Speculative remarks on quadratic divergence in scalar mass and naturalness

Speculative remarks on quadratic divergence in scalar mass and naturalness

Kazuo Fujikawa

Institute of Quantum Science, Nihon University

1. Quadratic divergence and SUSY

2. Quadratic divergence and dimensional regularization

3. Subtractive renormalization and naturalness

Quadratic divergence of the Higgs mass is the basis of the precise re-evaluation of the Standard Model and pos- sible physics beyond it.

This issue is related to the notion of ”naturalness”, and also to the presence or absence of SUSY at LHC. One may attempt a last minute consideration of this problem before the experiments at LHC.

I thus look at the problem from a personal point of view.

Quadratic divergence : Example:

L = 1

2 ∂ _µ φ(x)∂ ^µ φ(x) − 1

2 m ² ₀ φ(x) ² − 1

4! λ ₀ φ(x) ⁴ which gives one-loop mass correction

λ ₀ 2

∫ d ⁴ k (2π) ⁴

1

k ² + m ² ₀ = λ ₀ 32π ²

∫ _Λ

0

dk ² k ² k ² + m ² ₀

= λ ₀

32π ² [Λ ² − m ² ₀ ln Λ ² + m ² ₀

m ² ]

Thus the (renormalized) two-point effective potential is Γ ₂ (k, m ₀ ) = k ² + m ² ₀ + λ ₀

32π ² [Λ ² − m ² ₀ ln Λ ² + m ² ₀ m ² ₀ ]

since the wave function renormalization factor Z = 1 to this order. The conventional multiplicative renormaliza- tion suggests the replacement

m ² ₀ = Z _m m ² with

Z _m = 1 + λ ₀

32π ² ln Λ ² + m ² ₀

µ ²

We then have

Γ ₂ (k, m ₀ ) = k ² + m ² + λ ₀

32π ² [Λ ² − m ² ₀ ln µ ²

m ² ₀ ] + O(λ ² ₀ )

= k ² + m ² (

1 − λ

32π ² ln µ ² m ²

)

+ λ ₀

32π ² Λ ² + O(λ ² ₀ ) This quantity needs to be finite if the multiplicative renor-

malization works. One thus needs to take care of the quan-

dratic divergence term.

One may understand this relation in the following way:

(i) The mass of the scalar field becomes of the order of the cut-off mass. Namely, the scalar mass is forced to be enormous once the quantum corrections are included.

This leads to the problem of the ”hierarchy” problem. The Higgs particle, which needs to be around 1 TeV or less to preserve the so-called tree unitarity, cannot stay 1 TeV even if one chooses the tree level mass at around 1 TeV.

(ii) One may subtract the quadratic divergence altogether

so that one treats the scalar mass by multiplicative renor-

malization. This is the scheme which SUSY and also di-

mensional regulatization enforce. Alternatively, one may just subtract the quadratic divergence order by order by adding a suitable counter term.

The application of the dimensional regularization or the simple subtraction is generally called ”unnatural”, since there appears to be no symmetry or dynamical reason to remove the enormously large induced mass.

In contrast, SUSY scheme is called natural since the basic super symmetry ensures the multiplicative renormal- ization of the scalar mass just as the fermion mass.

(iii) Alternatively, one may notice that the scalar mass

is not precisely specified by the starting Lagrangian since the large mass could in principle be induced by quantum effects even if the bare mass is zero.

This possibility may suggest that one may have some

more freedom to treat the scalar mass. We make some

speculative comments on this problem later.

In this talk, we first discuss SUSY and then dimensional regularization.

We comment on SUSY as a hybrid version of the conven- tional renormalization and the dynamical view of Sakata.

The dimensional regularization is discussed in connec- tion with the classical scaling symmetry argument of W.

Bardeen. We then suggest a more realistic space-time view

of the dimensional regularization.

1. SUSY and quadratic divergence

The simplest model of supersymmetry in 4-dimensional space-time is the Wess-Zumino model.

J. Wess and J. Bagger, ”Supersymmetry and supergrav- ity”, (Princeton Univ. Press, 1992).

L = i∂ _n ψ ¯ σ ¯ ⁿ ψ + A ^? ¤ A + F ^? F +[m(AF − 1

2 ψψ) + g (AAF − ψψA) + h.c.]

which gives rise to the A ⁴ coupling if one integrates out

the auxiliary field F .

Nonrenormalization theorem

The Wess-Zumino model is made finite by a uniform wave function renormalization without even the finite renormal- ization of m and g when renormalized at the vanishing momenta.

All order proof:

Component formulation: J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310. ”higher derivative regu- larization” .

