
1 1 |uη | ± |uη | , 2 2
Z (0)|λ= 1 = 0.96555187 ± 0.00140990 3
The central value corresponds to the standard Stieltjes approximation which is given, for the rank η defined ∀n, |uη | 6 |un | η−1
Z (0) =
X k =0
D. Greynat
uk +
1 |uη | . 2
Asymptotic expansions and Mellin-Barnes representation
13
Introduction Mellin Transform A physical example MDMT Asymptotic analysis Scheme
Z (0) Formal approach
Analytic approach Mellin-Barnes Representation
Mapping
MB - Remainder
Exp. improvement
Perturbative Expansion
Remainder (tail)
Non-perturbative Expansion
Iterative procedure Hyperasymptotic expansions D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Hyperasymptotic Theory
Analytic approach
D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Hyperasymptotic Theory
Initial remarks Construction of the remainder The iterative procedure
Mellin-Barnes Hyperasymptotic Theory Main Idea Construct an exponentially small remainder with its inverse Mellin-Barnes representation (non perturbative in λ). The first step is to obtain an inverse Mellin-Barnes representation of Z (0): −s ˆ λ 4 λ 4 ds φ Γ(s) , e − 4! φ = 2iπ 4! c+iR
with h0, +∞i and | arg λ| < 1 Z (0) = √ π
π . 2
Then we obtain the following results
ˆ
ds 2iπ
c+iR
with 0,
1 4
and | arg λ|
0 and |b0 | < ∞ so that if a0 + b0 |λ|
n=
then Rn is exponentially small in λ (non perturbative). Here, a
|Rn | = O |λ|→0
e
0 − |λ|
2a0 3
a0 |λ|
!
and for a0 = 32 , we have the Optimal Truncation Scheme i.e. the smallest remainder − 3 |Rn | = O e 2|λ| . n→∞
D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Hyperasymptotic Theory
Initial remarks Construction of the remainder The iterative procedure
The iterative procedure
Main Idea It is possible to construct more and more exponentially small remainders order by order in the expansion of Rn .
With the duplication formula 22z−1 1 Γ(2z) = √ Γ(z)Γ z + 2 π we can rewrite the remainder as Rn =
1 √
π 2
ˆ −c+n+iR
dt 2iπ
3 2λ
−t
1 3 π Γ t+4 Γ t+4 sin πt Γ(t + 1)
We introduce now the main tool : the inverse factorial expansion.
D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Hyperasymptotic Theory
Initial remarks Construction of the remainder The iterative procedure
The inverse factorial expansion Barnes’ lemma Γ(s + a)Γ(s + b) = Γ(s + c)
ˆ iR
dt Γ(t + c − a)Γ(t + c − b)Γ(s + ϑ − t)Γ(−t) 2iπ Γ(c − a)Γ(c − b)
with ϑ = a + b − c. Inverse factorial expansion M−1 X (−1)j Γ(s + a)Γ(s + b) = (c − a)j (c − b)j Γ(s + ϑ − j) Γ(s + c) j! j=0 ˆ dt Γ(t + c − a)Γ(t + c − b)Γ(s + ϑ − t)Γ(−t) + 2iπ Γ(c − a)Γ(c − b) M+i R
M > Re (a − c) + δ, M > Re (b − c) + δ, M > Re (s + ϑ) for 0 < δ < 1 and | arg s| < π2 .
Olver 1995
D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Hyperasymptotic Theory
Initial remarks Construction of the remainder The iterative procedure
One has in our example,
m−1 X Γ t + 14 Γ t + 34 (−1)j Aj Γ(t − j) + = Γ(t + 1) j=0
where Aj =
1 j!
1 3 4 j 4 j
f (s) =
and
The remainder is then m−1 1 X Rn = − √ (−1)j Aj π 2 j=0
dt 2iπ
ds dt ∧ 2iπ 2iπ
ˆ −c+n+iR
3 2λ
ˆ
ds f (s) Γ(t − s) 2iπ
c+m+i R
Γ(−s)Γ s + 14 Γ s + 43 Γ 14 Γ 34
−t
π Γ (t − j) + Rn,m sin πt
−t
π f (s)Γ(t − s) sin πt
with 1 Rn,m = − √ π 2
ˆ (−c+n+iR) ×(−c+m+iR)
Yet, we have (m 6 n), |Rn,m | = O n→∞
D. Greynat
3 2λ
! −n−c 3 −n n−m m+c− 12 e (n − m) m 2λ Asymptotic expansions and Mellin-Barnes representation
Hyperasymptotic Theory
Initial remarks Construction of the remainder The iterative procedure
Optimal Truncation Scheme For m =
a1 |λ|
+ b1 , we have
|Rn,m | = O |λ|→0
|Rn | = O |λ|→0
Paris et al.
p e
90’s
a
|λ|e
1 a +ln 3−ln 2 − 0 |λ| a1|λ| (a0
a
0 − |λ|
2a0 3
a0 |λ|
− a1 )
a0 −a1 |λ|
!
