Four-loop beta functions in the Standard Model:Leading contributions and their impact on
vacuum stability
M. F. Zoller
in collaboration with K. G. Chetyrkin
DESY Zeuthen, particle physics seminar, 09/06/2016
Outline
I. Introduction: Vacuum stability and the evolution of couplings
II. SM β-functions at 4 loops
1. Details of the calculation: Renormalization, IR regularization
2. The problem of γ5-treatment
3. Leading 4 loop contributions to βgs [JHEP 1602 (2016) 095]
and leading 4 loop contributions to βyt and βλ [arXiv:1604.00853]
III. Connection to pure QCD results
IV. Current status of the the vacuum stability question in the SM
2
I. Vacuum stability and the evolution of couplings
The Standard Model: L = LQCD + LEW + LY ukawa + LΦ
gauge group: SUC(3)× SU(2)×UY (1) SSB−−−→ SUC(3)×UQ(1)
• fermions: quarks and leptons:
QL =
(
tb
)
L
, tR, bR; LL =
(
νττ
)
L
, τR; (+ light generations)
• gauge bosons:Aaµ (SUC(3), a = 1, . . . ,8)
W aµ (SU(2), a = 1,2,3) Dµ = ∂µ − ig1YfBµ − ig2
2σaW aµ − igsT aAaµ
Bµ (UY(1))
• scalar SU(2)-doublet: Φ =
(
Φ1
Φ2
)
SSB−−→
(Φ+
1√2(v+H + iχ)
)
3
Standard Model interactions:
• strong:
Aµa
gselectroweak:
W µa
g2
Bµ
g1
• Yukawa:
Φi
ytHiggs self-interaction:
Φi Φi
Φj Φj
λ
Classical Higgs potential:
V (Φ) = m2Φ†Φ+ λ(Φ†Φ
)2
|Φ(x)| =1√2(v+H(x))
Φcl(x) ≡ 〈0| Φ(x) |0〉= v√26= 0
v ≈ 246 GeV, M2H = −2m2 = 2λv2
0 50 100 150 200 250
ÈFÈ�GeV
VHÈFÈL v√
2
4
The effective Potential• Radiative corrections ⇒ Evolution of couplings λ, g1, g2, gs, yt, . . . and fields Φ, . . .
• Higgs potential → Veff (λ(Λ), gi(Λ), yt(Λ), . . .) [Φ(Λ)] [Coleman, Weinberg]
(Λ: scale up to which the SM is valid)
m H<m m in
m H>m m in
ÈFclÈ
Vef
fHÈF
clÈL
v√
2
5
For Φcl ∼ Λ≫ v:
V RGeff [Φ] = λ(Λ)Φ4(Λ) +O(λ2(Λ), g2i (Λ))
Φ(Λ) = Φcl·exp
−
log
(Λ2
µ20
)
∫
0
dt′γΦ(λ(t′), gi(t′))
(Λ: scale up to which the SM is valid, µ0: starting point for running, e.g. µ0 =Mt)
[Altarelli, Isidori; Ford, Jack, Jones]
Stability of SM vacuum ⇔ λ(Λ) > 0[Cabibbo; Sher; Lindner; Ford]
6
Evolution of couplings X ∈ {λ, g1, g2, gs, yt, . . .}
β-functions: µ2 ddµ2
X(µ2) = βX[λ(µ2), yt(µ2), gi(µ2), . . .]
