Beyond Standard Model and Asymptotically

Safe Gravity

Jan H. Kwapisz

3 July 2019 Warsaw

Beyond General Relativity, Beyond Cosmological Standard Model

1

Table of Contents

1. Standard Model of Particle

Physics

2. Asymptotic safety

3. Beyond Standard Model

4. Summary

2

Standard Model of Particle Physics

Standard Model of particle Physics

Figure 1: Standard Model Interactions

3

Standard Model of particle Physics

• No experiments particle physics contradicting the Standard

Model predictions

• Experimental problems: dark matter, baryon asymmetry, lack

of right handed neutrinos

• Theoretical problems: strong CP problem, U(1) Landau pole,

hierarchy problem

4

Beyond the Standard Model

• Grand Unification Theories

• Supersymmetry: Minimal Supersymmetric Standard Model,

Supergravity, Superstrings

• No new physics scale between EW and Planck scale: minimal

extensions of SM

5

Asymptotic safety

What is asymptotic safety?

Renormalisation group equations (running of the couplings):

k∂gi (k)

∂k= βi ({gi (k)}) . (1)

Asymptotic safety:

• Generalisation of asymptotic freedom. A UV fixed interacting

point.

• Hypothesis: Gravity is asymptotically safe

S. Weinberg (1979) , see Tim Morris talk

• Phenomenological consequence: prediction of irrelevant

couplings, allowed range for relevant couplings

6

Matter beta functions with gravitational corrections

The matter beta functions are supplemented by the gravitational

corrections:

β(gj) = βSM(gj) + βgrav (gj , k), (2)

where due to universal nature of gravitational interactions:

βgrav (gj , k) =ajk

2

M2P + ξk2

gj . (3)

The ξ ≈ 0.024 for Standard Model and depends on cosmological

constant and MP fixed points. Depending on a sign of aj , there is

a repelling/attracting fixed point at 0.

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ξ dependence on fields

Constant ξ is field dependent

ξ =1

16πG ∗N,

with

G ∗N ≈ −12π

NS + 2ND − 3NV − 46≥ 0,

where NS ,ND ,NV are number of fermions, scalars and vector

particles. So most of the GUTs and MSSM were excluded. The

minimal extensions seems to be compatible with AS. Also the

predicted Higgs mass can differ depending on the model.

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Standard Model with gravitational corrections

• With the assumption of asymptotic safety of gravity the Higgs

mass (self coupling) was calculated as mH = 126± few GeV

two years before the detection.

M. Shaposhnikov and C. Wetterich (2010) arXiv:0912.0208.

• The top mass was also predicted, “yielding a Higgs mass of

Mh = 132 GeV in our simple truncation. Higher-order

interactions might reconcile global stability of the potential

with a Higgs mass agreeing with the experimental result.”

A. Eichhorn, A. Held (2010) arxiv:1707.01107

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Beyond Standard Model

Higgs Portal Models

• Sterile complex (real) scalar φ coupled to Higgs doublet:

Lscalar = (DµH)†(DµH) + (∂µφ?∂µφ)− V (H, φ). (4)

V (H, φ) = −m21H†H −m2

2φ?φ+ λ1(H†H)2

+λ2(φ?φ)2 + 2λ3(H†H)φ?φ. (5)

• The scalar particles combined from two states:

m21 = λ1v

2H + λ3v

2φ, m2

2 = λ3v2H + λ2v

2φ, (6)

and the lighter one identified with Higgs particle.

10

Model 1: Additional sterile quarks (KSVZ axion)

We add one scalar and electro-weak sterile heavy quarks:

LY = Yqijφ

Nq∑i=1

Q̄ iLQ

jR , (7)

with the UA symmetry and discrete symmetry Q iL → −Q i

L,

Q iR → Q i

R , φ→ −φ, due to SU(3) instanton effects the “phase”

of φ becomes massive: axion particle. We take Yqij = δijYq.

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Model 2: Conformal Standard Model

Include right handed neutrinos coupled to φ with the coupling yM :

L 3 1

2YMji φN

jαN iα, (8)

where YMij = yMδij . To resolve the baryogengesis problem, via

resonant leptogenesis, the right handed neutrinos have to be

unstable:

MN = yMvφ/√

2 > m2. (9)

Furthermore the CSM can resolve the SM problems like: hierarchy

problem, inflation and has dark matter candidate: minoron with

mass: v2/MP .

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Conditions for low energy coupling values and ai values

Impose two conditions:

• absence of Landau poles

• λ1(µ) > 0, λ2(µ) > 0, λ3(µ) > −√λ2(µ)λ1(µ).

And take:

agi = −1, ayt = −0.5, aλ1 = +3 = aλ2 = +3, aλ3 = +3. (10)

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The running of couplings

Consider the following couplings: g1, g2, g3, yt , yM = yq, λ1, λ2, λ3.

The matter beta functions are β̂ = 16π2β:

β̂g1 = 416 g3

1 ,

β̂g2 = −196 g3

2 ,

β̂g3 = (−11 + 4 + 23Nqθ(Mq − µ))g3

3 ,

β̂yt = yt(

92y

2t − 8g2

3 − 94g

22 − 17

12g21

),

β̂yM = 52y

3M ,

β̂λ1 = 24λ21 + 4λ2

3 − 3λ1

(3g2

2 + g21 − 4y2

t

)+ 9

8g42 + 3

4g22 g

21 + 3

8g41 − 6y4

t ,

β̂λ2 =(20λ2

2 + 8λ23 + 6λ2y

2M − 3y4

M

),

β̂λ3 = 12λ3 [24λ1 + 16λ2 + 16λ3

−(9g2

2 + 3g21

)+ 6y2

M + 12y2t

],

(11)

Nq number of sterile quarks and Mq is its mass.14

Gauge and Yukawa couplings running

The low-energy values at µ0 = 173.34 are: g1(µ0) = 0.35940,

g2(µ0) = 0.64754, g3(µ0) = 1.1888 and yt(µ0) = 0.95113.

