Towards the gauge beta function at O (1 / N 2 f ) and O (1 / N 3 f ) - - PowerPoint PPT Presentation

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Towards the gauge beta function at O (1 / N 2 f ) and O (1 / N 3 f ) - - PowerPoint PPT Presentation

Nov 25, 2023 16 likes •578 views

Towards the gauge beta function at O (1 / N 2 f ) and O (1 / N 3 f ) Manuel Reichert Bridging perturbative and non-perturbative physics, Primosten, 07. October 2019 CP 3 -Origins, SDU Odense, Denmark Nicola Dondi, Gerald Dunne, MR, Francesco

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Towards the gauge beta function at O(1/N2

f ) and O(1/N3 f )

Manuel Reichert Bridging perturbative and non-perturbative physics, Primosten, 07. October 2019

CP3-Origins, SDU Odense, Denmark Nicola Dondi, Gerald Dunne, MR, Francesco Saninno: arXiv:1903.02568 Nicola Dondi, MR, Francesco Saninno: in preparation

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Which matter systems are asymptotically safe in d = 4?

  • Gauge-Yukawa theories at large Nf & Nc (perturbatively) [Litim, Sannino ’14]
  • How far does this extend to small Nc?
  • Test gauge theories at large Nf non-perturbatively

Standard QCD picture:

  • Small Nf : asymptotic freedom &

confinement in the IR

  • Medium Nf : asymptotic freedom

& IR Banks-Zaks fixed point

  • Large Nf : asymptotic freedom lost

→ asymptotic safety?

2 3 4 5 6 7 10 20 30 40 50 60 70 Nc Nf IR conformal Safe QCD [Antipin, Sannino ’17]

1

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Beta functions of (S)QED and (S)QCD

β(K) = β(0)(K) + β(1)(K) Nf + . . .

1 2 3 4 5 6 7 8 5 10 K β(1)

QED QCD SQED SQCD

UV fixed point for QED & QCD Landau pole for SQED & SQCD

2

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How physical are these fixed points?

  • The fermion mass anomalous dimension goes to zero in QCD and to

infinity in QED

[Antipin, Sannino ’17]

  • Hints for FP in QCD at medium Nf from resummations with Meijer

G-functions

[Antipin, Maiezza, Vasquez ’18]

  • Lattice studies inconclusive so far

[Leino, Rindlisbacher, Rummukainen, Sannino, Tuominen ’19]

  • Poles might be resummable within the 1/Nf series

[Alanne, Blasi, Dondi ’19]

3

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SLIDE 5

How to go beyond 1/Nf

  • The next orders in the 1/Nf expansion would test the physical

nature of the FP

  • No known resummation formula for two bubble-chains,

needed for 1/N2

f and higher orders

  • Can we extract the location of the pole, the residuum, etc.,

with a finite amount of coefficients?

4

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How to go beyond 1/Nf

  • The next orders in the 1/Nf expansion would test the physical

nature of the FP

  • No known resummation formula for two bubble-chains,

needed for 1/N2

f and higher orders

  • Can we extract the location of the pole, the residuum, etc.,

with a finite amount of coefficients? Two methods:

  • Large-order behaviour of expansion coefficients
  • Pad´

e approximants

4

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Large-order behaviour: Darboux’s Theorem

The nearby singularity determines the large-order growth of the expansion coefficients an. E.g. for expansion around z = 0

  • pole of order p at z0 (f (z) ∼ φ(z)(1 − z/z0)p + finite)

an ∼ 1 zn

  • n + p − 1

n

  • φ(z0) + . . .
  • logarithmic branch cut at z0 (f (z) ∼ φ(z) ln(1 − z/z0) + finite)

an ∼ 1 zn 1 nφ(z0) + . . .

5

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Large-order behaviour: Darboux’s Theorem

The nearby singularity determines the large-order growth of the expansion coefficients an. E.g. for expansion around z = 0

  • pole of order p at z0 (f (z) ∼ φ(z)(1 − z/z0)p + finite)

an ∼ 1 zn

  • n + p − 1

n

  • φ(z0) + . . .
  • logarithmic branch cut at z0 (f (z) ∼ φ(z) ln(1 − z/z0) + finite)

an ∼ 1 zn 1 nφ(z0) + . . . Expectation for QED FQED =

n fn xn

fn ∼

  • R0

2 15 n + R1 2 21 n + R2 2 27 n + . . .