We can optimize the remainders in two ways, first satisfying the condition on Rn and then optimizing Rn,m or optimizing directly Rn,m :
OTS 1
3 a0 = 2
OTS 2
a0 = 3
3 a1 = 4 3 a1 = 2
D. Greynat
Rn = O(·) e
3 − 2|λ|
−
p
Rn,m = O(·) |λ|e
3 (1+ln 2) − 2|λ|
p
|λ|e
3 − |λ|
Asymptotic expansions and Mellin-Barnes representation
Hyperasymptotic Theory
Initial remarks Construction of the remainder The iterative procedure
Hyperasymptotic expansion of Z (0) at first hyperasymptotic level
Z (0) =
n−1 X
(−1)k Ak
k =0
3 2λ
k
m−1 1 X − √ (−1)j Aj π 2 j=0
1 − √ π 2
ˆ
dt 2iπ
−c+n+iR
ds dt ∧ 2iπ 2iπ
ˆ (−c+n+iR) ×(−c+m+iR)
D. Greynat
3 2λ
3 2λ
−t
−t
π Γ (t − j) sin πt
π f (s)Γ(t − s) sin πt
Asymptotic expansions and Mellin-Barnes representation
Hyperasymptotic Theory
Initial remarks Construction of the remainder The iterative procedure
OTS1 at first hyperasymptotic level h
Z (0) =
i
3 −1 2|λ|
X
(−1)k Ak
k =0
m−1 1 X (−1)j Aj − √ π 2 j=0
+O
p
|λ|e
−
3 2λ
k ˆ
h i 3 −c+ 2|λ| +iR
3(1+ln 2) 2|λ|
dt 2iπ
3 2λ
−t
π Γ (t − j) sin πt
OTS2 at first hyperasymptotic level Z (0) =
h
i
3 −1 |λ|
X
(−1)k Ak
k =0
1 − √ π 2 +O
h
p
i
3 2λ
k
3 −1 2|λ|
X
j
(−1) Aj
j=0
|λ|e
3 − |λ|
ˆ
h i 3 −c+ |λ| +iR
D. Greynat
dt 2iπ
3 2λ
−t
π Γ (t − j) sin πt
Asymptotic expansions and Mellin-Barnes representation
Hyperasymptotic Theory
Initial remarks Construction of the remainder The iterative procedure
Iterations... We have shown that it is possible to write the remainder as m−1 1 X Rn = − √ (−1)j Aj π 2 j=0
1 − √ π 2
where f (s) =
ˆ
dt 2iπ
−c+n+iR
3 2λ
ds dt ∧ 2iπ 2iπ
ˆ (−c+n+iR) ×(−c+m+iR)
−t
π Γ (t − j) sin πt
3 2λ
−t
π f (s)Γ(t − s) sin πt
1 3 Γ(−s)Γ s + 14 Γ s + 43 1 Γ s+ 4 Γ s+ 4 π √ = − Γ(s + 1) sin πs Γ 14 Γ 34 π 2
Yet applying the Inverse factorial expansion one has Functional equation 0 m −1 π X 1 (−1)l Al Γ(s − l) + f (s) = − √ π 2 sin πs l=0
ˆ c+m0 −i R
dt f (t) Γ(s − t) 2iπ
Then using this expression of f in Rn one may get straightforwardly a hyperasymptotic expansions at an arbitrary hyperasymptotic level. D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Resummation Resurgence phenomenon Numerics
Formal approach
D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Resummation Resurgence phenomenon Numerics
The starting point is now the expression of Z (0) as a divergent series k k n−1 ∞ k k 1 1 λ λ 1 X (−1) Γ 2 + 2k 1 X (−1) Γ 2 + 2k Z (0) ∼ √ +√ λ→0 k! 6 k! 6 π k =0 π k =n {z } | {z } | . . =Rn =S Pert. n−1
Using the duplication formula,
Rn =
k ∞ 1 3 1 XΓ k+4 Γ k+4 2λ √ − Γ (k + 1) 3 π 2 k =n
We can apply the Inverse factorial expansion for m−1 X Γ k + 14 Γ k + 34 (−1)j Aj Γ(k − j) + = Γ(k + 1) j=0
1 where Aj = j!
