Standard Model and Planck Scale -3 years after Higgs -

January 31 2015

at KEK

Hikaru Kawai

(Kyoto Univ.)

Based on the collaborations with Yuta Hamada and Kin-ya Oda: arXiv: 1501.04455 Hamada’s poster

SM without SUSY is very good. From string theory point of view, SUSY breaking scale can be anywhere. So the simplest guess is that SUSY breaks at the Planck scale.

Outline

In other words, SM physics may be directly connected to the Planck scale physics. One concrete example is the Higgs inflation.

However we need a new principle by which Higgs mass and the cosmological constant are explained.

Suppose the underlying fundamental theory, such as string theory, has the momentum scale mS and the coupling constant gS .

The naturalness problem

Then, by dimensional analysis and the power counting of the couplings, the parameters of the low energy effective theory are given as follows:

dimension 2 (Higgs mass)

dimension 4 (vacuum energy or cosmological constant)

naturalness problem (cont.’d)

dimension 0 (gauge and Higgs couplings)

dimension -2 (Newton constant) 2

2 .SN

S

gGm

1 2 32

, , ,

.S

H S

g g g ggλ

2 22 0 .H S Sg mm +

2 40 .Ssg mλ −⋅ +

( ) ( )222 2 2 18100GeV 10 GeVH S Sm g m

( ) ( )44 4 182 ~ 3meV 10 GeVSmλ

unnatural ! →

unnatural ! !→

tree

SUSY as a solution to the naturalness problem

Bosons and fermions cancel the UV divergences:

However, SUSY must be spontaneously broken at some momentum scale MSUSY , below which the cancellation does not work.

+ bosons fermions

⇒ 2 2SUSY .Hm M

+ ⇒ 0.λ =

⇒ 2 0.Hm =

⇒ 4SUSY .Mλ

(cont’d)

Therefore, if MSUSY is close to mH , the Higgs mass is naturally understood, although the cosmological constant is still a big problem. However, no signal of new particles is observed in the LHC below 1 TeV. It is better to think about the other possibilities.

Possibility of desert

The first thing we should know is whether the SM is valid to the string/Planck scale or some new physics should come in. If it is the former case, there is a possibility that the SM is directly connected to the Planck scale physics.

Is the SM valid to the Planck/string scale?

In order to answer the question, we consider the SM Lagrangian with cutoff momentum Λ, and determine the bare parameters in such a way that the observed parameters are recovered. If no inconsistency arises, it means that the SM can be valid to the energy scale Λ.

.+

Y. Hamada, K. Oda and HK: arXiv:1210.2358 , arXiv:1305.7055 , 1308.6651

Bare parameters as a function of the cutoff

• 2-loop RGE • no Landau pole

[1405.4781]

[Hamada, Oda, HK ,1210.2538, 1308.6651]

mHiggs =125.6 GeV

21 2

116

Iπ

Λ

Higgs self coupling λ

λ takes a small value at high energy scale. If Mt=171GeV, λ=βλ=0 at 1017-18GeV.

[Hamada, Oda, HK,1210.2538, 1308.6651]

[1405.4781]

Bare mass If Mt=170GeV, the bare mass becomes 0 at MP.

Hamada, Oda, HK,1210.2538]

Three quantities,

become close to zero around the Planck/string scale.

Triple coincidence

( ), ,B B Bmλλ β λ

Such situation was predicted by Froggatt and Nielsen ’95. Multiple Point Criticality Principle (MPP)

This means that the Higgs potential becomes zero around the Planck/string scale.

V

𝑚𝑠 𝜙

[ ] [ ]( )exp .d Sϕ ϕ−∫

In the ordinary quantum theory, we start with the path integral of the form of canonical ensemble:

On the other hand in the statistical mechanics, the most fundamental concept is the micro canonical ensemble

[ ] [ ]( ) ,Z d H Eϕ δ ϕ= −∫and the canonical ensemble follows in the thermodynamic limit:

[ ] [ ]( ) [ ] [ ]( )exp / .d H E d H Tϕ δ ϕ ϕ ϕ− ⇒ −∫ ∫

Overview of MPP

In the micro canonical ensemble we control extensive variables, while in the canonical ensemble we control intensive variables. Example: Consider a given number of water molecules put in a box with a given volume.

water

vapor

E 𝑬𝒄

vapor

In the micro canonical ensemble the energy is controlled.

If 𝑬 < 𝑬𝒄, the system is automatically on the critical line. Coexisting phase is obtained without fine tuning. Intensive variables are tuned “miraculously”.