  • 5
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SLIDE 9

Large-order behaviour of FQED

10 20 30 40 50 60 −0.4 −0.2 0.2 0.4 n

fn+1 fn 2 15

Ratio test fn+1

fn

reveals location of the first pole

6

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Large-order behaviour of FQED

10 20 30 40 50 60 −4,000 −2,000 2,000 4,000 n

15

2

n+1 fn −

28 45π2

7

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Large-order behaviour of FQED

25 30 35 40 45 50 55 60 −0.5 0.5 1 1.5 n

15

2

n+1 fn −

28 45π2

With the knowledge of the pole the residuum is computable

7

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Large-order behaviour of FQED

10 20 30 40 50 60 −0.4 −0.2 0.2 0.4 n

˜ fn+1 ˜ fn 2 21

Subtracting the first pole reveals the second pole ˜ fn = fn + 28 45π2 15 2 −n−1

8

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Large-order behaviour of FQED

10 20 30 40 50 60 −40 −20 n

log10 |(n + 1)c(1)

n |

log10 |(n + 1)c(1)

n |large-order approximation

After ∼ 30 terms the large-order behaviour sets in (for subleading behaviour later)

9

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SLIDE 14

How many coefficients are needed?

1 2 3 4 10 20 30 40 50 npole coefficients

QED QCD SQED SQCD

”Closer” to the origin → less coefficients are needed

10

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Pad´ e methods

Analytic continuation of truncated Taylor series by ration of two polynomials FQED(x) ≈

M

  • n=0

fn xn − → P[R,S](x) = PR(x) QS(x) with R + S = M.

11

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Pad´ e methods

Analytic continuation of truncated Taylor series by ration of two polynomials FQED(x) ≈

M

  • n=0

fn xn − → P[R,S](x) = PR(x) QS(x) with R + S = M. Rewriting of resummed FQED(x) FQED(x) ∼ Γ(1 + x

3)

Γ( 1

2 + x 3)

sin2 πx

3

  • cos

πx

3

  • Pad´

e approximant with 2R ≈ S should lead to best results.

11

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SLIDE 17

Pad´ e approximants of FQED

2 4 6 8 10 12 14 16 18 20 −0.5 0.5 1 x

FQED 12

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SLIDE 18

Pad´ e approximants of FQED

2 4 6 8 10 12 14 16 18 20 −0.5 0.5 1 x

FQED M = 20 12

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SLIDE 19

Pad´ e approximants of FQED

2 4 6 8 10 12 14 16 18 20 −0.5 0.5 1 x

FQED M = 20 M = 30 12

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SLIDE 20

Pad´ e approximants of FQED

2 4 6 8 10 12 14 16 18 20 −0.5 0.5 1 x

FQED M = 20 M = 30 M = 40 12

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SLIDE 21

Pad´ e approximants of FQED

2 4 6 8 10 12 14 16 18 20 −0.5 0.5 1 x

FQED M = 20 M = 30 M = 40 M = 50 12

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SLIDE 22

Pad´ e approximants of FQED

2 4 6 8 10 12 14 16 18 20 −0.5 0.5 1 x

FQED M = 20 M = 30 M = 40 M = 50 M = 60 12

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SLIDE 23

Pad´ e approximants of FQED

2 4 6 8 10 12 14 16 18 20 −0.5 0.5 1 x

FQED M = 20 M = 30 M = 40 M = 50 M = 60

  • Need ∼ 30 coefficients to resolve first singularity

(similar to large-order growth analysis)

  • Can resolve function beyond the first singularity

12

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SLIDE 24

QED beta function

1/Nf :

13

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SLIDE 25

QED beta function

1/Nf : 1/N2

f (subset): 13

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SLIDE 26

QED beta function

1/Nf : 1/N2

f (subset):