Superfield formulation: K. Fujikawa and W. Lang, Nucl.

Phys. B88 (1975) 61. ”vanishing momenta”

The quadratic divergence is thus controlled by SUSY by a cancellation of contributions of fermions and bosons.

But no simple regularization of gauge invariant SUSY models is known at this moment.

(i) No superfield formulation for general supersymmetric gauge theory.

(ii) Higher derivative regularization does not work for one loop diagrams

(iii) Dimensional regularization has complications in chiral

symmetry

(iv) Lattice regularization has difficulties with the Leibniz rule among others.

Those complications may be of fundamental nature or may be just of technical nature. In any case, the (non- perturbative) lattice treatment of supersymmetric gauge theory, for example, has no clear perspective at this mo- ment despite of many attempts.

(The non-gauge Wess-Zumino model is treated on the

lattice: K. Fujikawa, Nucl. Phys. B636 (2002) 80.)

Deep implication of supersymmetry

The extension of the notion of space-time (superspace):

(x ⁰ , x ¹ , x ² , x ³ ) → (x ⁰ , x ¹ , x ² , x ³ , θ _α , θ ¯ _{α ˙} )

with two-component Grassmann numbers θ _α , θ ¯ _{α ˙} , and su- persymmetry is the translation in these coordinates

φ(x ^µ − 2iθσ ^µ ζ ¯ − iησ ^µ ζ, θ ¯ + η, θ ¯ + ¯ ζ ) (0.1) A. Salam and J. Strathdee, Nucl. Phys.B76 (1974) 477.

”Infinitesimal extension of the notion of 4-dimensions”.

Many interesting model analyses of SUSY, and many experts on the subject. We just wait for the results of LHC.

If super-particles are discovered at LHC, it would pro- vide a ”materialistic level” of solution to the quadratic divergence, in the parlance of Sho-ichi Sakata, who at- tempted to replace renormalization by Bose-Fermi cancel- lation.

S. Sakata and O. Hara, Prog. Theor. Phys. 2 (1947)

30.

2. Dimensional regularization and quadratic divergence

The basic idea of the dimensional regularization is to extend the space-time dimensionality slightly away from D = 4, and then take the limit D = 4 after the actual calculation.

G. ’t Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189.

C.G. Bollini and J.J. Giambiagi, Nuovo Cim. B12 (1972)

20.

Example:

λ ₀ 2

∫ d ⁴ k (2π) ⁴

1

k ² + m ² ₀ → λ ₀ 2

∫ d ^D k (2π) ^D

1

k ² + m ² ₀ and ∫

d ^D k 1

k ² + m ² ₀ =

∫

d ^D k

∫ _∞

0

dse ^{− s(k}

^+m

⁾

=

∫ _∞

0

ds( π

s ) ^D/2 e ^{− sm}

= π ^D/2 (m ² ₀ ) ^{D/2 − 1} Γ(1 − D/2)

where

xΓ(x) = Γ(1 + x), Γ(1) = 1 Near ² = 2 − D/2 ' 0, we use

A ^{− ²} ' 1 − ² ln A, Γ(² − 1) = 1

²(² − 1) Γ(1 + ²) ' 1 − ²γ

²(² − 1)

= − 1

² + γ − 1

− ⁰

We thus have λ ₀

2

∫ d ⁴ k (2π) ⁴

1

k ² + m ² ₀ ' (λ ₀ µ ^{− 2²} )

32π ² m ² ₀ [ − 1

² + γ − 1 − ln( πµ ² 4m ² ₀ )]

The correspondence with the momentum cut-off is 1

² = 1

2 − D/2 ↔ ln Λ ² µ ²

and we do not encounter the quadratic divergence. Note

that the combination λ ₀ µ ^{− 2²} is demensionless even away

To my knowledge, the absence of the quadratic diver- gence in dimensional regularization was first noted by S.

Weinberg when the fine tuning problem associated with quadratic divergence was first pointed out by L. Susskind.

It is well-known that the dimensional regularization has certain complications with the definition of γ ₅ , and thus the treatment of the chiral anomaly, for example, is subtle.

W. Siegel, Phys, Lett. 94B (1980) 37.

In the context of gauge theory, the dimensional regular-

ization is expected to work consistently in anomaly-free gauge theory.

W. Bardeen argued that the massless scalar theory has a scaling symmetry in the classical level. It may thus be natural to use a regularization such as dimensional regula- tor which formally preserves this naive scaling symmetry.

But this scaling symmetry is generally broken by quantum

Weyl anomaly.

