On a Singular Solution in Higgs Field (IV) -Energy Flow toward a Femt-Scale Structure

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On a Singular Solution in Higgs Field (IV) -Energy Flow toward a Femt-Scale Structure

Kazuyoshi Kitazawa

To cite this version:

Kazuyoshi Kitazawa. On a Singular Solution in Higgs Field (IV) -Energy Flow toward a Femt-Scale

Structure. The 61st National Congress of Theoretical and Applied Mechanics, Mar 2012, Tokyo,

Japan. pp.4, �10.11345/japannctam.61.0.225.0�. �hal-01399634�

1

On a Singular Solution in Higgs Field (IV) - Energy Flow toward a Femt-Scale Structure.

Kazuyoshi KITAZAWA, Mitsui Chemicals

FAX: 03-6253-4244, E-mail: [email protected]

The mass of Standard Model (SM) Higgs boson (H⁰) is re-examined under fluid mechanical consideration of micro (femt-scale) Reynolds number in Higgs boson sea for the process of Higgs mechanism. In this analysis, two gauge particles (W and Z bosons) are adopted as representatives to describe the process through their each mass acquisition. The mass value of fluid mechanical H⁰ (f.m.-H⁰) is calculated relativistically at 128.6 GeV/c², which is a little (6.6 per-cent) larger than the mass of the asymptotic solution (theoretical mass) of Higgs field. This difference of mass value shows that there would be some extent of excess in sectional area’s evaluation for f.m.-H⁰. Because, in this numerical calculation we assumed that f.m.-H⁰ in Higgs boson sea is sphere. While our theoretical mass of H⁰ had a shape of truncated-Octahedron (tr-O) which inscribes to the sectional circle of f.m.-H⁰. So we are able to reduce this excess of mass since drag force against Higgs boson, whose projection area differs between H⁰ and f.m.-H⁰, corresponds to acquired mass by Higgs mechanism.

１．Introduction

The value of Higgs boson mass, has long been sought by both theoretically and experimentally until now.^1),2) In preceding paper ³⁾ the author discussed the mass and the basic structure of SM Higgs boson with reference to bound top quark-pair by obtaining asymptotic solution for their equation of motion of nonlinear partial differential equation of Klein-Gordon type.

In this paper, we regard Higgs mechanism as a fluid dynamic- critical phenomenon in which the vacuum (the sea of Higgs boson condensate) is disturbed by gauge particle and then gives a mass to it. In fluid mechanics ⁴⁾, non-dimensional Reynolds number (Re) can describe such a critical point, on which a disturbance begins. Therefore we shall start to express an analogue of Re which assumed to be an equal number at critical point for all gauge particle.

２．Analysis and the Result 2.1 Formulation

We treat from now on a flow around the gauge particle which has flown in the sea of Higgs boson condensate shown in Fig.1.

Then we assume we could define an analogue of micro Re at a critical (that is, a common-disturbed) point at which the flow of gauge particle (energy) begins a little separation (breaking full contact) with H⁰, considering special relativistic effect ⁵⁾ as

(i) (i) u (i)

( .)

H(i) H(i) H(i)

d u

const.

(m n )

crit



 

^＊



＊

Re

＊ ⁽¹⁾

where

H(i)

(i) th

(i) th ( .)

2

(i) (i)

u

d :'expansive size (diameter)' of i gauge particle u : critical velocity of i (massive) gauge particle at Re

: 1 1 (u c) , c: velocity of light : viscosity of the sea of Hig

crit





 

＊ ＊

＊

＊

H(i)

H(i)

th

gs boson condensate m : Higgs boson mass, produces i gauge particle mass n : number density of Higgs boson in condensate

Since we may approximate that



_H(i)

n

_H(i) 

const

for any i th gauge particle, in eq.(1);

H(i) (i) (i) u (i)

m m u

^＊



＊ 

const,

(2)

where

0 2

(i) (i) (i)

m



collision number toward H with u

^＊ 

d .

Conservation equation of momentum between Higgs boson and gauge particle, with initial velocity of zero for Higgs boson, is

H(i) H(i) uH(i ) (i) (i) (i) u (i)

m u 



E c



m u

^＊



＊ (3)

where

th

2 H(i)

th H(i)

(i )

(i)

uH

light velocity of Higgs boson after collision with i gauge particle

enery of i gauge particle before collision with Higgs boson

u : f

: 1 1 (u c) E :

  

We will here express an equation of energy balance such as

H

2 2

(i) (i) (i) (i) (i) (i ) H(i) H(i) H(i)

'

u u

1 1

E E m ( m E )

2 2

u u

  ^

  ^＊  

where (4)

(i) th

H(i)

E ' enery of gauge particle after collision E nergy increase in Higgs boson after collision

: i



: e

Next, we could have an equation which governs the process of collision and thus mass producing,

Fig. 1 Flow of gauge particle in Higgs sea.