⇒ Coupled system of differential equations with initial conditions:
µ2 d
dµ2λ(µ2) = βλ[λ(µ
2), yt(µ2), gi(µ
2)], λ(µ20) = λ0,
µ2 d
dµ2yt(µ
2) = βyt[λ(µ2), yt(µ
2), gi(µ2)], yt(µ
20) = yt0,
µ2 d
dµ2gs(µ
2) = βgs[λ(µ2), yt(µ
2), gi(µ2)], gs(µ
20) = gs0,
µ2 d
dµ2g2(µ
2) = βg2[λ(µ2), yt(µ
2), gi(µ2)], g2(µ
20) = g20,
µ2 d
dµ2g1(µ
2) = βg1[λ(µ2), yt(µ
2), gi(µ2)], g1(µ
20) = g10
︸ ︷︷ ︸ ︸ ︷︷ ︸
Calculated in MS-scheme,power series in couplings
Experimental data matched toMS-scheme
7
Evolution of SM couplings
λ(μ)g1(μ)g2(μ)yt(μ)gs(μ)
5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
Log10[μ/GeV]
Co
up
lin
gs
8
Evolution of λ(µ)
1 loop
2 loop
3 loop
5 10 15-0.02
0.00
0.02
0.04
0.06
0.08
0.10
Log10[μ/GeV]
λ(μ
)
9
II. Standard Model β-functions up to 4 loops
2 loop
[M. Fischler, C. Hill (1981); D. Jones (1982); M. Machacek, M. Vaughn (1983,1984,1985);I. Jack, H. Osborn (1984,1985)] [M. Fischler, J. Oliensis (1982); M. Machacek, M. Vaughn(1984); C. Ford, I. Jack, D. Jones (1992); M. Luo, Y. Xiao (2003)]
3 loop
• for gauge couplings g1, g2, gs [L. Mihaila, J. Salomon, M. Steinhauser (2012);A. Bednyakov, A. Pikelner, Velizhanin (2012)]
• for Yukawa couplings yt, yb, yτ, etc. [K. Chetyrkin, M.Z. (2012);A. Bednyakov, A. Pikelner, Velizhanin (2013)]
• for the Higgs self-coupling λ (and the mass parameter m2) [K. Chetyrkin, M.Z.(2012 and 2013); A. Bednyakov, A. Pikelner, Velizhanin (2013)]
4 loop
• βgs(gs) [T. van Ritbergen, J. Vermaseren, S. Larin (1997); M. Czakon (2005)]
• βgs(gs, yt, λ) [A. Bednyakov, A. Pikelner (2015); M.Z. (2015)]
• βλ ∝ y4t g6s [S. Martin (2016); K. Chetyrkin, M.Z. (2016)]
• βyt ∝ ytg8s and βm2 ∝ y2tg6s[K. Chetyrkin, M.Z. (2016)]
10
Calculation of βλ(λ, yt, gs, g2, g1)
= + + + . . .+
δZ(4Φ)1
= finite
︸ ︷︷ ︸
∝ y4t(16π2)
︸ ︷︷ ︸
∝ λ2
(16π2)
[
= finite
]
⇒ Field strength renormalization constant Z(2Φ)2
L4Φ = (−λ+ δZ(4Φ)1 )
(
Φ†Φ)2
!= −λB [Φ
†BΦB]
2 = −(λ+ δZλ)
[
Z(2Φ)2 Φ†Φ
]2
⇒ δZλ =λ−δZ(4Φ)
1(
Z(2Φ)2
)2 − λ =∞∑
n=1
an(λ,yt,gs,g2,g1)εn
use µ2 ddµ2
λB ≡ 0 ⇒ βλ =
[
λ ∂∂λ + 1
2
∑
igi
∂∂gi− 1
]
a1
11
2 and 3 loop orders:
∝ y4t λ
(16π2)2∝ y4t g
2s
(16π2)2∝ y4t g
4s
(16π2)3
∝ y8t(16π2)3
∝ y4t g42
(16π2)3∝ y4t g
4s
(16π2)3
4 loop order:
H H
H H
H H
H H
∝ y4t g6s
(16π2)4
12
Analogously the Higgs propagator:
H H H H
∝ y2t g6s
(16π2)4
13
And for the top-Yukawa vertex and top propagator:
t t
H
t t
∝ ytg8s
(16π2)4∝ g8s
(16π2)4
14
Regularization of IR singularities[Misiak and Munz (1995); Larin, van Ritbergen, Vermaseren (1996);
Chetyrkin, Misiak and Munz (1998)]
Problem: IR divergencies in diagrams with 4 external Φ:
q
+
q
• Introduce auxiliary mass M2 in every propagator denominator:
1(q+p)2
→ 1(q+p)2−M2.
