3-loop gauge coupling β functions in the SM
Frontiers in Perturbative Quantum Field Theory, Bielefeld, September 10-12, 2012
Matthias Steinhauser — TTP Karlsruhe | in collaboration with Luminita Mihaila and Jens Salomon
0.05
0.045
α 1, α 2, α 3
0.04
0.035
0.03
0.025
0.02
0.015
0.01
2
4
6
8
10
12
14
16
log10(µ/GeV)
KIT – University of the State of Baden-Wuerttemberg and
National Laboratory of the Helmholtz Association
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
0.05
0.045
α1 , α2 , α3
0.04
0.035
0.03
0.025
0.02
0.015
0.01
2
4
6
8
10
12
14
16
log10(µ/GeV)
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
3
Gauge coupling (non) unification
60
1/αi
40
20
0
0
5
10
log10(µ/GeV)
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
15
3
Vacuum stability
Quartic Higgs coupling at the Planck scale
182
mpole
[GeV]
t
180 EW vacuum:
178 unstable
176
95%CL
174
172
metastable
170
168
stable
ILC
LHC
166
Ingrediants:
Tev⊕lHC
164
– 2-loop effective potential,
120
122
124
126
128
MH [GeV]
– 3-loop beta functions for couplings,
– 2-loop matching relation for initial values of couplings
130
132
[..., 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
d αs (µ)
= β(αs )
dµ2 π
α 3
α 2
α 2 αs
s
s
s
β0 + β1 +
β2 +
β3 + . . .
= −
π
π
π
π
µ2
β0 =
11
2
CA − Tnf
3
3
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
[Gross,Wilczek’73; Politzer’73]
5
QCD
d αs (µ)
= β(αs )
dµ2 π
α 3
α 2
α 2 αs
s
s
s
β2 +
β3 + . . .
β0 + β1 +
= −
π
π
π
π
µ2
β0 =
11
2
CA − Tnf
3
3
β1
[Jones’74; Caswell’74; Egorian,Tarasov’78]
β2
[Tarasov,Vladimirov,Zharkov’80; Larin,Vermaseren’93]
β3
[van Ritbergen,Vermaseren,Larin’97; Czakon’04]
[Gross,Wilczek’73; Politzer’73]
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 =
α2 =
α
5
3 cos2 θW
[SU(5)-like normalization]
α
sin2 θW
α3 = αs
Yukawa:
αx
=
αmx2
2
2 sin2 θW MW
x ∈ {u , d , c , s, t , b, e, µ, τ }
Higgs:
α7 =
λ
4π
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
[L = −λ(Φ† Φ)2 + . . .]
6
SM: β functions
µ2
d αi
= βi (α1 , α2 , α3 , αt , αb , ατ , . . . , λ)
dµ2 π
αibare = µ2ǫ Zαi αi
Zαi charge renormalization constant
MS scheme

−1
X
α
∂
Z
∂
Zαi
α
α
i
α
i
i
i
βj  1 +
β i = − ǫ +
π
Zαi
∂αj
Zαi ∂αi

j 6=i
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 boson
broken 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
Zα3
B
Zα2
Zα1
B
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
B
B
B
W±
B
B
γ, Z
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
# loops
BB
W3 W3
gg
cg c̄g
cW3 c̄W3
φ+ φ−
BBB
W1 W2 W3
ggg
cg c̄g g
cW1 c̄W2 W3
φ + φ − W3
1
14
17
9
1
2
10
34
34
21
2
2
24
Lorenz gauge
2
3
410
45 926
502
55 063
188
17 611
12
521
42
2 480
429
46 418
2 172 358 716
2 216 382 767
946 118 086
66
4 240
107
10 577
2 353 387 338
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
4
7 111 021
8 438 172
2 455 714
46 390
251 200
6 918 256
73 709 886
79 674 008
20 216 024
460 389
1 517 631
77 292 771
13
Number of diagrams
# loops
γB γB
γB Z B
Z BZ B
W +B W −B
gB gB
1
13
13
20
18
10
BFG
2
3
416
61 968
604 100 952
1064 183 465
