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