(p external, q internal momenta)
• Expand in external momenta p
⇒ massive tadpole integrals
15
Exact decomposition of propagators: (p external, q internal momenta)
1
(q+ p)2=
1
q2 −M2+−p2 − 2q·p−M2
q2 −M2
1
(q+ p)2
Recursion:
1
(q+ p)2=
1
q2 −M2+−p2 − 2q·p(q2 −M2)2
+(−p2 − 2q·p)2(q2 −M2)3
︸ ︷︷ ︸
used in this method
− M2
(q2 −M2)2+M2(M2 +2p2 +4q·p)
(q2 −M2)3︸ ︷︷ ︸
∝M2
+(−p2 − 2q·p−M2)3
(q2 −M2)31
(q+ p)2︸ ︷︷ ︸
contributes only to finite part
Exact result independent of M2 ⇒ restore terms ∝M2 in exact decomposition
by introducing all possible M2-counterterms in our expansion
M2
2 δZ(2g)M2 AaµA
aµ, M2
2 δZ(2W)
M2 W aµW
a µ, M2
2 δZ(2B)
M2 BµBµ,M2
2 δZ(2Φ)
M2 Φ†Φ
16
Example:
At 1 loop:l
= l2
εCl2 +
M2
εCM2+ finite
At 2 loop: +
M2δZ(2Φ)M2 = −M2
εCM2
17
Renormalization: all counterterms needed at lower loop orders
In lower loop diagrams:
• Vertex V multiplied by (1 + h δZV )
• Wavefunction and auxiliary mass counterterms in propagators:
→ + h + h2 + h3
• for fermions in the full SM: (PL δZL + PR δZR)
with PL = 12 (1− γ5), PR = 1
2 (1 + γ5)
18
Automation
• Generation of diagrams → QGRAF [Noguira]
• SU(2)×UY (1) group factors → Form code [M.Z.]
• SUC(3) group factors → COLOR [Van Ritbergen, Schellekens, Vermaseren]
• find topologies → GEFICOM [Chetyrkin, M.Z.] and
Q2E, EXP [Seidesticker, Harlander, Steinhauser]
• Feynman rules, projectors, counterterms, fermion traces, expansion in
external momenta → FORM [Vermaseren] code [Chetyrkin, M.Z.]
• massive tadpole integrals → up to 3 loop: MATAD [Steinhauser];
Reduction at 4 loop: FIRE5 (C++ version) [Smirnov] (based on IBP)
⇒ use 19 Master integrals [Czakon et al]
19
p1
p2
p3
p4p5
p6 p7
p8
p9 p1
p2
p3
p4p5
p6 p7
p8
p10
planar topology: tad4lp non-planar topology: tad4lnp
independent loop momenta: p1, p2, p3, p4
⇒ p5 = p4 − p1, p6 = p2 − p1, p7 = p3 − p2,p8 = p3 − p4, p9 = p4 − p2, p10 = p4 + p2 − p1 − p3
Scalar products: p1 · p2 = −12
(
p26 − p21 − p22)
etc.
p2j →M2 − iDj
with scalar propagators Di =1i
1M2−p2i
All scalar four-loop diagrams are linear combinations of
TAD4l(n1, . . . , n10) :=∫
dDp1∫
dDp2∫
dDp3∫
dDp410∏
i=1Dnii
20
Results µ2 ddµ2
λ(µ) = βλ =∞∑n=1
1(16π2)n
β(n)λ (in the MS-scheme)
β(4)λ
=y4tg6sdR
{
C3F
(
−2942
3+ 160ζ5 +288ζ4 +48ζ3
)
+TFC2F
(
−64+ nf
(
+562
3− 160ζ4 +
32
3ζ3
))
+CAC2F
(3584
3+ 720ζ5 + 32ζ4 −
3304
3ζ3
)
+CATFCF
(5888
9− 160ζ5 +352ζ3 + nf
(
−2644
243+ 128ζ4 +16ζ3
))
+C2ACF
(
−121547
243− 520ζ5 − 88ζ4 +
1880
3ζ3
