A LPHA Collaboration October 21, 2020 References, s by the ALPHA - - PowerPoint PPT Presentation

Perturbative tests at high energies, using lattice results by the ALPHA collaboration

Stefan Sint (Trinity College Dublin) work in collaboration with: Mattia Bruno, Mattia Dalla Brida, Patrick Fritzsch, Tomasz Korzec, Alberto Ramos, Stefan Schaefer, Hubert Simma, Rainer Sommer

LPHA

A

Collaboration

October 21, 2020

References, αs by the ALPHA collaboration: “Determination of the QCD Λ-parameter and the accuracy of perturbation theory at high energies,” Mattia Dalla Brida, Patrick Fritzsch, Tomasz Korzec, Alberto Ramos, S.S., Rainer Sommer [ALPHA Collaboration],

• Phys. Rev. Lett. 117, no. 18, 182001 (2016) arXiv:1604.06193 [hep-ph].

“A non-perturbative exploration of the high energy regime in Nf = 3 QCD ,” Mattia Dalla Brida, Patrick Fritzsch, Tomasz Korzec, Alberto Ramos, S.S., Rainer Sommer [ALPHA Collaboration],

• Eur. Phys. J. C 78 (2018) no.5, 372 arXiv:1803.10230 [hep-lat].

“Slow running of the Gradient Flow coupling from 200 MeV to 4 GeV in Nf = 3 QCD,” Mattia Dalla Brida, Patrick Fritzsch, Tomasz Korzec, Alberto Ramos, S.S, Rainer Sommer [ALPHA Collaboration],

• Phys. Rev. D 95, no. 1, 014507 (2017), arXiv:1607.06423 [hep-lat].

⇒ “QCD Coupling from a Nonperturbative Determination of the Three-Flavor Λ Parameter,” Mattia Bruno, Mattia Dalla Brida, Patrick Fritzsch, Tomasz Korzec, Alberto Ramos, Stefan Schaefer, S. S., Hubert Simma Rainer Sommer [ALPHA Collaboration],

• Phys. Rev. Lett. 119, no. 10, 102001 (2017), arXiv:1706.03821 [hep-lat].

Topics: Results for the SF coupling between 1/L0 ≈ 4GeV and O(100) GeV Extraction of L0Λ(3) & tests of perturbation theory Summary

The QCD Λ-parameter vs. αs(µ) = ¯ g2(µ)/4π

The coupling αs(µ) can be traded for its associated Λ-parameter: Λ = µϕ(¯ g(µ)) = µ b0¯ g2(µ)− b1

2b2

0 e

− 1 2b0¯ g2(µ) exp

• −

¯

g(µ)

dg

• 1

β(g) + 1 b0g3 − b1 b2

0g

• exact solution of Callan-Symanzik equation:

µ ∂

∂µ + β(¯

g) ∂

∂¯ g

• Λ = 0

Number Nf of massless quarks is fixed. If the coupling ¯ g(µ) non-perturbatively defined so is its β-function! β(g) has asymptotic expansion β(g) = −b0g3 − b1g5 − b2g7.. b0 = (11 − 2

3 Nf)/(4π)2,

b1 = (102 − 38

3 Nf)/(4π)4,

. . . b0,1 are universal, scheme-dependence starts with 3-loop coefficient b2. Scheme dependence of Λ almost trivial: g2

X(µ) = g2 Y(µ) + cXYg4 Y(µ) + ...

⇒ ΛX ΛY = ecXY/2b0 ⇒ can use ΛMS as reference (even though the MS-scheme is purely perturbative!)

A family of SF couplings I

Dirichlet b.c.’s in Euclidean time, abelian boundary values Ck, C′

k:

Ak(x)|x0=0 = Ck(η, ν), Ak(x)|x0=L = C′

k(η, ν)

Ck = i L

• η − π

3

ην − η

2

−ην − η

2 + π 3

• ,

C′

k = i

L

• −η − π

ην + η

2 + π 3 η 2 − ην + 2π 3

• ⇒ induce family of abelian, spatially constant background fields Bµ with

parameters η, ν (→ 2 abelian generators of SU(3)): Bk(x) = Ck(η, ν) + x0 L

• C′

k(η, ν) − Ck(η, ν)

, B0 = 0. Induced background field is unique up to gauge equivalence Effective action e−Γ[B] =

• D[A, ψ, ψ]e−S[A,ψ,ψ],

Γ[B] =

1 g2

Γ0[B] + Γ1[B] + O(g2

0)

Define 1 ¯ g2

ν(L) =

∂ηΓ[B] ∂ηΓ0[B]

• η=0

= ∂ηS ∂ηΓ0[B]

• η=0

⇒ 1-parameter family of SF couplings as response of the system to a change of a colour-electric background field. [L¨

uscher et al. ’92]

A family of SF couplings II

ν-dependence is explicit, obtained by computing ¯ g2 ≡ ¯ g2

ν=0 and ¯

v at ν = 0: 1 ¯ g2

ν

= 1 ¯ g2 − ν¯ v relation between couplings at ν and ν = 0 gives exact ratio: rν = Λ/Λν = exp(−ν × 1.25516) The β-function is known to 3-loops: (4π)3 × b2,ν = −0.06(3) − ν × 1.26 N.B.: values ν of O(1) look perfectly fine! infrared cutoff (finite volume) ⇒ no renormalons; secondary minimum of the action: exp(−2.62/α) ≃ (Λ/µ)3.8 Cutoff effects: O(a4) at tree-level, but O(a) effects from the boundaries:

subtracted perturbatively variation of coefficients treated as systematic error, continuum extrapolations ∝ a2