2

(i) (i) (i)

H(i) 4 3

u

m E

=const.

m



＊

(5) This relation is derived to consider that energy density of

‘gauge particle space participates in collision’ (



_(i)^{) is}

(i)

E

(i)

V =const.

(i)



 (5a)

where

(i)

2

th H (i )

H(i)

2 2

(i ) H (i) (i)

2

H(

(i) (i)

(i) (i)

u

V : collision volume of i energy space 4d c (5b) d : 'expansive size (diameter)' of Higgs boson

: contact time of collision d ( d ) (5c)

d m , d

 

 







＊

) 2

i) H(i

2

m 3 (5d)



Therefore, we afterward get eq.(5) by substituting eqs.(5b), (5c) and (5d) into eq.(5a). Here, _(i)



u＊ in eq.(5) is a relativistic effect, comes from the time



_{(i )}of contact (that is, to cover the surface of Higgs boson with massless (the source of) gauge particle which has at first some thickness₀, and then, to release

‘mass-acquired’ gauge particle) in eq.(5c). The process by which elements of mass are to be produced is described as Fig.2.

At last, we put one condition: it must always be satisfied that

H0 H(i)

m



m



const.

(6)

2.2 Result of Numerical Analysis

We shall here after adopt W and Z bosons ⁶ as two gauge particles for numerical analysis to seek SM Higgs boson mass.

Eqs.(2),(3),(5) and (6) above are to be simultaneously solved.

After calculation, we could surely get only one well-converged ⁷ numerical solution:

H0

m  128.6 GeV/c

2 (7)

which is a little (6.6 per-cent) larger than the mass of the asymptotic solution ³ (theoretical mass) of Higgs field (120.611 GeV/c²), which is consistent with LEP’s latest preferred value ² and also has not been excluded by latest results of LHC for SM Higgs boson mass ⁸.

The difference of mass value above suggests that there would be some extent of excess in sectional area’s evaluation for f.m.-H⁰. Because, in the numerical calculation we assumed that the figure of f.m.-H⁰ in Higgs boson sea is spherical. On the other hand, we have seen that our theoretical mass of H⁰ would have a shape of truncated-Octahedron ³ (tr-O) which inscribes to

the sectional circle of f.m.-H⁰ as shown in Fig.3. In addition, tr-O does not circumscribe to the sectional circle. So we see this excess of mass is to be almost cancelled since drag force against Higgs boson, whose projection area differs between H⁰ and f.m.-H⁰ of same diameter, corresponds to the acquired mass regarding gauge particle (W or Z) by Higgs mechanism.

３．Appendix – A Condition of Mass Growing

We here compute a condition of mass growing from the energy, considering the relativistic energy conservation equation:

E

²

  m c ^

²



²

 (pc)

²

, where m ^   m

_{ }m 0 (8)

Then, ^{2 2}

2

m 1 E (pc)

c

   

(9)

So choosing plus sign and to take

dm dt

 

0

, we have

dE

2

dp

E c p 0

dt dt

  (10)

When

E



const.

we put

dE dt



0

and as

p  0

, we have a condition:

with p= m’u,

dp

or

du dm

0, 0.

dt dt dt

1 u

1

m









⁽¹¹⁾

Thus when at last mass growing stops we may surely write,

0 and

u

0

m   m = u ,

(12)

where

m :

₀ aquired mass o fgauge particle ,

u

₀

: its

velocity.

References

1) T. Schücker: Higgs-mass predictions, ArXiv:0708.3344v8 [hep-ph] 14 Dec, 2011.

2) http://lepewwg.web.cern.ch/lepewwg/, 2011

3) K.-Y. Kitazawa: On a Singular Solution in Higgs Field, Theoretical and Applied Mechanics Japan, 57, pp.217-225, 2009; ditto (II), ibid, 58, pp.61-70, 2010.

4) L. D. Landau, Fluid Mechanics, Butterworth-Heinemann, 1987; A. Sommerfeld: Mechanics of Deformable Body, Academic Press, 1964.

5) W. Pauli: Theory of Relativity, Dover, 1981.

6) K. Nakamura et al: (Particle Data Group), J. Phys. G, 37, 075021, pp.1-1422, 2010.

7) ²

H _( ) H _( 1)

m

_n

 m

_n

 0.006 GeV/c

8) https://twiki.cern.ch/twiki/bin/view/AtlasPublic/HiggsPubli cResults; also,

https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsRes ultsHIG

Fig. 3 tr-O inscribed to sectional circle of f.m.-H⁰.

Fig. 2 Mass conversion process around Higgs boson.