NP from PT? · 2018-06-20 · extrapolation 0 0.002 0.004 0.006 0.008 0.01 t 0.997 0.998 0.999 1...

NP from PT?

L Del Debbio

Higgs Centre for Theoretical PhysicsUniversity of Edinburgh

L Del Debbio nspt Paris, June 2018 1 / 31

work in collaboration with

F Di Renzo, G Filaci


large orders in perturbation theory

Bender & Wu (1969)(− d2

dx2+

1

4x2 +

1

4λx4

)Φ(x) = E(λ)Φ(x)

limx→±∞

Φ(x) = 0

perturbative solution

E0(λ) =1

2+

∞∑n=1

Anλn


factorial growth of the coefficients

An ∼ (−)n+1(6/π3

)1/2Γ

(n+

1

2

)3n

=⇒ asymptotic series ∫ ∞0

dwew/t

1− w=

∞∑n=0

n!tn+1

3

4

5

6

7

8

9

10

0.02 0.04 0.06 0.08

-0.1

0.1

0.2

0.3

0.4


asymptotic series

∣∣∣∣∣R(λ)−N∑n=1

pnλn

∣∣∣∣∣ < KN+1λN+1

pn ∼n→∞

Kann!nb

minimized truncation error

KN ∝ aNN !N b =⇒ n(λ) ∼ 1

|a|λsize of the remainder

Knλn ∼n1

e−1/(|a|λ)


resummation

Borel transform

R(λ) =

∞∑n=1

rnλn =⇒ B[R](t) =

∞∑n=0

rntn

n!

Borel sum

R(λ) =

∫ ∞0

dt e−t/λB[R](t)

rn = KanΓ(n+ 1 + b) =⇒ B[R](t) =KΓ(1 + b)

(1− at)1+b

→ singularities in the Borel plane


ambiguities

deform the contour of the Borel integral

Im R(λ) = ∓πKae−1/(aλ) (aλ)−b

aRe t

Im t

O


nonperturbative contributions

∫d4x e−iqx〈Tjµ(x)jν(0)〉 = i

(qµqν − q2gµν

)Π(Q2)

D(Q2) = 4π2 d

dQ2Π(Q2) ' CF

π

∑n

αn+1cnβn0 n!

for an asymptotic free theory

α(Q) ∼ − 1

β0 logQ2/Λ2

ambiguity in the Adler function

δD(Q2) ∝ e2/(β0α(Q)) ∼(

Λ

Q

)4

a defect of perturbation theory? link to nonperturbative physics?


check using NSPTstochastic quantization for pure gauge theory

SG [U ] = − β

2Nc

∑

Tr(U + U

†)

stochastic process

∂

∂tUµ(x; t) = i

[−∇xµSG[Uµ(x; t)]− ηµ(x; t)

]Uµ(x; t)

= Fµ(x; t)Uµ(x; t)

limt→∞〈O[U(t)]〉 =

1

Z

∫DU e−SG[U ]O[U ] .


perturbative expansion

NSPT is defined by the perturbative expansion of the stochastic process

Uµ(x; t) = U (0)µ (x; t) +

∑k=1

β−k/2U (k)µ (x; t)

Fµ(x; t) =∑k=1

β−k/2F (k)µ (x; t)

explicitly

U (1)(t+ ε) = U (1)(t)− F (1)(t)

U (2)(t+ ε) = U (2)(t)− F (2)(t) +1

2F (1)(t)2 − F (1)(t)U (1)(t)

. . .


implementation in GRID

SU(3) gauge theory, nf = 2, 84 volume, TBC

Langevin dynamics

τ0 0.002 0.004 0.006 0.008 0.01

(1)

W

2.014−

2.012−

2.01−

2.008−

2.006−

2.004−

2.002−

2− datalinear fit

error bandσ1=0τdata extrapolated at

analytic finite-volume prediction

τ0 0.002 0.004 0.006 0.008 0.01

(2)