SSF in the continuum limit

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 1 1.2 1.4 1.6 1.8 2 2.2 [σ(u) − u]/u u Final result

• ne-loop

two-loop three-loop L/a = 6 L/a = 8 L/a = 12 0.05 0.1 0.15 0.2 0.25 0.5 1 1.5 2 2.5 3 3.5 [σ(u) − u]/u2 u This work Nf = 0 (ALPHA) Nf = 2 (ALPHA) Nf = 3 (PACS-CS) Nf = 4 (ALPHA)

⇒ Significantly improved precision compared to previous work with Nf = 0, 2, 3, 4

Computation of L0Λ

Define L0 implicitly by ¯ g2(L0) = 2.012 = u0 Use the non-perturbative continuum step scaling function σ(u): un−1 = σ(un), n = 1, . . . , ⇒ un = ¯ g2 2−nL0

• At scale 2−nL0 obtain L0Λ using the perturbative β-function:

L0Λ = 2n b0¯ g2(2−nL0)− b1

2b2

0 e

− 1 2b0¯ g2(2−nL0)

× exp

• −

¯

g(2−nL0)

dg

• 1

β(g) + 1 b0g3 − b1 b2

0g

• Do the same for schemes ν = 0 using the continuum relation:

1 ¯ g2

ν(L0) =

1 2.012 − ν × 0.1199(10) ⇒ check accuracy of perturbation theory: L0Λ must be independent of ν and number of steps, n !

Result for L0Λ

0.029 0.03 0.031 0.032 0.033 0.034 0.005 0.01 0.015 0.02 0.025 0.03 L0Λ α2 Final Result ν = −0.5 ν = 0 (Fit B) ν = 0 (Fit C) ν = 0.3

All results agree around α = 0.1, we quote L0Λ = 0.0303(7) ⇒ L0ΛNf=3

MS

= 0.0791(19) (error 2.4%) Recall L0 ≡ Lswi is defined implicitly by ¯ g2(L0) = 2.012.

Alternative test via the MS-scheme I

Idea: Perturbatively match the SF coupling to the MS-coupling then evaluate the Λ-parameter using the 5-loop β-function Relation between couplings, allowing for a scale factor s: 4παMS(s/L) ≡ ¯ g2

MS(L/s) = ¯

g2

ν(L) + pν 1(s)¯

g4

ν(L) + pν 2(s)¯

g6

ν(L) + O(¯

g8) Same as earlier, except now in the MS scheme: ΛMSL0 = sL0

L ϕMS

• ¯

gMS(L/s) = s 2nϕMS

• ¯

g2

ν(L) + pν 1(s)¯

g4

ν(L) + pν 2(s)¯

g6

ν(L)

• ,

expect to see independence of the number of steps n, scale factor s and parameter ν. Look at ν = 0, depdendence on n and s. Note: The neglected order for Λ: ∆g2 dϕ

dg2 ∝ ∆g2 {gβ(g)}−1 = ∆g2 × O(g−4)

⇒ truncation error: O(g8) × O(g−4) = O(g4) = O(α2).

Alternative test via the MS-scheme II

α(sq) = αν(q) + cν

1(s)α2 ν + cν 2(s)α3 ν(q) + ...,

pν

i = cν i /(4π)i

parameters: ν = 0, s∗ ≈ 3 0.07 0.075 0.08 0.085 0.09 0.005 0.01 0.015 0.02 0.025 0.03 L0ΛMS α2 Final result s ≈ s⋆/3 s ≈ s⋆/2 s ≈ s⋆ s ≈ 2s⋆ s ≈ 3s⋆

−2 2 4 6 8 2 4 6 8 10 s c1(s) c2(s)

Choice of scale factor is important, coefficients can get large. “fastest apparent convergence” principle: c1(s∗) = 0 which means s∗ = ΛMS/Λ = 2.612 ≈ 3 seems like a good idea.

Alternative test via the MS-scheme III

α(sq) = αν(q) + cν

1(s)α2 ν + cν 2(s)α3 ν(q) + ...,

pν

i = cν i /(4π)i

parameters: ν = −0.5, s∗ ≈ 5 0.07 0.075 0.08 0.085 0.09 0.005 0.01 0.015 0.02 0.025 L0ΛMS α2 Final result s = 1 s ≈ s⋆/2 s ≈ s⋆ s ≈ 2s⋆ s ≈ 3s⋆

−2 2 4 6 8 2 4 6 8 10 12 14 s cν=−0.5

1

(s) cν=−0.5

2

(s)

Alternative test via the MS-scheme IV

variation of the scale factor s ∈ [s∗/2, 2s∗]

0.075 0.08 0.085 0.09 0.005 0.01 0.015 0.02 0.025 0.03 L0ΛMS α2 Final result ν = −0.5 ν = 0 ν = 0.3

⇒ may significantly underestimate the systematic error!

Summary, tests of perturbation theory

The determination of αs is well-suited for the lattice approach; The systematics can be well controlled by combining technical tools developed

• ver the last 25 years:

finite volume renormalization schemes and recursive step-scaling methods non-perturbative Symanzik improvement perturbation theory adapted to finite volume; relation between SF and MS-coupling known to 2-loop order!

⇒ Completely solves the problem of large scale differences; perturbation theory at low energies can be avoided! Turning this around: many opportunities to test perturbation theory at high energies! ⇒ with hindsight: estimates of perturbative truncation errors require some luck!