G

L S

T

p

[ ] [ ]( ) .d S Cϕ δ ϕ −∫

Let’s imagine that quantum theory is defined by a path integral of the micro canonical ensemble type

It is equivalent to the ordinary quantum mechanics, if the space-time volume is sufficiently large. If C is in some region, the coupling constants are automatically tuned in such a way that the vacuum energy degenerates with another vacuum. In other words, it is more natural to have degenerate vacua.

Higgs inflation Question: Can the Higgs field play the roll of inflaton?

Higgs inflation is possible if nature tunes one-parameter.

Hamada, Oda, HK: arXiv:1308.6651, 1403.5043

Higgs inflation (1) conservative approach We trust the effective potential only below the cutoff scale, and try to make bound on the parameters. Hamada, Oda and HK: arXiv: 1308.6651 (2) radical approach We trust the flat potential including the inflection point. We assume that nature does fine tunings if they are necessary. We introduce a non-minimal coupling. Hamada, Oda, Park and HK, arXiv: 1403.5043 “Higgs inflation still alive” Cook. Krauss, Lawrence , Long and Sabharwal, arXiv: 1403.4971 “Is Higgs Inflation Dead? ” Bezrukov and Shaposhnikov, arXiv: 1403.6078 “Higgs inflation at the critical point”

mH = 125.6 GeV

Higgs potential

𝜑[GeV]

( ) ( ) 4

4V

λ ϕϕ ϕ=

The effective potential we have seen can be trusted only below the cutoff scale Λ.

Above Λ it is not described by SM but we need string theory.

Conservative approach

SM string

𝚲

At present we do not have enough knowledge above Λ.

But there is a possibility that the Higgs potential in the stringy region is almost constant, and the inflation occurs in this region.

In order for this scenario to work, the necessary conditions are

( ) *

0 for ,

.

SM

SM

d VdV V

ϕϕ

> < Λ

Λ <where

24 65 4*

* *3 1.3 10 GeV .

2 0.11S

PA rV r Mπ = = × ×

Necessary conditions

125.6 GeV0.1

Hmr

==

Higgs inflation is possible if the cutoff scale is ∼ 1017GeV .

Radical approach We assume that the Higgs potential can be trusted up to the inflection point region.

The naïve guess is that inflation occurs if the parameters are tuned such that the inflection point almost becomes a saddle point.

𝜑[GeV]

inflection1η ϕ ϕ< ⇒

However, this inflection point can not produce a realistic inflation:

( )*sufficient in this region small N V ϕ′⇒

* too large /SA V ε⇒

𝜑[GeV]

We introduce a non-minimal coupling as the original Bezrukov-Shaposhnikov’s. Then a realistic Higgs inflation is possible.

2Rξ ϕ

( )2 2

, .1 /

h hP

hVh M

ϕ ϕξ

=+

In the Einstein frame the effective potential becomes

Non minimal Higgs inflation with flat potential

• 𝜉 need not be so large. It can be as small as 10.

• 𝑟 is no longer 1/𝑁∗2, and can be as large as 0.2.

Important difference from the original B-S

Higgs potential is small and flat around the string scale from the beginning.

We do not have to lower the potential by the coupling.

The only role of the non-minimal coupling is to flatten the potential for the large field region.

𝝃 need not be very large

Maximum value of V is 𝑉 ∼ 𝜆 𝜑ℎ 𝜑ℎ4 ∼ 𝜆 𝜑ℎ 𝑀𝑃4/𝜉2.

The situation so far is the same as the original BS.

What is new here is that 𝜆 𝜑 becomes small around the string scale: 𝜆 𝜑ℎ∗ ∼ 10−6 .

This allows rather small values for 𝜉 ∼ 10 .

65 4** 1.3 10 GeV .

0.11rV = × ×

𝒓 can be any value in 𝟎.𝟎𝟎 ≤ 𝒓 ≤ 𝟎.𝟎 .

In this case 𝑟 is no longer 1/𝑁∗2 . Instead possible values of 𝑟 is governed by the height of the potential:

𝑉∗ varies from ∼ 2 × 1064 GeV4 to ∼ 2 × 1065 GeV4,

when 𝑚𝐻 varies from 125.5 GeV to 126.5 GeV . (next slide)

65 4** 1.3 10 GeV .

0.11rV = × ×

Our scenario is still alive even if 𝒓 ∼ 𝟎.𝟎𝟎 .

mH = 125.6 GeV

𝜑[GeV]

example

𝑚𝐻 = 126.4 GeV

𝑚𝑡 :

𝜉 = 7 ℎ∗ = 0.90 MP

tuned around ∼ 171.5 GeV such that the height of the potential is 1 percent larger than that of the saddle point case.

Then we have 𝑟∗ = 0.19, 𝑁∗= 58 , V∗

𝜖∗= 5.0 × 10−7 , 𝑛𝑆∗ = 0.955 .

time evolution of the example

• Higgs potential becomes almost flat around the string scale.