)
+T 2FCF
(
−256
9nf + n2
f
(
−128
3ζ3 +
10912
243
))}
+O(y6t) +O(λ) +O(g2) +O(g1)
21
Numericsfor Mt = 173.39 GeV, αs(MZ) = 0.1181, MH = 125.09 GeV
At µ =Mt:
β(1)λ (µ = Mt)
(16π2)= −1.0× 10−2,
β(2)λ (µ = Mt)
(16π2)2= −2.3× 10−5,
β(3)λ (µ = Mt)
(16π2)3= +1.1× 10−5,
β(4)λ (µ = Mt)
(16π2)4= +1.3× 10−5
22
Numericsfor Mt = 173.39 GeV, αs(MZ) = 0.1181, MH = 125.09 GeV At µ = 109GeV:
β(1)λ (µ = 109GeV)
(16π2)= −1.4× 10−3,
β(2)λ (µ = 109GeV)
(16π2)2= +8.0× 10−8,
β(3)λ (µ = 109GeV)
(16π2)3= +3.1× 10−7,
β(4)λ (µ = 109GeV)
(16π2)4= +6.9× 10−8
23
Results µ2 ddµ2
yt(µ) = βyt =∞∑
n=1
1(16π2)n
β(n)yt (in the MS-scheme)
β(4)yt
=ytg8s
{dabcdF
dabcdA
dR
(32− 240ζ3) + nf
dabcdF
dabcdF
dR
(−64+ 480ζ3)
+C4F
(1261
8+ 336ζ3
)
− CAC3F
(15349
12+ 316ζ3
)
+C2AC2
F
(34045
36− 440ζ5 +152ζ3
)
+ C3ACF
(
−70055
72+ 440ζ5 −
1418
9ζ3
)
+nfTFC3F
(280
3+ 480ζ5 − 552ζ3
)
+ nfCATFC2F
(8819
27− 80ζ5 +264ζ4 − 368ζ3
)
+nfC2ATFCF
(65459
162− 400ζ5 − 264ζ4 +
2684
3ζ3
)
+n2fT 2
FC2
F
(
−304
27− 96ζ4 +160ζ3
)
+ n2fCAT
2FCF
(
−1342
81+ 96ζ4 − 160ζ3
)
+n3fT 3
FCF
(664
81− 128
9ζ3
)}
+O(y3t) +O(λ) +O(g2) +O(g1).
with dabcdF
= 16Tr
(T aT bT cT d + T aT bT dT c + T aT cT bT d + T aT cT dT b + T aT dT bT c + T aT dT cT b
)
24
Numericsfor Mt = 173.39 GeV, αs(MZ) = 0.1181, MH = 125.09 GeV
At µ =Mt:
β(1)yt (µ = Mt)
(16π2)= −2.4× 10−2,
β(2)yt (µ = Mt)
(16π2)2= −2.9× 10−3,
β(3)yt (µ = Mt)
(16π2)3= −1.2× 10−4,
β(4)yt (µ = Mt)
(16π2)4= +5.9× 10−6
25
βgs at 4 loop from the ghost-gluon-vertex
µ 0
q
projector:qµ
q2
q
already scalar ∝ q2
q
µ1 µ2= + + . . .
projector:1
D−1 gµ1µ2·Ptr + qµ1qµ2·Plong
26
Treatment of γ5
γ5 = iγ0γ1γ2γ3 = i4!εµνρσγ
µγνγργσ in D = 4: {γ5, γµ} = 0 and γ25 = 1
• external fermion line:· · ·
γ5
• fermion loop (internal line):γ5
µ1 µ2
µ3 µ4
k2 k1
Tr(. . .) ∝ # εµ1µ2µ3µ4 + # εµ1µ2αβ kα1 k
β2 + . . .
⇒ At least 4 free Lorentz structures needed, else diagram=0
27
Treatment of γ5 in the Zyt calculation
γ5
∝ PL = 12(1− γ5)
use projector ∝ γ5 on
external fermion line,
apply
γ5 = i4!εµνρσγ
µγνγργσ with ε0123 = 1
and use
εµ1µ2µ3µ4εν1ν2ν3ν4 = −g[µ1ν1gµ2ν2g
µ3ν3g
µ4]ν4
(error of O(ε))
⇒ Pole part OK if Feynman integrals have only 1ε poles. (
√)
28
Treatment of γ5 in the gluon propagator
µ1 µ2
ν1 ν2
l1
l2
PL
PR
PR
PL
• Use εµ1µ2µ3µ4εν1ν2ν3ν4 = −g[µ1ν1gµ2ν2g
µ3ν3g
µ4]ν4 (1 + ε · LABEL)
⇒ LABEL drops out in final result, only 1ε poles remain.
• Anticommuting γ5 to different points in different diagrams changes result
in this case!
