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 A LPHA Collaboration October 21, 2020

References, α s 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 N f = 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 N f = 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 /L 0 ≈ 4GeV and O(100) GeV Extraction of L 0 Λ (3) & tests of perturbation theory Summary

g 2 ( µ ) / 4 π The QCD Λ -parameter vs. α s ( µ ) = ¯ The coupling α s ( µ ) can be traded for its associated Λ -parameter: � ¯ g 2 ( µ ) � − b 1 g ( µ ) 1 1 b 0 g 3 − b 1 1 � � �� 2 b 2 − g 2 ( µ ) exp g ( µ )) = µ � 0 e 2 b 0 ¯ Λ = µϕ (¯ b 0 ¯ − dg β ( g ) + b 2 0 g 0 exact solution of Callan-Symanzik equation: � µ ∂ g ) ∂ � ∂µ + β (¯ Λ = 0 ∂ ¯ g Number N f 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 ) = − b 0 g 3 − b 1 g 5 − b 2 g 7 .. b 0 = (11 − 2 3 N f ) / (4 π ) 2 , b 1 = (102 − 38 3 N f ) / (4 π ) 4 , . . . b 0 , 1 are universal, scheme-dependence starts with 3-loop coefficient b 2 . Scheme dependence of Λ almost trivial: Λ X g 2 X ( µ ) = g 2 Y ( µ ) + c XY g 4 = e c XY / 2 b 0 Y ( µ ) + ... ⇒ Λ Y ⇒ can use Λ MS 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 C k , C ′ k : A k ( x ) | x 0 =0 = C k ( η, ν ) , A k ( x ) | x 0 = L = C ′ k ( η, ν ) � � � � η − π 0 0 C k = i k = i 0 0 − η − π 3 C ′ ην − η , 0 ην + η 2 + π 0 0 0 3 2 L L − ην − η η 2 − ην + 2 π 0 0 2 + π 0 0 3 3 ⇒ induce family of abelian, spatially constant background fields B µ with parameters η, ν ( → 2 abelian generators of SU( 3 )): B k ( x ) = C k ( η, ν ) + x 0 � k ( η, ν ) − C k ( η, ν ) � C ′ , B 0 = 0 . L Induced background field is unique up to gauge equivalence Effective action � e − Γ[ B ] = D [ A, ψ, ψ ]e − S [ A,ψ,ψ ] , 1 Γ 0 [ B ] + Γ 1 [ B ] + O( g 2 Γ[ B ] = 0 ) g 2 0 Define � � 1 ∂ η Γ[ B ] � ∂ η S � � � ν ( L ) = = � � g 2 ¯ ∂ η Γ 0 [ B ] ∂ η Γ 0 [ B ] � � η =0 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 g 2 ≡ ¯ g 2 ν -dependence is explicit, obtained by computing ¯ ν =0 and ¯ v at ν = 0 : 1 = 1 g 2 − ν ¯ v g 2 ¯ ¯ ν relation between couplings at ν and ν = 0 gives exact ratio: r ν = Λ / Λ ν = exp( − ν × 1 . 25516) The β -function is known to 3-loops: (4 π ) 3 × b 2 ,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( a 4 ) at tree-level, but O( a ) effects from the boundaries: subtracted perturbatively variation of coefficients treated as systematic error, continuum extrapolations ∝ a 2

SSF in the continuum limit 0 . 24 0 . 25 Final result This work one-loop N f = 0 (ALPHA) 0 . 22 two-loop N f = 2 (ALPHA) three-loop N f = 3 (PACS-CS) 0 . 2 0 . 2 L/a = 6 N f = 4 (ALPHA) L/a = 8 L/a = 12 0 . 18 [ σ ( u ) − u ] /u 2 0 . 15 [ σ ( u ) − u ] /u 0 . 16 0 . 14 0 . 1 0 . 12 0 . 1 0 . 05 0 . 08 0 . 06 0 1 1 . 2 1 . 4 1 . 6 1 . 8 2 2 . 2 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 u u ⇒ Significantly improved precision compared to previous work with N f = 0 , 2 , 3 , 4

Computation of L 0 Λ Define L 0 implicitly by g 2 ( L 0 ) = 2 . 012 = u 0 ¯ Use the non-perturbative continuum step scaling function σ ( u ) : g 2 � 2 − n L 0 � u n − 1 = σ ( u n ) , n = 1 , . . . , ⇒ u n = ¯ At scale 2 − n L 0 obtain L 0 Λ using the perturbative β -function: g 2 (2 − n L 0 ) � − b 1 1 2 b 2 − 2 n � g 2 (2 − n L 0 ) 0 e 2 b 0 ¯ L 0 Λ = b 0 ¯ � ¯ g (2 − n L 0 ) 1 b 0 g 3 − b 1 1 � � �� × exp − dg β ( g ) + b 2 0 g 0 Do the same for schemes ν � = 0 using the continuum relation: 1 1 ν ( L 0 ) = 2 . 012 − ν × 0 . 1199(10) g 2 ¯ ⇒ check accuracy of perturbation theory: L 0 Λ must be independent of ν and number of steps, n !