1 3 4 j 4 j
and
D. Greynat
ˆ
ds f (s) Γ(k − s) 2iπ
c+m+i R
Γ(−s)Γ s + 14 Γ s + 43 f (s) = Γ 14 Γ 34
Asymptotic expansions and Mellin-Barnes representation
Resummation Resurgence phenomenon Numerics
Resummation and terminant function The remainder is then Rn =
k ∞ m−1 X −2λ 1 X √ Γ(k − j) (−1)j Aj 3 π 2 j=0 k =n k ˆ ∞ X ds −2λ 1 Γ(k − s) + √ f (s) 2iπ 3 π 2 c+m+i R
k =n
One can now perform a Borel resummation on the two infinite sums in Rn , "∞ k # n X 2λ 3 −2λ = Γ(n − j) − Λn−j−1 Γ(k − j) B 3 3 2λ k =n
The function Λ` is known as a terminant function, here Λ` (x) = x
`+1
ˆ
∞
where Γ(a, x) =
Dingle 1973
x
e Γ(−`, x)
dy y a−1 e −y
x
is the incomplete Gamma function. D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Resummation Resurgence phenomenon Numerics
Therefore performing the summation, one has the expansion (m 6 n) k n−1 k 1 1 X (−1) Γ 2 + 2k λ Z (0) = √ k! 6 π k =0 (−1)n + √ e π 2 +
3 2λ
(−1)n √ e π 2
m−1 X j=0
3 2λ
j
(−1) Aj Γ(n − j) ˆ
c+m+i R
ds 2iπ
2 3λ
3 Γ −n + j + 1, 2λ 3 f (s) Γ(n − s) Γ −n + s + 1, 2λ
2λ 3
−s
j
where the sub-dominant terms appear explicitly through the expression of the tail of the series. Using the MB representation of the incomplete Gamma function we have exactly the hyperasymptotic expansion of Z (0) at first level obtained in the analytic approach.
D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Resummation Resurgence phenomenon Numerics
Resurgence phenomenon If we consider a few terms in the perturbative expansion of Z (0) Z (0) = 1 −
1 35 2 385 3 λ+ λ − λ + ··· 8 384 3072
we have the same coefficient in the non-perturbative expansion of Rn as (−1)n 3 Rn = − √ e 2λ π 2 3 1 3 × 1 Γ(5)Γ −4, − λ Γ(4)Γ −3, 2λ 8 2λ 35 2 3 385 3 3 + λ Γ(3)Γ −2, − λ Γ(2)Γ −1, +··· 384 2λ 3072 2λ This resurgence phenomenon also appears in higher order hyperasymptotic levels.
D. Greynat
Asymptotic expansions and Mellin-Barnes representation
Resummation Resurgence phenomenon Numerics
Numerical analysis
OTS 1
OTS 2
Pertur. expa.
OTS 2
m
m0
m00
1 2 3
3 ]=4 [ 2|λ| 3 [ 2|λ| ]=4 3 [ 2|λ| ]=4
3 ]=2 [ 4|λ| 3 [ 4|λ| ]=2 3 [ 4|λ| ]=2
3 [ 8|λ| ]=1 3 [ 8|λ| ]=1
3 [ 8|λ| ]=0
1 2 3
3 [ |λ| ]=9 9 [ 2|λ| ] = 13 6 [ |λ| ] = 18
3 [ 2|λ| ]=4 3 [ |λ| ] = 9 9 [ 2|λ| ] = 13
3 [ 2|λ| ]=4 3 [ |λ| ]=9
3 [ 2|λ| ]=4
Z (0) 0.965560481.. 0.96555187
Mathematica
OTS 1
n
Sn
Sm
0.9638 0.9638 0.9638 0.9696 1.0573 -27.696
0.0017 0.0017 0.0017 -0.0040 -0.0917 28.662
Sm 0
Sm00
0.000041 0.000041
0
±0.001410990
1 2 3 1 2 3
0.96552297 0.96556492 0.96556492 0.965562911 0.965560477 0.965560486
D. Greynat
0.0000061 -0.0001292
9 × 10−9
Asymptotic expansions and Mellin-Barnes representation
Resummation Resurgence phenomenon Numerics
• Numerical stability
Olver et al.
’95
It has been proven that it is possible to obtain numerical stability by choosing another OTS, taking n+1−j aj = n+1 For the third hyperasymptotic level we have n=8
m0 = 4
m=6
m00 = 2
and Z (0) = 0.965560480 • The non-Borel summable sector For λ = − 13 , we have Mathematica Stieljes S5 +
S2NP
1.05995021 − 0.00758472 i
1.06098837 + 0 i ± 0.00140990 1.05990083 − 0.00752794 i D. Greynat
Asymptotic expansions and Mellin-Barnes representation
C ONCLUSIONS The Mellin Barnes representation is a perfect tool to obtain asymptotic expansions of Feynman diagrams containing one or several scales. It assures the calculation at any order and under certain conditions the exact representation. The Mellin Barnes representation is also a powerful instrument to deal with perturbative divergent series and their resummation. It allows also through the hyperasymptotic method to get non perturbative contributions directly from the perturbative expansion itself.
D. Greynat
Asymptotic expansions and Mellin-Barnes representation