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
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
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 3
Gauge coupling (non) unification
log10(µ/GeV)
1/α i
0
20
40
60
0 5 10 15
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 3
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]
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 4
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]
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 5
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.
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 5
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 + . . .]
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 6
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
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 7
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]
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 8
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
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 9
Computation of Zαi
(i) choose vertex involving αi
(ii) compute Zvrtx
(iii) compute wave function ren. Zwf,k
(iv) Zαi =(
Zvrtx
ΠZwf,k
)2
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 10
Vertices
Lorenz gauge Background field gauge
Zα3B B
Zα2B B W±
Zα1 BB B γ,Z
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 11
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 12
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 12
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 12
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 12
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 12
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
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 13
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
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 13
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]
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 14
γ5
anomaly cancellation in fermiontriangle
different for Green’s functions withexternal fermions
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 15
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]
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 16
Automation
QGRAF
q2e
exp
MATAD/MINCER
FeynArts
QGRAF [Nogueira’91]
q2e/exp [Harlander,Seidelsticker,Steinhauser’97;Seidelsticker’97]
MINCER [Larin,Tkachov,Vermaseren’91]
MATAD [Steinhauser’00]
FeynArts3 [Hahn’01]
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 16
Automation
QGRAF
q2e
exp
MATAD/MINCER
FeynArtsToQ2ENEW
FeynArts
QGRAF [Nogueira’91]
q2e/exp [Harlander,Seidelsticker,Steinhauser’97;Seidelsticker’97]
MINCER [Larin,Tkachov,Vermaseren’91]
MATAD [Steinhauser’00]
FeynArts3 [Hahn’01]
FeynArtsToQ2E [Salomon’12]
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 16
Automation
QGRAF
q2e
exp
MATAD/MINCER
FeynRules
FeynArtsToQ2ENEW
FeynArts
QGRAF [Nogueira’91]
q2e/exp [Harlander,Seidelsticker,Steinhauser’97;Seidelsticker’97]
MINCER [Larin,Tkachov,Vermaseren’91]
MATAD [Steinhauser’00]
FeynArts3 [Hahn’01]
FeynArtsToQ2E [Salomon’12]
FeynRules [Christensen,Duhr’09]
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 16
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!)
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 17
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
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 18
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
]}
1 loop:α2
2
(4π)2
{
− 863 + 16nG
3
}
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 18
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
]}
trT̂ , trB̂, trL̂: Yukawa couplingsexample: keep only αt , αb ατ
➪ trT̂ → αt , trB̂ → αb, trL̂ → ατ
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 18
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
]}
2 loops: ∝ α1, α2, α3, αt , αb, ατ
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 18
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
]}
3 loops: αiαj(
. . .)
+ λαi(
. . .)
+ λ2(
. . .)
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 18
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
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 19
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
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 19
Numerics
log10(µ/GeV)
∆α3/
α 3
10-5
10-4
10-3
10-2
2 4 6 8 10 12 14 16
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 19
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%
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 20
Conclusions
βα1 , βα2 , βα3 in SM to 3 loops
fundamental quantity of SM
automated setup
“3 loops − 2 loops” ∼ experimental uncertainty
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM 21
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