• Higgs inflation is possible if one of the parameters is tuned so that the potential nearly has a saddle point.

What we have seen so far.

Can we say something beyond 𝑀𝑃 by string

theory?

Higgs potential in string theory

Higgs potential for large field values can be evaluated, and turns out to be compatible with MPP.

Hamada, Oda , HK: arXiv: 1501.04455

To be concrete we take heterotic string, and consider a 4 dimensional perturbative vacuum that is non-supersymmetric and tachyon-free.

Because the Higgs particle is a real scalar and massless at the tree level, its emission vertex is written as

𝑉 𝑘 = 𝑂(𝑧, 𝑧̅)𝑒𝑖𝑖⋅𝑋,

where 𝑂(𝑧, 𝑧̅) is a real (1,1) operator.

Higgs potential in string theory

The Higgs potential for the field value 𝜙 can be obtained by evaluating the partition function for the world sheet action

.

In general 𝑂(𝑧, 𝑧̅) has a nontrivial OPE with itself, and the calculation of the partition function is not simple.

Here we consider the case where 𝑂(𝑧, 𝑧̅) is factorized to (1,0) and (0,1) operators:

.

Then we can rewrite them by using free scalars

and the partition function is easily calculated.

★ Higgs comes from gauge-Higgs unification.

Emission vertex of gauge field 𝐴𝜇𝑎 is

𝜕𝑧𝑋𝜇 𝑗𝑎 𝑧̅ 𝑒𝑖𝑖⋅𝑋 .

Then the emission vertex of 𝐴5𝑎 is

𝜕𝑧𝑋5 𝑗𝑎 𝑧̅ 𝑒𝑖𝑖⋅𝑋 .

★ Generic cases in the Fermionic construction.

★ Higgs comes from an untwisted sector in the orbifold construction.

This is rather generic.

A few cases where 𝑶 is factorized

As the simplest example we consider the case

𝑂 𝑧, 𝑧̅ = 𝜕𝑧𝑋5 𝜕�̅�𝑋5,

where 𝑋5 is a compactified dimension with the compactification radius 𝑅.

Then the world sheet action to consider is

𝑆 = 𝑆0 + 𝜙�𝑑2𝑧 𝑂(𝑧, 𝑧̅) = �𝑑2𝑧(1 + 𝜙)𝜕𝑧𝑋5 𝜕�̅�𝑋5 + ⋯ .

This comes back to the original 𝜙 = 0 form by the redefinition

𝑋′ 5 = 1 + 𝜙 𝑋5.

But the compactification radius is changed from 𝑅 to

𝑅𝑅= 1 + 𝜙 𝑅.

Simplest example - radion potential-

The radion potential for large field can be evaluated by considering the dimensional reduction of 5D gravity to 4D:

𝑆𝑒𝑒𝑒 ~∫𝑑4𝑥 −𝑔𝑅𝑅(ℛ −12𝑔𝜇𝜇

𝑅′2𝜕𝜇𝑅′𝜕𝜈𝑅′ − 𝒞),

(5)2

0,

0MN

gg

Rµν

= ′ ( )5 (5) (5) ,effS d x g− −∫ R C

(1 ) .R Rφ′ = +

Converting it to the Einstein frame

𝑔𝜇𝜈 = 𝑅′𝑔𝜇𝜈𝐸 , we have

𝑆𝑒𝑒𝑒 ~�𝑑4𝑥 −𝑔𝐸(ℛ𝐸 −12𝑔𝐸𝜇𝜈

𝑅′2𝜕𝜇𝑅′𝜕𝜈𝑅′ −

𝒞𝑅′

).

The Einstein frame potential is runaway for large radius if cosmological constant in 5D is positive. This is true for all orders. It follows only from the low energy effective action in 5D ( )5 (5) (5) .effS d x g− −∫ R C

Background for 𝜕𝑧𝑌𝜕�̅�𝑍 corresponds to the Lorentz boost for the momentum lattice for Y and Z.

3cases: runaway, periodic, chaotic

General 𝟏,𝟎 × 𝟎,𝟏 case

In the runaway case, a new space dimension is generated and opened up from the internal degree of freedom.

In string theory there is no essential difference between internal degrees of freedom and the space-time coordinates. “Color is space. Space is color.”

𝜙

𝑉

5D 4D

We consider 9D toy model instead of 4D theory.

Compactify the tenth direction of 10D 𝑺𝑶 𝟏𝟏× 𝑺𝑶(𝟏𝟏) heterotic string on S1 with radius 𝑟. The tenth component of the 10D gauge field becomes a scalar, whose emission vertex is

𝑉𝑎 = 𝜕𝑧𝑋10𝑗𝑎 𝑧̅ 𝑒𝑖𝑖⋅𝑋,

where 𝑗𝑎(𝑧̅) is 𝑆𝑂 16 × 𝑆𝑂(16) current.