1438 252 162
186
17 494
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
4
13 683 693
23 640 897
44 049 196
42 423 978
2 775 946
13
Vertex diagrams/loop integrals
set all masses to zero
set q1 = 0 or q2 = 0
q
2
q
1
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 fermion
triangle
different for Green’s functions with
external fermions
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
15
Automation
QGRAF
q2e
exp
QGRAF [Nogueira’91]
q2e/exp [Harlander,Seidelsticker,Steinhauser’97;Seidelsticker’97]
MINCER [Larin,Tkachov,Vermaseren’91]
MATAD [Steinhauser’00]
MATAD/MINCER
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
16
Automation
QGRAF
q2e
exp
FeynArts
QGRAF [Nogueira’91]
q2e/exp [Harlander,Seidelsticker,Steinhauser’97;Seidelsticker’97]
MINCER [Larin,Tkachov,Vermaseren’91]
MATAD [Steinhauser’00]
FeynArts3 [Hahn’01]
MATAD/MINCER
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
16
Automation
QGRAF
FeynArtsToQ2E
NEW
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]
FeynArtsToQ2E [Salomon’12]
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
16
Automation
QGRAF
FeynArtsToQ2E
NEW
q2e
exp
MATAD/MINCER
FeynArts
QGRAF [Nogueira’91] FeynRules
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(α33 αt ) to β3
[Steinhauser’98]
3-loop results for SU (3), SU (2), U (1) (only one gauge coupling)
3-loop λ terms [Curtright’80; Jones’80]
Z1,αi c c̄
[Pickering,Gracey,Jones’01]
Z
= √1,αi αi αi 3
( Z3,αi )
calculation for arbitrary ξQCD , ξW , ξB
➪ β functions are ξ independent
Zαi =
Z2c
√
Z3,αi
BBB vertex ➪ zero to 3 loops (= sum of 300 000 diagrams)
IR safe: introduce mass m 6= 0; asymptotic expansion for q 2 ≫ 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
α22
(4π)2
α22
+
(4π)3
6α1
α22
+
(4π)4
163α21
β2 =
−
−
+
−
5
8
243α2 trL̂
57trT̂ 2
4
+
−
16nG
3
518α2
− 6trT̂ − 6trB̂ − 2trL̂ + nG
3
+
561α1 α2
57trB̂
4
−
40
2
+
667111α22
432
533α1 trB̂
40
45(trB̂ )
44α21
15
+
−
13α1 α2
5
1660α22
−
5
8
4α1
5
+
196α2
3
+ 6α2 λ̂ − 12λ̂2 −
− 28α3 trB̂ −
19trL̂2
4
+ 16α3
+
593α1 trT̂
40
51α1 trL̂
8
5(trL̂)2
2
+
27trT̂ B̂
2
45(trT̂ )2
2
25648α22
27
176α23
27
9
nG : #45of generations
6α1 λ̂
729α2 trB̂
+ 15trB̂trL̂ +
2
+
−
+
2
+ 45trT̂ trB̂ + 15trT̂ trL̂ +
28α21
+ nG2 −
+
− 28α3 trT̂ −
+ nG −
3
400
729α2 trT̂
8
86
−
4α1 α3
15
+ 52α2 α3 +
500α23
3
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
18
Results: β2 as an example
α22
(4π)2
α22
+
(4π)3
β2 =
+
−
−
+
2
2
α
(4π)4
5
4
2
nG
2
1
16nG
3
518α2
− 6trT̂ − 6trB̂ − 2trL̂ + nG
3
+
561α1 α2
+
57trB̂ 2
4
+
667111α
−
40
− 28α3 trT̂ −
28α21
533α1 trB̂
40
45(trB̂ )2
2
44α21
15
45
1 loop:
+
13α1 α2
5
1660α22
−
α22 27
(4π)2
+
−
−
−
+
8
5
+
196α2
3
+ 6α2 λ̂ − 12λ̂2 −
− 28α3 trB̂ −
19trL̂2
4
+ 16α3
+
593α1 trT̂
40
51α1 trL̂
8
5(trL̂)2
2
+
27trT̂ B̂
2
2
27
+
5
4α1
45(trT̂ )2
176α23
9
6α1 λ̂
729α2 trB̂
+ 15trB̂trL̂ +
25648α22
86
3
2
2
432
+ 45trT̂ trB̂ + 15trT̂ trL̂ +
−
+
−
400
243α2 trL̂
57trT̂ 2
3
163α
8
8
86
6α1
729α2 trT̂
+ nG −
+
−
−
16nG
3
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
4α1 α3