• Move γ5 to reading point, e. g. external vertex [Korner, Kreimer, Schilcher]
29
Treatment of γ5 in the gluon propagator
µ1 µ2
ν1 ν2
l1
l2
PL
PR
PR
PL
Non-naive contribution to gluon self-energy from terms with one γ5 on each
fermion line:
• Move γ5 to external vertices → 1εg
4s y
4t T
2F
(43 +8ζ3
)
• Same result for γ5 γµext or γµext γ5 or 12 (γ5 γµext − γµext γ5) [Larin]
(but same prescription in all 72 affected diagrams)
• Ward identity manifest in the transversal structure of the gluon self-energy
respected!
30
Results µ2 ddµ2
gs(µ) = βgs =∞∑n=1
1(16π2)n
β(n)gs (in the MS-scheme)
β(4)gs
gs=+ g8s
(
−14975312
+1078361
324nf −
50065
324n2f −
1093
1458n3f
−1782ζ3 +3254
27ζ3nf −
3236
81ζ3n
2f
)
+ g6s y2t
(
−995918
+1625
54nf +136ζ3
)
+ g4s y4t
(423
2+
2
3− 60ζ3 +4ζ3
)
← non-naive part
+ g2s y6t
(
−4238− 3ζ3
)
− 15 g2s y4t λ+18 g2s y
2t λ
2
β(4)gs
β(1)gs (16π2)3
=2.26× 10−4︸ ︷︷ ︸
g8s
+2.47× 10−5︸ ︷︷ ︸
g6s y2t
−1.06× 10−5︸ ︷︷ ︸
g4s y4t (naive)
+4.17× 10−7︸ ︷︷ ︸
g4s y4t (non-naive)
+2.77× 10−6︸ ︷︷ ︸
g2s y6t
+1.06× 10−7︸ ︷︷ ︸
g2s y4t λ
−1.82× 10−8︸ ︷︷ ︸
g2s y2t λ
2
31
III. Connection to pure QCD results
t t
H
H H
H H
H H
Leff = LQCD −yt√2Htt− λ
4H4
only external H ⇒ insertion of scalar current js = tt in QCD diagrams.
βyt = yt γm(αs) + . . . ,
γΦ2 =y2t2γSSq (αs) + . . .
scalar correlator ΠS(q2,mt) = i∫
dDx eiqx〈0|T [js(x)js(0)] |0〉
µ2d
dµ2ΠS = −2γmΠS + γSSq Q2 + γSSm m2
t , Q2 = −q2
32
The scalar correlator
ΠSB(q2,mt) = i
∫
dDx eiqx〈0|T[
jBs (x)jBs (0)
]
|0〉, jBs = tBtB
is renormalized as
ΠS(q2,mt) = Z2mΠSB(q
2,mBt ) +
(
ZSS2 Q2 + ZSSm m2t
)
µ−2ε
⇒ µ2d
dµ2ΠS = −2γmΠS + γSSq Q2 + γSSm m2
t
Anomalous dimensions:
γm = −d lnZmd lnµ2
,
γSSq =dZSS2d lnµ2
+ (2γm − ε)ZSS2 ,
γSSm =dZSSmd lnµ2
+ (4γm − ε)ZSSm .
33
And for βλ?Should be connected to renormalization of
(2π)Dδ (p1 + p2 + p3 + p4) Γ({pi}, αs,mt, µ)
= Z4mZ
22
∫
d4x1 . . .d4x4
∏
i
eipi·xi
〈0|T [js(x1) js(x2) js(x3) js(x4)] |0〉.
Start with generating functional of (connected) Green’s functions
W (J) =∞∑
n=1
1
n!
∫
d4x1 . . .d4xnG
(n)(x1, . . . , xn) J(x1) . . . J(xn)
defined in
Z(L, J) = eiW(J) =∫
DΦeiS(Φ)+∫Φ·Jd4x, S(Φ) =
∫
L(Φ)d4x.
Quantum Action Principle [Lowenstein; Lam; Breitenlohner] relates proper-
ties of (regularized) Lagrangian and the full Green’s functions:
∂
∂mtW (J) ≡
∫
DΦeiS(Φ)+∫Φ·Jd4x ∂
∂mtS(Φ)
/Z(L, J),
34
Renormalized QCD Lagrangian with nl massless quarks and a massive top:
LQCD = − 1
4Z3 (∂µAν − ∂νAµ)2 −
1
2g Z
3g1 (∂µA
aν − ∂νAaµ) (Aµ ×Aν)a
− 1
4g2Z
4g1 (Aµ × Aν)2 + Z2
nl∑
i=1
ψi (i/∂ + gZψψg1 Z−12 /A)ψi
+ Z2 t (i/∂ + gZψψg1 Z−12 /A− Zmmt) t.