Figure 2: The running of g1, g2, g3, yt

With this Yukawa for pure Standard Model: MH = 136 GeV.

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Model I: aλ3 = +3

One gets λ3 = 0, but still the Higgs mass is affected due to change

in g3 running.

Can the model be

extended to Planck scale? Nq Mq Mh

yes 1 200 GeV 133 GeV

no 2 200 GeV -

yes 2 1000 GeV 133.7 GeV

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Model II:aλ2 = +3, aλ3 = −3

Then we calculate λ1(λ3, yM), λ2(λ3, yM). Take the tree level

relations for m1,m2 (the one loop pole matching gives corrections

of order 1 GeV):

m21 = λ1v

2H + λ3v

2φ, (12)

m22 = λ2v

2φ + λ3v

2H , (13)

with m1 = 126GeV, vH = 246 GeV. Then:

300GeV > m2 > 270GeV, (14)

and

yM > 0.8, 350GeV > MN > 300GeV. (15)

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Comparison with experimental data

• The excess of events with four charged leptons at E ∼ 325

GeV seen by the CDF and CMS Collaborations can be

identified with a detection of a new ‘sterile’ scalar particle

proposed by the Conformal Standard Model

K. Meissner H. Nicolai 2013 arXiv:1208.5653

• The hypothetical heavy boson mass is measured to be around

272 GeV (in the 270− 320 GeV range)

arXiv:1506.00612

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Summary

Thank you for your attention

Talk based on article: arxiv.org/abs/1810.08461

For details check also backup slidesTo contact me use my mail:

jkwapisz@fuw.edu.pl

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Backup slides

Non-standard possibilities for couplings

Non standard possibilities are:

• Asymptotic safety, limµ→∞ g 6= 0,∀iβi (g∗) = 0. Theory has a

UV fixed point. Example: Weinberg hypothesis: Gravity.

• Oscillating g . Theory has a limit cycle. Quantum mechanics:

−g/r2 potential [12].

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Softly Broken Conformal Symmetry

We consider an effective theory valid below Λ. We split the bare

parameters mass and self-coupling into renormalised parameters

and counter-terms:

m2B(Λ) = m2

R − fquad(Λ, µ, λR)Λ2 + m2Rg

(λR , log

(Λ

µ

)), (16)

where g(λR , log

(Λµ

))is some function. Assume that the

quadratic divergences depends only on bare couplings:

fquad(Λ, µ, λR) = fquad(λB(Λ)). (17)

So if fquad(λB) = 0 at certain scale, then the hierarchy problem is

solved.

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Conformal Standard Model and Softly Broken Conformal Sym-

metry

For CSM the f̂ quadi = 16π2f quadi are:

f̂ quad1 (λ, g , y) = 6λ1 + 2λ3 +9

4g2

2 +3

4g2

1 − 6y2t , (18)

f̂ quad2 (λ, g , y) = 4λ2 + 4λ3 − 3y2M . (19)

For the Conformal Standard Model Λ . MP is sufficient, while the

Standard Model requires Λ� MP .

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Inflation in Conformal Standard Model

Lagrangian in the Jordan frame [6]:

L = DµH†DµH+∂µφ∂

µφ∗−(M2

P + ξ1H†H + ξ2|φ|2

)2

R−VJ(H, φ),

(20)

with ξi > 0. We get the standard result that:

ns ' 1− 2

N' 0.97, (21)

and:

r ' 12/N2 ' 0.0033, (22)

however it requires that ξ1, ξ2 ∼ O(104).

23

Can we have λ3 6= 0? Take aλ3 < 0

(a) λ1 dependence on λ3, yM

(b) λ2 dependence on λ3, yM 24

Coefficients aλ2 = aλ3 = −3, set of allowed couplings λ2, λ3, yM

Figure 3: Maximal (left) and minimal (right) yM(λ3, λ2),aλ2 = −3, aλ3 = −3

25

Coefficient: aλ2 = aλ3 = −3, allowed λ1

Figure 4: Plot of λ1(λ2, λ3, yM)

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Coefficient: aλ3 = +3, aλ2 = +3

Still: λ3 = 0. So SM and φ decouple, but:

Figure 5: λ2 dependence on yM

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Coefficients aλ2 = aλ3 = −3

It constrains the second scalar mass as:

272 GeV < m2 < 328 GeV and yM > 0.71. (23)

The neutrinos masses:

MN = 342+41−41 GeV (24)

and they satisfy instability condition.

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Conditions on m2

One can parametrise the discrepancies from SM as:

tanβ =λ0 − λ1

λ3

vHvφ. (25)

Conditions:

• | tanβ| < 0.35.

• un-stability condition for the second particle: m2 > 2m1.

29

Conditions on m2(2)

1. We checked that the analysed parameters satisfy the Softly

Broken Conformal Symmetry requirements at MP with

couplings going to zero (but nowhere else).

2. We analyzed the running of the β functions for m2 and m1,

where we took ami = −1. It gives no new bounds on m2 and

lambda-couplings.

30

Higgs portal case: yM = 0.0

For yM = 0.0:

m2 = 160+103−100 GeV, (26)

so classically it is stable.

31

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32

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