Speculative comment on dimensional regular- ization

Physical basis of dimensional regularization may be:

The dimensionality of our space-time is infinitesimally away from the exact D = 4, namely, the dimension of our space-time is actually defined by the limit ² → 0 with

D = 4 − 2².

All the successful aspects of the Standard Model are kept

in tact in this understanding. It is an interesting problem

to examine the treatment of dimension-sensitive quantities

such as quantum anomalies in this speculation.

3. Speculative comment on subtractive renor- malization

We apply the higher derivative regularization, which re- places the Lagrangian in renormalized perturbation theory by

L = − 1

2 φ(x)[ −¤ + m ² ]( −¤ + M ²

M ² ) ² φ(x) − 1

4! λφ(x) ⁴

− 1

2 (Z − 1)φ(x)( −¤ )φ(x) − 1

2 (Z _m − 1)m ² φ(x) ²

− 1

4! (Z _λ − 1)λφ(x) ⁴ − 1

2 δm ² φ(x) ²

where we renormalized as φ ₀ (x) = √

Zφ(x), m ² ₀ Z = Z _m m ² , λ ₀ Z ² = Z _λ λ.

The term δm ² = Zδm ² ₀ is an extra term to make the theory finite, but otherwise the choice of δm ² ₀ is arbitrary.

The free propagator is given by h T φ(x)φ(y) i =

∫ d ⁴ k

(2π) ⁴ e ^{ik(x − y)} 1

k ² + m ² ( M ²

k ² + M ² ) ²

and the one-loop mass correction λ

2

∫ d ⁴ k (2π) ⁴

1

k ² + m ² ( M ²

k ² + M ² ) ²

= λ

32π ²

∫ _∞

0

dk ² k ²

k ² + m ² ( M ²

k ² + M ² ) ²

= λ

32π ² [M ² + m ² M ²

M ² − m ² − m ² ( M ²

M ² − m ² ) ² ln M ² m ² ] A possible choice of the mass countern term is

(Z _m − 1)m ² + δm ²

= − λ

2 [M ² + m ² M ²

2 − ² − m ² ( M ²

2 − ² ) ² ln M ²

2 ]

and thus for large M ² , we have Zm ² ₀ + δm ² = m ² − λ

32π ² [M ² + m ² − m ² ln M ² m ² ].

Z = 1 to this order in the present model.

In contrast, in dimensional regulator where we do not directly encounter the quadratic divergence

λ ₀ 2

∫ d ⁴ k (2π) ⁴

1

k ² + m ² ₀ ' (λ ₀ µ ^{− 2²} )

32π ² m ² ₀ [ − 1

² + γ − 1 − ln( πµ ²

4m ² ₀ )]

The correspondence with the momentum cut-off is 1

² = 1

2 − D/2 ↔ ln Λ ² µ ² and no quadratic divergence.

The bare mass to this order is thus given by m ² ₀ = m ² + λ

32π ² m ² 1

² ∼ m ² + λ

32π ² m ² ln Λ ²

µ ²

To maintain this analogy in the higher derivative regu- larization, alternative choice is

m ² ₀ = m ² + λ

32π ² m ² ln M ² µ ² δm ² = − λ

32π ² M ²

In this case, the renormalized mass with one-loop correc- tion becomes

m ² + λ

32π ² [m ² − m ² ln µ ²

m ² ]

which is close to the dimensional regulator.

Our basic question is if it is possible to choose δm ² = − λ

32π ² (M ² − µ ² )

In this case, the renormalized (effective) mass with one- loop correction becomes

m ² + λ

32π ² [µ ² + m ² − m ² ln µ ² m ² ]

which does not vanish even if m ² ₀ = 0. This means that we deal with a possibility which is not controlled by the starting bare Lagrangian.

Cf., H. Sonoda, arXiv0710.1662v2 on ”exact renormal-

A weaker form of this version is to ask if m ² + λ

32π ² [m ² − m ² ln µ ²

m ² ] + λ(µ ₀ ) 32π ² µ ² ₀ with a fixed µ ₀ is possible.

