3-loop gauge couplingβ functions in the SM · Gauge coupling (non) uniﬁcation log10(µ/GeV) α 1...

3-loop gauge coupling β functions in the SMFrontiers in Perturbative Quantum Field Theory, Bielefeld, September 10-12, 2012

Matthias Steinhauser — TTP Karlsruhe | in collaboration with Luminita Mihaila and Jens Salomon

KIT – University of the State of Baden-Wuerttemberg and

National Laboratory of the Helmholtz Association

www.kit.edu

log10(µ/GeV)

α 1, α 2, α 3

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

2 4 6 8 10 12 14 16

http://www.kit.edu

Outline

Motivation

Calculation of 106 Feynman diagrams

Results

Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 2

Gauge coupling (non) unification

log10(µ/GeV)

α 1, α

2, α

3

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

2 4 6 8 10 12 14 16


Gauge coupling (non) unification

log10(µ/GeV)

1/α i

0

20

40

60

0 5 10 15


Vacuum stability

Quartic Higgs coupling at the Planck scale

Tev⊕lHCLHCILCstable

stable

meta-

unstable

EW vacuum:

95%CL

MH [GeV]

mpole

t[G

eV

]

132130128126124122120

182

180

178

176

174

172

170

168

166

164

Ingrediants:– 2-loop effective potential,– 3-loop beta functions for couplings,– 2-loop matching relation for initial values of couplings[..., Bezrukov,Kalmykov,Kniehl,Shaposhnikov’12; Degrassi,Di Vita,Elias-Miro,Espinosa,Giudice,Isidori,Strumia’12;

Alekhin,Djouadi,Moch’12]


QCD

µ2 d

dµ2

αs(µ)

π= β(αs)

= −(αs

π

)2[

β0 +αs

πβ1 +

(αs

π

)2β2 +

(αs

π

)3β3 + . . .

]

β0 =11

3CA − 2

3Tnf [Gross,Wilczek’73; Politzer’73]


QCD

µ2 d

dµ2

αs(µ)

π= β(αs)

= −(αs

π

)2[

β0 +αs

πβ1 +

(αs

π

)2β2 +

(αs

π

)3β3 + . . .

]

β0 =11

3CA − 2

3Tnf [Gross,Wilczek’73; Politzer’73]

β1 [Jones’74; Caswell’74; Egorian,Tarasov’78]

β2 [Tarasov,Vladimirov,Zharkov’80; Larin,Vermaseren’93]

β3 [van Ritbergen,Vermaseren,Larin’97; Czakon’04]

β2 computed in may different ways.Check of β3 with indepent method still missing.


Couplings in the SM

Gauge:

α1 =5

3

α

cos2 θW[SU(5)-like normalization]

α2 =α

sin2 θWα3 = αs

Yukawa:

αx =αm2

x

2 sin2 θW M2W

x ∈ {u, d , c, s, t, b, e, µ, τ}

Higgs:

α7 =λ

4π[L = −λ(Φ†Φ)2 + . . .]


SM: β functions

µ2 d

dµ2

αi

π= βi(α1, α2, α3, αt , αb, ατ , . . . , λ)

αbarei = µ2ǫZαiαi Zαi charge renormalization constant

MS scheme

βi = −

ǫαi

π+

αi

Zαi

∑

j 6=i

∂Zαi

∂αjβj

(

1 +αi

Zαi

∂Zαi

∂αi

)−1

Calculation of β1, β2, β3 requires:

Zα1 , Zα2 , Zα3 to 3 loops

βt , βb, βτ to 1 loop αt,b,τ dependence of Zα1 , Zα2 , Zα3 starts at 2 loops

βλ to tree level λ dependence of Zα1 , Zα2 , Zα3 starts at 3 loops


SM: known results

Gauge

1 loop: [Gross,Wilczek’73; Politzer’73]

2 loops: [Jones’74; Caswell’74; Tarasov,Vladimirov’77; Egorian,Tarasov’79; Jones’81; Fischler,Hill’82; Machacek,Vaughn’83;