We then introduce a background 𝐴 for one component 𝑎.

A toy model

Example of GHU

No SUSY

periodic in A increasing in r

One-loop potential in Jordan frame

Runaway for large r True for all orders

One-loop potential in Einstein frame

We have seen that we have a runaway vacuum in addition to our vacuum.

Eternal inflation at domain wall

Domain wall between two vacua:

✴ For a given random initial condition.

If the curvature at the maximum is of order one in the Planck unit, the domain wall is wide enough that

✴DW supports eternal inflation.

✴ A solution to horizon problem.

Eternal inflation at false vacuum is also possible.

Higgs inflation works. Need another inflaton.

Solution to CC problem

If we assume MPP, our vacuum should degenerate with the runaway vacuum, which has vanishing CC.

So the CC of our vacuum should be zero.

This is too good.

CC of our universe is not exactly zero ~ 2.2 𝑚𝑒𝑉 4.

𝜙

𝑉

In general a system follows quantum mechanics. In the classical limit, it is described by the least action principle.

However the value of the action is not completely determined because of the quantum fluctuation. Classical action is defined only within this error.

The fluctuation is evaluated as

Quantum fluctuation of action

(Δ𝑆)2 ∼ ∫ 𝑑4𝑥ℒ 𝑥 ∫ 𝑑4𝑦ℒ 𝑦 ∼ 𝑉 × 𝑀𝑃4 .

𝑉: space-time volume

(𝜕𝜙)2 (𝜕𝜙)2

If 𝑉 = ∞, Δℒ = 0.

For a de Sitter space with Hubble parameter 𝐻, it is natural to take 𝑉 ∼ 𝐻−4.

So we have Δℒ ∼ 𝑀𝑃2 ⋅ 𝐻2.

In particular the CC is determined up to the error ∼ 𝑀𝑃

2 ⋅ 𝐻2 ∼ 𝑚𝑒𝑉 4.

The classical Lagrangian density is determined

up to an ambiguity Δℒ ∼ 𝑀𝑃2

𝑉 .

Predictions on the Higgs portal DM

MPP imposes constraints on the Higgs portal DM.

Hamada, Oda, HK: arXiv: 1404.6141

Higgs portal dark matter We consider 𝒁𝟐 invariant Higgs portal DM:

ℒ = ℒ𝑺𝑺 +𝟏𝟐𝝏𝝁𝑺

𝟐 −𝟏𝟐𝒎𝑺

𝟐𝑺𝟐 −𝝆𝟒!𝑺𝟒 −

𝜿𝟐𝑺𝟐𝑯†𝑯

mHiggs =125GeV mtop =172.892 GeV

κ ρ

The effect of 𝑺 on 𝝀 is opposite to that of top quark.

Top Yukawa lowers 𝝀 for smaller values of 𝝁, while 𝜿 increases 𝝀 almost uniformly in 𝝁.

increasing Top Yukawa increasing 𝜿 𝝁

𝝀 Suppose we have an inflection point at some scale 𝚲:

𝝀 𝚲 = 𝟎, 𝜷 𝝀 𝚲 = 𝟎.

If top mass is increased, 𝝀 becomes negative at some scale. Then we can recover the inflection point by increasing 𝜿. The scale of the inflection point decrease.

From the natural abundance of DM, the DM mass should be related to 𝜿:

𝒎𝑫𝑺 ∼ 𝟎𝟎𝟎𝐆𝐆𝐆 ×𝜿𝟎.𝟏

Naturalness and Big Fix

Even if CC problem is solved, we still have to explain the magnitude of the weak scale. In terms of the ordinary field theory it involves a fine tuning, but it may become natural if one consider modification of field theory such as MPP.

There are several attempts to slightly extend the framework of the local field theory in order to explain such fine tunings. • multiple point criticality principle Froggatt, Nielsen, Takanishi • baby universe and maximal entropy principle Coleman Okada, Hamada, Kawana, HK Kawana’ s poster • classical conformality Meissner, Nicolai, Foot, Kobakhidze, McDonald, Volkas Iso, Okada, Orikasa • asymptotic safety Shaposhnikov, Wetterich

• It seems we have nothing other than a minor modification of SM below the string scale.

• It is possible that the fine tunings result from a slight modification of the conventional quantum theory.

• For example, we can consider the possibility that the couplings are fixed to maximize the entropy of the universe.

• From this point of view Higgs inflation is natural.

• If DM is the Higgs portal scalar, its mass is predictable.

• Cosmological constant may be understood by combining stringy potential and MPP.

Summary