W1.089−

1.088−

1.087−

1.086−

1.085−

1.084−

1.083−

datalinear fit

error bandσ1=0τdata extrapolated at

analytic finite-volume prediction


understanding the algorithm

critical mass with Wilson fermions

Γ(ap = 0, amc, β) = amc + Σ(ap = 0, amc, β) = 0

computed as a perturbative series

Σ(ap = 0, am, β) =∑n

γn(am)β−n

=⇒ amc =∑n

mnβ−n

iterative solution

mK+1 = γK+1(MK), MK =

K∑n=1

mnβ−n


critical mass

SU(2) gauge theory, nf = 2, 124 volume, TBC


convergence of the stochastic process


numerical instabilities

0 2 4 6 8 10 12 14 16t

4000−

3000−

2000−

1000−

0

1000

2000

3000

2410×

39p

quenched, L=8

=2 Wilson fermions, L=8fN

0 1 2 3 4 5 6 7 8 9t

10000−

5000−

0

5000

10000

2110×

36p

=2 staggered fermions, L=32fN

=2 staggered fermions, L=48fN


renormalons & condensates: plaquette

P =1

6NcL4

∑

Re Tr (1− U) .

〈P 〉MC =

∞∑n=0

cnαn+1 +

π2

36CG(α) a4〈G2〉+O

(a6)

CG(α)−1 =36

π2

(1 +

β1

β0

α

4π

)

asymptotic behaviour

cn ' NP

(β0

2πdF 2

)n Γ(n+ 1 + dF 2b)

Γ(1 + dF 2b)

1 +

dF 2b

n+ dF 2b+ . . .

[Bali et al 14]


http://arxiv.org/abs/1401.7999

measurements

〈P 〉pert =

∞∑n=0

pn β−n−1

0 500 1000 1500 2000 2500 3000 3500 4000

Number of configurations

126

128

130

132

134

136

138

140

142

144

310×

=0.

005)

τ(10p

0 500 1000 1500 2000 2500 3000

Number of configurations

20−

10−

0

10

20

30

401810×

=0.

01)

τ(30p


correlations

0.6−

0.4−

0.2−

0

0.2

0.4

0.6

0.8

1

2 7 12 17 22 27 322

7

12

17

22

27

32


extrapolation

0 0.002 0.004 0.006 0.008 0.01τ

0.997

0.998

0.999

1

1.001

1.002

1.003

=0.

005)

τ( 0 /

p0p

4=484 =2 staggered fermions, Lf

, k=0: SU(3) with Nk

Plaquette p

data

linear fit

=0τdata extrapolated at

infinite-volume prediction

0 0.002 0.004 0.006 0.008 0.01τ

0.998

0.999

1

1.001

1.002

=0.

005)

τ( 1 /

p1p

4=484 =2 staggered fermions, Lf

, k=1: SU(3) with Nk

Plaquette p


renormalon behaviour

cn ' NP

(β0

2πdF 2

)n Γ(n+ 1 + dF 2b)

Γ(1 + dF 2b)

1 +

dF 2b

n+ dF 2b+ . . .

or equivalently

pnnpn−1

=β0

8π

[1 +

β1

2β20

4

n+O

(1

n2

)]where

β0 =11

3Nc −

2

3nf

β1 =34

3N2c − nf

(13

3Nc −

1

Nc

).


renormalon behaviour – pure gauge

[Bali et al 14]



renormalon behaviour – with fermions

0 5 10 15 20 25 30 35n

0.2−

0

0.2

0.4

0.6

0.8

1

1.2

)n-

1/(

npnp

L=24

L=28

L=32

L=48

LO

NLO


minimal term and scaling of the condensate

(n+ dF 2b)β0α

2πdF 2

= exp

[− 1

2(n+ dF 2b)+ . . .