Result for L 0 Λ 0 . 034 Final Result ν = − 0 . 5 ν = 0 (Fit B) 0 . 033 ν = 0 (Fit C) ν = 0 . 3 0 . 032 L 0 Λ 0 . 031 0 . 03 0 . 029 0 0 . 005 0 . 01 0 . 015 0 . 02 0 . 025 0 . 03 α 2 All results agree around α = 0 . 1 , we quote L 0 Λ N f =3 L 0 Λ = 0 . 0303(7) ⇒ = 0 . 0791(19) ( error 2 . 4% ) MS g 2 ( L 0 ) = 2 . 012 . Recall L 0 ≡ L swi is defined implicitly by ¯

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 : g 2 g 2 ν ( L ) + p ν g 4 ν ( L ) + p ν g 6 g 8 ) 4 πα MS ( s/L ) ≡ ¯ MS ( L/s ) = ¯ 1 ( s )¯ 2 ( s )¯ ν ( L ) + O(¯ Same as earlier, except now in the MS scheme: �� � Λ MS L 0 = sL 0 � g MS ( L/s ) � = s 2 n ϕ MS g 2 ν ( L ) + p ν g 4 ν ( L ) + p ν g 6 L ϕ MS ¯ ¯ 1 ( s )¯ 2 ( s )¯ 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 : dg 2 ∝ ∆ g 2 { gβ ( g ) } − 1 = ∆ g 2 × O( g − 4 ) ∆ g 2 dϕ ⇒ truncation error: O( g 8 ) × O( g − 4 ) = O( g 4 ) = O( α 2 ) .

Alternative test via the MS -scheme II α ( sq ) = α ν ( q ) + c ν 1 ( s ) α 2 ν + c ν 2 ( s ) α 3 p ν i = c ν i / (4 π ) i ν ( q ) + ..., parameters: ν = 0 , s ∗ ≈ 3 Final result s ≈ s ⋆ s ≈ s ⋆ / 3 s ≈ 2 s ⋆ s ≈ s ⋆ / 2 s ≈ 3 s ⋆ 8 0 . 09 c 1 ( s ) c 2 ( s ) 6 0 . 085 4 L 0 Λ MS 2 0 . 08 0 0 . 075 − 2 0 2 4 6 8 10 0 . 07 s 0 0 . 005 0 . 01 0 . 015 0 . 02 0 . 025 0 . 03 α 2 Choice of scale factor is important, coefficients can get large. “fastest apparent convergence” principle: c 1 ( s ∗ ) = 0 which means s ∗ = Λ MS / Λ = 2 . 612 ≈ 3 seems like a good idea.

Alternative test via the MS -scheme III α ( sq ) = α ν ( q ) + c ν 1 ( s ) α 2 ν + c ν 2 ( s ) α 3 p ν i = c ν i / (4 π ) i ν ( q ) + ..., parameters: ν = − 0 . 5 , s ∗ ≈ 5 Final result s ≈ s ⋆ s = 1 s ≈ 2 s ⋆ s ≈ s ⋆ / 2 s ≈ 3 s ⋆ 8 0 . 09 c ν = − 0 . 5 ( s ) 1 c ν = − 0 . 5 ( s ) 6 2 0 . 085 4 L 0 Λ MS 2 0 . 08 0 0 . 075 − 2 0 2 4 6 8 10 12 14 s 0 . 07 0 0 . 005 0 . 01 0 . 015 0 . 02 0 . 025 α 2

Alternative test via the MS -scheme IV variation of the scale factor s ∈ [ s ∗ / 2 , 2 s ∗ ] 0 . 09 Final result ν = − 0 . 5 ν = 0 ν = 0 . 3 0 . 085 L 0 Λ MS 0 . 08 0 . 075 0 0 . 005 0 . 01 0 . 015 0 . 02 0 . 025 0 . 03 α 2 ⇒ may significantly underestimate the systematic error!

Summary, tests of perturbation theory The determination of α s is well-suited for the lattice approach; The systematics can be well controlled by combining technical tools developed over 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!