But ∂∂mt
W (J) already not finite at J = 0!
∫
DΦeiS(Φ)+Φ·J(∫
ZmZ2 tt(x)dx
)
/Z(L, J)
35
Full QCD Lagrangian [Spiridonov]:
LfullQCD = LQCD −EB0with vacuum energy
EB0 = µ−2ǫ(E0(µ)− Z0(αs)m4t (µ)),
∂4
∂m4tW (J) with L = LfullQCD
⇒ Γ({pi}, αs,mt, µ)− i4!Z0 = finite
36
In all orders in αs:
βλ = y4t γ0(αs) + higher orders in yt, g2, g1, . . .
with the anomalous dimension of the vacuum energy
γ0 = (4γm − ε)Z0 + (β(αs)− ε)αs∂Z0
∂αs
defined as
dE0
d lnµ2= γ0(αs)m
4t .
Relation between 2-point and 4-point scalar correlator at q = 0 via the action
principle:
⇒ γ0 =1
12γSSm .
37
IV. Current status of vacuum stability
Starting values for SM couplings
[Sirlin, Zucchini; Hempfling, Kniehl; Jegerlehner et al; Bezrukov et al; Buttazzo et al; Kniehl, Veretin]
Example: The Higgs propagator
1 PI self energy:
+ + . . . =: =: Σ(p2,mH , gi)
full propagator:
+ + + . . . = ip2−m2
H−Σ(p2,mH ,gi)
Physical mass MH ⇔ pole of full propagator
For p2 =M2H ⇒ m2
H −Σ(p2,mH , gi) = M2H with mH =
√2λ v
Same for Mt, MW , MZ, GF ⇒ Solve for λ, v, yt, etc.
38
Starting values for SM couplings
Experimental input
Mt =Mt = 174.6pole ± 1.9 GeV.
MMCt ≈ 173.34± 0.76 GeV ⇒ Mt = 173.39+1.12
−0.98 GeV
[Moch et al] via MSR mass [Hoang et al]
MH ≈ 125.09± 0.24 GeV
αs ≈ 0.1181± 0.0013
MS-couplings:
gs(Mt) = 1.1652± 0.0035(exp),
yt(Mt) = 0.9374+0.0063−0.0062 (exp)± 0.0005 (2 loop matching),
λ(Mt) = 0.1259± 0.0005(exp)± 0.0003 (2 loop matching),
g2(Mt) = 0.6483,
g1(Mt) = 0.3587
39
Evolution of λ(µ)
Mt=(173.39-0.98) GeVMt=(173.39+1.12) GeVyt (Mt)=0.9374 ± 0.0005
yt (Mt)=0.9374
6 8 10 12 14 16 18
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
Log10 [μ/GeV]
λ(μ
)
40
Evolution of λ(µ)
yt (Mt)=0.9374 ± 0.0005
3 loop
partial 4 loop
9.4 9.5 9.6 9.7 9.8 9.9-0.0015
-0.0010
-0.0005
0.0000
0.0005
0.0010
0.0015
Log10 [μ/GeV]
λ(μ
)
41
Summary
• Stability of SM vacuum ↔ λ > 0
• β-functions in the SM at 3 loops for full SM and now partial 4 loop
• Leading contribution to βλ and βyt connected to pure QCD current corre-
lators to all orders.
• Limiting problem for full 4 loop β-functions and 3 loop matching in the
SM: γ5
• SM vacuum looks not stable (metastable), Mt largest source of uncertainty
42
Evolution of λ(µ)
Mt=(174.6 ± 1.9) GeV
yt (Mt)=0.9440 ± 0.0005
yt (Mt)=0.9440
6 8 10 12 14 16 18
-0.02
0.00
0.02
0.04
Log10 [μ/GeV]
λ(μ
)
43
Evolution of λ(µ)
Mt=173.39+1.12 GeV
Mt=173.39-0.98 GeV
MH=125.09-0.24 GeV
MH=125.09+0.24 GeV
αs=0.1181-0.0013
αs=0.1181+0.0013
2 loop
3 loop
6 8 10 12 14 16 18-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
Log10[μ/GeV]
λ(μ
)
44
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