]and hence:

SP (α) =

n∑n=0

cnαn+1

π2

36CG(α) 〈G2〉 =

1

a4[〈P 〉MC − SP (α)]

ambiguity

δSP (α) =√ncnα

n+1


determination of the minimal term

pnβ−n−1 < pn+1β

−n−2

data for L = 48, β = 5.3

27 28 29 30 31 32 33 34 35 36

n

4−10

3−10

2−10

1−10

1

0.4775− 0.477− 0.4765− 0.476− 0.4755−)β(PS

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Monte Carlo results

existing results for staggered fermions[Tamhankar et 99, Heller et al 94]

0 0.05 0.1 0.15 0.2 0.25 0.3m

0.46

0.48

0.5

0.52

0.54

0.56

MC

< 1

-P >

=5.6β=5.5β=5.415β=5.35β=5.3β

0 0.05 0.1 0.15 0.2 0.25 0.3m

1.5

2

2.5

3

3.5

4

4.5

5

/a 0r

=5.6β=5.5β=5.415β=5.35β=5.3β



condensate

a4〈G2〉 =36

π2CG(α)−1 [〈P 〉MC − SP (α)]

5.25 5.3 5.35 5.4 5.45 5.5 5.55 5.6 5.65

β

2

2.5

3

3.5

4

>G

<O

4 0r

0 0.01 0.02 0.03 0.04 0.0540/r4a

0

0.02

0.04

0.06

0.08

0.1)]β(P

)-S

β(M

C [<

P>

2 π)/β(

-1 G36

C

4)0

+ 3.1(2) (a/r-5 10⋅+ 5(58)

6)0

- 6(2) (a/r4)0

+ 3.4(4) (a/r-5 10⋅+ 7(68)


thoughts on IR conformality

• pn obey the expected renormalon behaviour

• study of lattice systematics

• relation with QCD condensates

• if IR conformality - then the condensates all vanish as m→ 0

• difference in the behaviour of the perturbative coefficients at large p?


Euler integration

one step of numerical integration + one step of stochastic gauge fixing

Uµ(x)′ = e−Fµ(x;t) Uµ(x; t) ,

Uµ(x; t+ τ) = ew[U ′](x)Uµ(x)′e−w[U ′](x+µ) ,

where the force term is

Fµ(x; t) =τ

β∇xµSG[U(t)] +

√τ

βηµ(x; t) .


TBC

periodicity up to gauge transformations

Uµ(x+ Lν) = ΩνUµ(x)Ω†ν

consistency condition

ΩµΩν = zµν ΩνΩµ , zµν ∈ ZNzµν = exp inµν2π/N

used TBC in the (1, 2) plane

Ω1 =

e−i2π3 0 0

0 1 0

0 0 ei2π3

Ω2 =

0 1 00 0 11 0 0

,

corresponding to z12 = exp(i2π

3

).


NSPT for gauge + fermions

force for the Langevin evolution

F =(εΦa +

√εηa)T a

Φa =[∇axµSG − Re

(ξ†(∇axµM

)M−1ξ

)]expand the fermionic force in a perturbative series

Uµ(x; t) = U (0)µ (x; t) +

∑p>0

β−p/2U (k)µ (x; t)

=⇒ M = M (0) +∑p>0

β−p/2M (p)

M−1 = M (0) −1 +∑p>0

β−p/2M (p) −1


adding smell

• adjoint fermionsψ(x+ Lν) = Ωνψ(x)Ω†ν

same transformation as for gauge fields

• fundamental fermionsadd smell degrees of freedom

ψ(x+ Lν)ir = (Ων)ij ψ(x)js

(Λ†ν

)sr

• correct number of degrees of freedom

exp (nfTr logM) 7→ exp(nfN

Tr logM)


NP from PT? · 2018-06-20 · extrapolation 0 0.002 0.004 0.006 0.008 0.01 t 0.997 0.998 0.999 1...

Documents

Transcript of NP from PT? · 2018-06-20 · extrapolation 0 0.002 0.004 0.006 0.008 0.01 t 0.997 0.998 0.999 1...