Master integral known / not know

13

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SLIDE 27

Beyond 1/Nf : nested diagrams

Nested sub-part of beta function: gauge & RG scale independent

1/Nf

Computation up to K 44

14

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SLIDE 28

Beyond 1/Nf : nested diagrams

Nested sub-part of beta function: gauge & RG scale independent

1/Nf

Computation up to K 44 At O(1/N3

f )

Computation up to K 32

14

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Ratio test at O(1/N2

f )

β(2)

nested =

  • n

bnK n

5 10 15 20 25 30 35 40 −0.5 0.5 1 n

bn+1 bn

New finite radius of convergence

15

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SLIDE 30

Ratio test at O(1/N2

f )

β(2)

nested =

  • n

bnK n

28 30 32 34 36 38 40 42 0.3 0.31 0.32 0.33 0.34 n

bn+1 bn

New finite radius of convergence but extreme slow convergence

15

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SLIDE 31

Richardson extrapolation

Enhance the convergence of the series an = s + A n + B n2 + C n3 + . . .

16

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SLIDE 32

Richardson extrapolation

Enhance the convergence of the series an = s + A n + B n2 + C n3 + . . . First Richardson (B = C = . . . = 0) R(1)an ≡ s = (n + 1)an+1 − nan

16

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SLIDE 33

Richardson extrapolation

Enhance the convergence of the series an = s + A n + B n2 + C n3 + . . . First Richardson (B = C = . . . = 0) R(1)an ≡ s = (n + 1)an+1 − nan Second Richardson (C = . . . = 0) R(2)an ≡ s = 1 2

  • (n + 2)2an+2 − 2(n + 1)2an+1 + n2an
  • 16
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Richardson extrapolation

Enhance the convergence of the series an = s + A n + B n2 + C n3 + . . . First Richardson (B = C = . . . = 0) R(1)an ≡ s = (n + 1)an+1 − nan Second Richardson (C = . . . = 0) R(2)an ≡ s = 1 2

  • (n + 2)2an+2 − 2(n + 1)2an+1 + n2an
  • For oscillating series: Shanks transformation

16

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SLIDE 35

Ratio test at O(1/N2

f )

28 30 32 34 36 38 40 42 0.3 0.31 0.32 0.33 0.34 n

bn+1 bn

Bare series: K ∗ = 3.14

17

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SLIDE 36

Ratio test at O(1/N2

f )

28 30 32 34 36 38 40 42 0.3 0.31 0.32 0.33 0.34 n

bn+1 bn

R(1) bn+1

bn 1 3

Bare series: K ∗ = 3.14 First Richardson: K ∗ = 3.003

17

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Ratio test at O(1/N2

f )

28 30 32 34 36 38 40 42 0.3 0.31 0.32 0.33 0.34 n

bn+1 bn

R(1) bn+1

bn

R(2) bn+1

bn 1 3

Bare series: K ∗ = 3.14 First Richardson: K ∗ = 3.003 Second Richardson: K ∗ = 3.00008

17

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Residue at O(1/N2

f )

30 32 34 36 38 40 42 44 −0.53 −0.52 −0.51 −0.5 −0.49 n

3nn2bn R(2)[3nn2bn] − 1

2

Bare series: 3nn2bn = −0.512 Second Richardson: 3nn2bn = −0.500007

18

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SLIDE 39

Subleading behaviour

˜ bn = bn + 1 2 1 3n 1 n2

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SLIDE 40

Subleading behaviour

˜ bn = bn + 1 2 1 3n 1 n2

32 34 36 38 40 42 0.28 0.3 0.32 0.34 n

bn+1 bn

R(2) bn+1

bn 1 3

Bare series: K ∗ = 3.215 Second Richardson: K ∗ = 3.0003

19

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SLIDE 41

Subleading behaviour

˜ bn = bn + 1 2 1 3n 1 n2

32 34 36 38 40 42 44 −0.52 −0.51 −0.5 −0.49 n

3nn3˜ bn R(2)[3nn3˜ bn] − 1

2

Bare series: 3nn3˜ bn = −0.512 Second Richardson: 3nn3˜ bn = −0.500007

20

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Large-order behaviour

Large-order behaviour bn ∼ −1 2 1 3n 1 n2 + 1 n3 + . . .