Jack,Osborn’84]

partial 3 loops: [Curtright’80; Jones’80; Steinhauser’98; Pickering,Gracey,Jones’01]

3 loops: [Mihaila,Salomon,Steinhauser’12]

Yukawa

2 loops: [Fischler,Oliensis’82; Machacek,Vaughn’83; Jack,Osborn’84]

partial 3 loops: [Chetyrkin,Zoller’12]

Higgs

2 loops: [Machacek,Vaughn’84; Jack,Osborn’84; Ford,JackJones’92; Luo,Xiao’02]

partial 3 loops: [Chetyrkin,Zoller’12]


2 approaches

I. Lorenz gauge,no spontaeous symmetry breaking,B and W bosons

II. Background field gauge,➪ only 2-point functions of background gauge bosonbroken phase of SM,γ, W and Z bosons


Computation of Zαi

(i) choose vertex involving αi

(ii) compute Zvrtx

(iii) compute wave function ren. Zwf,k

(iv) Zαi =(

Zvrtx

ΠZwf,k

)2


Vertices

Lorenz gauge Background field gauge

Zα3B B

Zα2B B W±

Zα1 BB B γ,Z


Number of diagrams

Lorenz gauge# loops 1 2 3 4

BB 14 410 45 926 7 111 021W3W3 17 502 55 063 8 438 172

gg 9 188 17 611 2 455 714cg c̄g 1 12 521 46 390

cW3 c̄W3 2 42 2 480 251 200φ+φ− 10 429 46 418 6 918 256BBB 34 2 172 358 716 73 709 886

W1W2W3 34 2 216 382 767 79 674 008ggg 21 946 118 086 20 216 024

cg c̄gg 2 66 4 240 460 389cW1 c̄W2W3 2 107 10 577 1 517 631φ+φ−W3 24 2 353 387 338 77 292 771


Number of diagrams

BFG# loops 1 2 3 4γBγB 13 416 61 968 13 683 693γBZ B 13 604 100 952 23 640 897Z BZ B 20 1064 183 465 44 049 196

W+BW−B 18 1438 252 162 42 423 978gBgB 10 186 17 494 2 775 946


Vertex diagrams/loop integrals

1q

2q

set all masses to zero

set q1 = 0 or q2 = 0

UV structure not changed

make sure that no IR poles are introduced

➪ “MINCER integrals” [Larin,Tkachov,Vermaseren’91]


γ5

anomaly cancellation in fermiontriangle

different for Green’s functions withexternal fermions


Automation

QGRAF

q2e

exp

MATAD/MINCER

QGRAF [Nogueira’91]

q2e/exp [Harlander,Seidelsticker,Steinhauser’97;Seidelsticker’97]

MINCER [Larin,Tkachov,Vermaseren’91]

MATAD [Steinhauser’00]


Automation

QGRAF

q2e

exp

MATAD/MINCER

FeynArts





FeynArts3 [Hahn’01]


Automation

QGRAF

q2e

exp

MATAD/MINCER

FeynArtsToQ2ENEW

FeynArts






FeynArtsToQ2E [Salomon’12]


Automation

QGRAF

q2e

exp

MATAD/MINCER

FeynRules

FeynArtsToQ2ENEW

FeynArts






FeynArtsToQ2E [Salomon’12]

FeynRules [Christensen,Duhr’09]


Checks

2-loop results for βYukawa [Fischler,Oliensis’82; Machacek,Vaughn’83; Jack,Osborn’84]

3-loop QCD β function [Tarasov,Vladimirov,Zharkov’80; Larin,Vermaseren’93]

3-loop O(α33αt) to β3 [Steinhauser’98]

3-loop results for SU(3), SU(2), U(1) (only one gauge coupling)[Pickering,Gracey,Jones’01]

3-loop λ terms [Curtright’80; Jones’80]

Zαi =Z1,αi cc̄

Z2c√

Z3,αi

=Z1,αiαiαi

(√

Z3,αi )3

calculation for arbitrary ξQCD, ξW , ξB

➪ β functions are ξ independent

BBB vertex ➪ zero to 3 loops (= sum of 300 000 diagrams)

IR safe: introduce mass m 6= 0; asymptotic expansion for q2 ≫ m2

➪ NO ln(m2/µ2) terms!(Note: up to 35 sub-diagrams/diagram!)