15
+ 52α2 α3 +
500α23
3
18
Results: β2 as an example
α22
(4π)2
α22
+
(4π)3
6α1
α22
+
(4π)4
163α21
β2 =
−
−
+
−
5
3
8
243α2 trL̂
+
−
400
729α2 trT̂
8
86
16nG
3
518α2
− 6trT̂ − 6trB̂ − 2trL̂ + nG
3
+
561α1 α2
57trB̂ 2
4
−
40
− 28α3 trT̂ −
+
+
667111α22
432
533α1 trB̂
40
45(trB̂ )2
2
57trT̂ 2
−
+
6α1 λ̂
5
729α2 trB̂
8
+ 15trB̂trL̂ +
4α1
5
+
196α2
3
− 28α3 trB̂ −
4
+ 16α3
+ 6α2 λ̂ − 12λ̂2 −
19trL̂2
+
593α1 trT̂
40
51α1 trL̂
8
5(trL̂)2
2
+
27trT̂ B̂
2
45(trT̂ )2
45trT̂ trB̂ + 15trT̂ trL̂ +
tr4T̂ , +trB̂,
trL̂: Yukawa couplings
2
2
13α1 α2only25648
αα22 b α4τα1 α3
28
α
example:
keep
α
,
1
t
+
+
−
+ nG −
15
5
27 trL̂ →15
➪
ατ
trT̂ → αt , trB̂ → αb , + nG2 −
44α21
45
−
1660α22
27
−
+ 52α2 α3 +
500α23
3
176α23
9
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
18
Results: β2 as an example
α22
(4π)2
α22
+
(4π)3
6α1
α22
+
(4π)4
163α21
β2 =
−
−
+
−
5
8
243α2 trL̂
57trT̂ 2
4
+
−
16nG
3
518α2
− 6trT̂ − 6trB̂ − 2trL̂ + nG
3
+
561α1 α2
57trB̂
4
−
40
2
+
667111α22
432
533α1 trB̂
40
45(trB̂ )
44α21
15
+
−
13α1 α2
5
1660α22
−
5
729α2 trB̂
8
+ 15trB̂trL̂ +
2
+
−
+
6α1 λ̂
2
+ 45trT̂ trB̂ + 15trT̂ trL̂ +
28α21
+ nG2 −
+
− 28α3 trT̂ −
+ nG −
3
400
729α2 trT̂
8
86
4α1
5
+
196α2
3
+ 6α2 λ̂ − 12λ̂2 −
− 28α3 trB̂ −
19trL̂2
4
+ 16α3
+
593α1 trT̂
40
51α1 trL̂
8
5(trL̂)2
2
+
27trT̂ B̂
2
45(trT̂ )2
2
25648α22
27
176α23
−
4α1 α3
15
+ 52α2 α3 +
500α23
3
45 ∝ α27, α , α 9, α , α , α
2 loops:
1
2
3
t
b
τ
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
18
Results: β2 as an example
α22
(4π)2
α22
+
(4π)3
6α1
α22
+
(4π)4
163α21
β2 =
−
−
+
−
5
8
243α2 trL̂
57trT̂ 2
4
3
+
−
400
729α2 trT̂
8
86
16nG
3
518α2
− 6trT̂ − 6trB̂ − 2trL̂ + nG
3
+
561α1 α2
40
− 28α3 trT̂ −
+
57trB̂
4
2
+
−
667111α22
432
533α1 trB̂
40
45(trB̂ )
−
+
2
+ 45trT̂ trB̂ + 15trT̂ trL̂ +
6α1 λ̂
5
729α2 trB̂
8
2
+ 15trB̂trL̂ +
4α1
5
+
196α2
3
+ 6α2 λ̂ − 12λ̂2 −
− 28α3 trB̂ −
19trL̂2
4
+ 16α3
+
593α1 trT̂
40
51α1 trL̂
5(trL̂)2
2
8
+
27trT̂ B̂
2
45(trT̂ )2
2
2
13α1 α2 25648α22
4α1α3
2 α α + 500α3
+ nG3 −
+
+
−. . . ++λ52
2. . 3.
.
.
.
+
λα
loops:
α
α
i
i
j
15
5
27
15
3
28α21
44α21
1660α22
176α23
+ nG2 −
−
−
45
27
9
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
18
Numerics
0.05
0.045
α1 , α2 , α3
0.04
0.035
0.03
0.025
0.02
0.015
0.01
2
4
6
8
10
12
14
16
log10(µ/GeV)
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
19
Numerics
x 10
-1
0.237
0.2368
α1 , α2
0.2365
0.2363
0.236
0.2358
0.2355
0.2353
0.235
12.98
13
13.02
13.04
13.06
13.08
log10(µ/GeV)
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
19
Numerics
10
∆α3/α3
10
10
10
-2
-3
-4
-5
2
4
6
8
10
12
14
16
log10(µ/GeV)
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
19
Dominant terms
Relative contribution to the difference
“3-loop running” − “2-loop running” (µ = MZ → µ = 1016 GeV)
α1
α12 α32
> 90%
α2
α22 α32
α24
α23 α3
α34
α33 αt
α33 α2
α32 αt2
α32 α22
α32 α2 αt
+56%
+37%
+13%
α3
α3
α35
Matthias Steinhauser— 3-loop gauge coupling β functions in the SM
+137%
−112%
+45%
+28%
+17%
−16%
−58%
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