Bare perturbation theory

To analyze this issue, it is convenient to use the bare per- turbation theory defined by the higher derivative regular- ization. One thus examines

L = − 1

2 φ ₀ (x)[ −¤ + m ² ₀ ]( −¤ + M ²

M ² ) ² φ ₀ (x) − 1

4! λ ₀ φ ₀ (x) ⁴

− 1 λ _{0 2 2 2 2}

where ˜ ∆ _sub (m ² ₀ , λ ₀ , M ² ) is chosen such that all the in- duced mass terms proportional to M ² are cancelled. When one renormalizes

φ ₀ (x) = √

Zφ(x), m ² ₀ = Z _m

Z m ² , λ ₀ = Z _λ

Z ² ,

the constant ˜ µ ² ₀ may be chosen such that λ ₀

32π ² µ ˜ ² ₀ Z = λ(µ) 32π ²

Z _λ

Z ² µ ˜ ² ₀ Z ≡ λ(µ)

32π ² µ ˜ ² (µ) ≡ µ ˆ ² (µ)

Note that the induced mass parameter ˜ µ ² (µ) is indepen- dent of the bare mass m ² ₀ or the renormalized mass m ² . The choice ˜ µ ² (µ) = 0 corresponds to the dimensional reg- ulator and the choice ˜ µ ² (µ) ∼ M ² corresponds to the natural value of the induced mass. This suggests that a more general renormalization scheme is feasible.

As for the renormalization group analysis, we have

h T φ ₀ (x ₁ )...φ ₀ (x _n ) i =

∫

D φ ₀ φ ₀ (x ₁ )...φ ₀ (x _n ) exp {−

∫

d ⁴ x L}

and

µ d

dµ h T φ ₀ (x ₁ )...φ ₀ (x _n ) i

= µ d dµ

( ( √

Z ) ⁿ h T φ(x ₁ )...φ(x _n ) i )

= 0

or in terms of the generating function of 1PI(single particle irreducible) vertices

µ d

dµ Γ _n(0) (x ₁ , ...., x _n )

= µ d dµ

( ( √

Z ) ^{− n} Γ _n (x ₁ , ...., x _n ) )

= 0

In terms of the Fourier transformed function Γ _n (p _i ; m(µ), λ(µ), µ(µ), µ) ˆ

=

( √ Z (µ)

√ Z (µe ^t ) ) _n

Γ _n (p _i ; m(µe ^t ), λ(µe ^t ), µ(µe ˆ ^t ), µe ^t ) or

Γ _n (p _i e ^t ; m(µ), λ(µ), µ(µ), µ) ˆ

=

( √ Z (µ)

√ Z (µe ^t ) ) _n

e ^2nt Γ _n (p _i ; m(µe ^t )e ^{− t} , λ(µe ^t ), µ(µe ˆ ^t )e ^{− t} , µ)

= exp[2nt − n

∫ _t

γ _φ (t ⁰ )dt ⁰ ]Γ _n (p _i ; m(µe ^t )e ^{− t} , λ(µe ^t ), µ(µe ˆ ^t )e ^{− t} , µ)

where we defined

√ Z (µ)

√ Z (µe ^t ) = exp[ −

∫ _t

0

γ _φ (t ⁰ )dt ⁰ ]

Note that the renormalization group running of m(µe ^t )and ˆ

µ(µe ^t ) is different m ² (µe ^t ) =

(

Z (µe ^t ) Z (µ)

) ( Z _m (µ) Z _m (µe ^t )

)

m ² (µ), ˆ

µ ² (µe ^t ) = (

Z (µe ^t ) Z (µ)

) ˆ

µ ² (µ).

This does not quite realize the renormalized (effective) mass with one-loop correction such as

m ² + λ

32π ² [m ² − m ² ln µ ²

m ² ] + λ

32π ² µ ² where ^λ

32π

µ ² replaces ˆ µ ² (µe ^t ). If this should be realized, we may choose the renormalization point µ not far away from the cut-off in space-time, and then the induced mass is not far away from cut-off. This renormalization may not be ”unnatural”.

We then lower the mass scale, and the physical scalar

significantly modified by the threshold effect m ² (m ² _f )[1 − λ

32π ² ln(

m ² _f

m ² )] + λ

32π ² m ² _f

where m _f is the characteristic fermion mass such as the top quark mass in the case of the Standard Model. Note that m ² (m ² _f )[1 − _32π ^λ

ln( ^m

2 f

m

)] is a (small) mass parameter in the present understanding.

The ”natural prediction” of such a scheme may be that the Higgs mass is of the order of the top quark mass.

We however need to avoid the Landau singularity in this

Conclusion

The consideration of the Higgs mass provides the food of thought in the analysis of the Standard Model. However, we do not have much time to enjoy this opportunity since the experimental results from LHC are already coming around the corner.

In conclusion, we made speculative remarks on

1. Quadratic divergence and SUSY

2. Quadratic divergence and dimensional regularization 3. Subtractive renormalization and naturalness

In this last comment 3, we examined if we can fix the

quadratic divergent mass term somewhere inbetween the

precise elimination (such as in dimensional regularization)

and the ”natural” value λΛ ² (as in the treatment of the

induced mass without any subtraction). This analysis is

incomplete and needs to be further refined.