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

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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

  1. 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

  2. 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].

  3. 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

  4. 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!)

  5. 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]

  6. 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

  7. 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

  8. 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 !

  9. 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 ¯

  10. 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 ) .

  11. 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.

  12. 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

  13. 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!

  14. 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!

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