Results: β2 as an example

β2 =α2

2

(4π)2

{

−86

3+

16nG

3

}

+α2

2

(4π)3

{

6α1

5−

518α2

3− 6trT̂ − 6trB̂ − 2trL̂ + nG

[

4α1

5+

196α2

3+ 16α3

]}

+α2

2

(4π)4

{

163α21

400+

561α1α2

40−

667111α22

432+

6α1λ̂

5+ 6α2λ̂− 12λ̂2 −

593α1trT̂

40

−729α2trT̂

8− 28α3trT̂ −

533α1trB̂

40−

729α2trB̂

8− 28α3trB̂ −

51α1trL̂

8

−243α2trL̂

8+

57trB̂2

4+

45(trB̂)2

2+ 15trB̂trL̂ +

19trL̂2

4+

5(trL̂)2

2+

27trT̂ B̂

2

+57trT̂ 2

4+ 45trT̂ trB̂ + 15trT̂ trL̂ +

45(trT̂)2

2

+ nG

[

−28α2

1

15+

13α1α2

5+

25648α22

27−

4α1α3

15+ 52α2α3 +

500α23

3

]

+ n2G

[

−44α2

1

45−

1660α22

27−

176α23

9

]}

nG: # of generations



β2 =α2

2

(4π)2

{

−86

3+

16nG

3

}

+α2

2

(4π)3

{

6α1

5−

518α2

3− 6trT̂ − 6trB̂ − 2trL̂ + nG

[

4α1

5+

196α2

3+ 16α3

]}

+α2

2

(4π)4

{

163α21

400+

561α1α2

40−

667111α22

432+

6α1λ̂

5+ 6α2λ̂− 12λ̂2 −

593α1trT̂

40

−729α2trT̂

8− 28α3trT̂ −

533α1trB̂

40−

729α2trB̂

8− 28α3trB̂ −

51α1trL̂

8

−243α2trL̂

8+

57trB̂2

4+

45(trB̂)2

2+ 15trB̂trL̂ +

19trL̂2

4+

5(trL̂)2

2+

27trT̂ B̂

2

+57trT̂ 2


45(trT̂)2

2

+ nG

[

−28α2

1

15+

13α1α2

5+

25648α22

27−

4α1α3

15+ 52α2α3 +

500α23

3

]

+ n2G

[

−44α2

1

45−

1660α22

27−

176α23

9

]}

1 loop:α2

2

(4π)2

{

− 863 + 16nG

3

}



β2 =α2

2

(4π)2

{

−86

3+

16nG

3

}

+α2

2

(4π)3

{

6α1

5−

518α2

3− 6trT̂ − 6trB̂ − 2trL̂ + nG

[

4α1

5+

196α2

3+ 16α3

]}

+α2

2

(4π)4

{

163α21

400+

561α1α2

40−

667111α22

432+

6α1λ̂

5+ 6α2λ̂− 12λ̂2 −

593α1trT̂

40

−729α2trT̂

8− 28α3trT̂ −

533α1trB̂

40−

729α2trB̂

8− 28α3trB̂ −

51α1trL̂

8

−243α2trL̂

8+

57trB̂2

4+

45(trB̂)2

2+ 15trB̂trL̂ +

19trL̂2

4+

5(trL̂)2

2+

27trT̂ B̂

2

+57trT̂ 2


45(trT̂)2

2

+ nG

[

−28α2

1

15+

13α1α2

5+

25648α22

27−

4α1α3

15+ 52α2α3 +

500α23

3

]