  • + O
  • 1

(x > 3)n

  • = −1

2 1 3n 1 n(n − 1)

21

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SLIDE 43

Large-order behaviour

Large-order behaviour bn ∼ −1 2 1 3n 1 n2 + 1 n3 + . . .

  • + O
  • 1

(x > 3)n

  • = −1

2 1 3n 1 n(n − 1) Resummation

∞

  • n=4

bnK n ∼ 1 6(K − 3) ln

  • 1 − K

3

  • + finite

21

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Large-order behaviour

Large-order behaviour bn ∼ −1 2 1 3n 1 n2 + 1 n3 + . . .

  • + O
  • 1

(x > 3)n

  • = −1

2 1 3n 1 n(n − 1) Resummation

∞

  • n=4

bnK n ∼ 1 6(K − 3) ln

  • 1 − K

3

  • + finite

Logarithmic branch cut but no pole!

21

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Nested beta function at 1/N2

f

0.5 1 1.5 2 2.5 3 0.1 0.2 K β(2)

nested

”Exact” nested beta function up to K = 3 Beta function ambiguous beyond K = 3 or magic cancellation needed

22

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SLIDE 46

Nested beta function at 1/N2

f beyond the first branch cut 1 2 3 4 5 6 7 8 −1.5 −1 −0.5 0.5 K β(2)

nested − branch-cut

P[20,20] P[21,21] P[22,22]

No singularity before K = 15/2 Positive pole at K = 15/2?

23

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SLIDE 47

Nested beta function at 1/N3

f 0.5 1 1.5 2 2.5 3 3.5 −0.4 −0.2 0.2 0.4 K β(3)

nested

P[13,13] P[14,14] P[15,15]

No singularity before K = 3 Branch cut at K = 3?

24

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SLIDE 48

Outlook 1: Conformal Pad´ e

[Costin, Dunne ’19]

Can we do more with the coefficients that we have?

25

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Outlook 1: Conformal Pad´ e

[Costin, Dunne ’19]

Can we do more with the coefficients that we have? Conformal map: K = 6z 1 + z2 ← → z = K/3 1 +

  • 1 − K 2/9

25

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SLIDE 50

Outlook 1: Conformal Pad´ e

0.5 1 1.5 2 2.5 3 0.1 0.2 K β(2)

nested

exact P[13,13] CP[13,13]

Improvement over standard Pad´ e Requires knowledge on the location of the branch cut

26

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Outlook 2: Renormalons

5 10 15 20 25 30 35 40 10−26 10−8 1010 1028 n

|bn|diagram |bn|counter |bn|full

Two factorially divergent contributions but the sum goes to zero Are we picking up renormalon contributions?

27

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SLIDE 52

Outlook 2: Renormalons

Borel transform of the finite part of 1/Nf

2 4 6 8 10 −10 −5 5 10 t B

  • finite part
  • 1/Nf
  • 28
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SLIDE 53

Outlook 2: Renormalons

Borel transform of the finite part of 1/Nf

2 4 6 8 10 −10 −5 5 10 t B

  • finite part
  • 1/Nf
  • Do we pick up the renormalon at t = 3?

Why not the renormalon at t = 6?

28

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SLIDE 54

Outlook 2: Renormalons

5 10 15 20 25 30 35 40 10−26 10−8 1010 1028 n

|bn|diagram |bn|counter |bn|full

Large-order behaviour an ∼ n! n3 3n

  • −3 − 9

1 ln(n)3

  • + . . .

29

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SLIDE 55

Summary and outlook

  • Large-order behaviour & Pad´

e methods constitute powerful tools

  • First partial result beyond O(1/Nf ) for QED:

New logarithmic branch cut at K ∗ = 3 without pole

  • Ideas: Conformal Pad´

e & tracking renormalons

  • Future: Remaining diagrams (Master integrals?) & QCD

30

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SLIDE 56

Summary and outlook

  • Large-order behaviour & Pad´

e methods constitute powerful tools

  • First partial result beyond O(1/Nf ) for QED:

New logarithmic branch cut at K ∗ = 3 without pole

  • Ideas: Conformal Pad´

e & tracking renormalons

  • Future: Remaining diagrams (Master integrals?) & QCD

Thank you for your attention

30