+ n2G

[

−44α2

1

45−

1660α22

27−

176α23

9

]}

trT̂ , trB̂, trL̂: Yukawa couplingsexample: keep only αt , αb ατ

➪ trT̂ → αt , trB̂ → αb, trL̂ → ατ



β2 =α2

2

(4π)2

{

−86

3+

16nG

3

}

+α2

2

(4π)3

{

6α1

5−

518α2

3− 6trT̂ − 6trB̂ − 2trL̂ + nG

[

4α1

5+

196α2

3+ 16α3

]}

+α2

2

(4π)4

{

163α21

400+

561α1α2

40−

667111α22

432+

6α1λ̂

5+ 6α2λ̂− 12λ̂2 −

593α1trT̂

40

−729α2trT̂

8− 28α3trT̂ −

533α1trB̂

40−

729α2trB̂

8− 28α3trB̂ −

51α1trL̂

8

−243α2trL̂

8+

57trB̂2

4+

45(trB̂)2

2+ 15trB̂trL̂ +

19trL̂2

4+

5(trL̂)2

2+

27trT̂ B̂

2

+57trT̂ 2


45(trT̂)2

2

+ nG

[

−28α2

1

15+

13α1α2

5+

25648α22

27−

4α1α3

15+ 52α2α3 +

500α23

3

]

+ n2G

[

−44α2

1

45−

1660α22

27−

176α23

9

]}

2 loops: ∝ α1, α2, α3, αt , αb, ατ



β2 =α2

2

(4π)2

{

−86

3+

16nG

3

}

+α2

2

(4π)3

{

6α1

5−

518α2

3− 6trT̂ − 6trB̂ − 2trL̂ + nG

[

4α1

5+

196α2

3+ 16α3

]}

+α2

2

(4π)4

{

163α21

400+

561α1α2

40−

667111α22

432+

6α1λ̂

5+ 6α2λ̂− 12λ̂2 −

593α1trT̂

40

−729α2trT̂

8− 28α3trT̂ −

533α1trB̂

40−

729α2trB̂

8− 28α3trB̂ −

51α1trL̂

8

−243α2trL̂

8+

57trB̂2

4+

45(trB̂)2

2+ 15trB̂trL̂ +

19trL̂2

4+

5(trL̂)2

2+

27trT̂ B̂

2

+57trT̂ 2


45(trT̂)2

2

+ nG

[

−28α2

1

15+

13α1α2

5+

25648α22

27−

4α1α3

15+ 52α2α3 +

500α23

3

]

+ n2G

[

−44α2

1

45−

1660α22

27−

176α23

9

]}

3 loops: αiαj(

. . .)

+ λαi(

. . .)

+ λ2(

. . .)


Numerics

log10(µ/GeV)

α 1, α

2, α

3

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

2 4 6 8 10 12 14 16


Numerics

log10(µ/GeV)

α 1, α

2

0.235

0.2353

0.2355

0.2358

0.236

0.2363

0.2365

0.2368

0.237

x 10-1

12.98 13 13.02 13.04 13.06 13.08


Numerics

log10(µ/GeV)

∆α3/

α 3

10-5

10-4

10-3

10-2

2 4 6 8 10 12 14 16


Dominant termsRelative contribution to the difference“3-loop running” − “2-loop running” (µ = MZ → µ = 1016 GeV)

α1 α21α

23 > 90%

α2 α22α

23 +56%

α42 +37%

α32α3 +13%

α3 α43 +137%

α33αt −112%

α33α2 +45%

α23α

2t +28%

α23α

22 +17%

α23α2αt −16%

α3 α53 −58%


Conclusions

βα1 , βα2 , βα3 in SM to 3 loops

fundamental quantity of SM

automated setup

“3 loops − 2 loops” ∼ experimental uncertainty


3-loop gauge couplingβ functions in the SM · Gauge coupling (non) uniﬁcation log10(µ/GeV) α 1...

Documents

Transcript of 3-loop gauge couplingβ functions in the SM · Gauge coupling (non) uniﬁcation log10(µ/GeV) α 1...