Kazuyoshi Kitazawa

On a Singular Solution in Higgs Field

A formula for mass of Standard Model Higgs boson is derived by considering certain asymptotic behavior for singular solution of equation of motion (EOM) of Higgs field via Euler-Lagrange equation, in which M H 0 is shown as a rest mass of Higgs boson mass of the field, which maintains Lorentz invariance. Where the asymptotic formula extracts a proper information near the singular solution (vacuum expectation value (vev)) from EOM. By modifying the mass formula to 'mass triangle' with H 0 production scheme of W/Z-fusion process and by obtaining mass representation at a stationary point, the value of M H 0 is determined at 120.611 GeV/c 2 , which is not excluded by latest experimentally preferred mass, and is consistent with simulation result for vector boson fusion.

INTRODUCTION

The value of Higgs boson mass has long been sought by both theoretically and experimentally until now. At this time, the values of 100 < M H < 130 GeV/c 2 with radiative correction (theoretically [START_REF] Okada | Upper bound of the lightest Higgs boson mass in the minimal supersymmetric standard model[END_REF] for MSSM Higgs boson) and 114 < M H 0 < 123 GeV/c 2 (experimentally 2) (68%CL), for SM Higgs boson), are known. However any theoretically exact formula or definite value for SM Higgs boson mass has not been shown yet, that is mainly from difficulty of obtaining the value of quartic (self-coupling) constant  .

Therefore, here we try to give a formula of SM Higgs boson mass by studying, at first, asymptotic behavior of singular solution for equation of motion (EOM) near vev, which is derived from Lagrangian density of Higgs scalar field (  ). This EOM is, mathematically, one of nonlinear Klein-Gordon equation (NLKG). [START_REF] Hörmander | Lectures on nonlinear hyperbolic differential equations[END_REF] Since EOM should have a unique singular solution ( 0)   of the field at vev where the Higgs scalar field has been extended, we study its behavior near the solution by considering certain asymptotic formula for it. Then we will extract an information without  from EOM as the asymptotic behavior, introducing an infinitesimal Grassmann number. And we express a formula of Higgs boson mass of the field () m  , keeping Lorentz invariance, in which M H 0 is shown as a rest mass. The formula is modified to 'mass triangle' with the relation of H 0 production scheme of W/Z-fusion processes which have been already described by Feynman diagram, to formulate a mass equation by certain parameter. Then by differentiating the mass formula regarding the parameter to obtain the mass representation at a stationary point, we can now get an expected solution  

22 0 0 H M 2 2 W Z W Z M M M M v    
which shows that the mass value is at 120.611

GeV/c 2 , then

4 0 0.119975 c   
, which is not excluded by Large Electron Positron Collider (LEP)'s latest preferred value and also is consistent with simulation results of A Toroidal LHC Apparatus (ATLAS), etc. for vector boson fusion (VBF). 4), 5) And we compute respective W  -and Z  -gauge boson fields with the value of this singular solution, and describe the potential V with Higgs scalar fields. Finally, canonical quantization and renormalizability of Higgs field are briefly reviewed.

FORMULATION AND THE RESULT 2.1. LAGLANGIAN DENSITY OF GAUGE FIELD AND EOM OF HIGGS FIELD

Since we will later treat the case of VBF in which only weak bosons relate, we start with well known Lagrangian density for gauge field of WY SU(2) U(1)  after spontaneous symmetry breaking and with unitary condition as follows [START_REF] Fujikawa | Theory of Gauge Field[END_REF] , to make a gauge invariant formulation of the theory where the gauge-boson masses arise.

2 2 † † 22 11 ( ) ( ) ( + ) ( ) 2 4 4 a a abc b c L D D W W gf W W B B                                  (1) ) (Z cos sin ) (Z cos sin ) ( 2 ) W Z W W W W g m v W W M W W G ig M A A W W ig W W A A W WW                                                                                      (Z cos sin ) (Z cos sin ) W W W W A W A Hermite conjugates                 22 2 2 ( ) (Z cos sin ) (Z cos sin ) , 42 W W W W gg W W W W W A W A                           (2)
where

3 12 1 ( ), , Z , 22 2 WZ gW g B gG W W iW M v M v G             3 2 2 2 2 , sin , 2 , W g W gB gg A e g m v G g g GG              (3)
abc f SU 2 2 : cf. eq.(1), : self-coupling constant of v      , : Weinberg angle W  (3) '
As it is hard to directly solve m  from eq.(2) itself, let us apply Euler-Lagrange equation onto L of eq.(2) regarding  , and then try to solve it; i.e.,  

0, LL x x                 (4)
Thus, after calculation, we get an EOM of NLKG for Higgs scalar field ( );

22 3 2 2 2 2 2 2 2 11 3 (Z ) (Z ) 0 2 4 2 WZ g v m W W G gM W W GM t                                                (5)

DERIVATION OF HIGGS MASS FORMULA

Though EOM (eq.( 5)) above is consistent with the type of PDE form: Klein-Gordon equation, which describes equation of motion of boson particle, still now there is no mathematically systematic method to calculate an exact solution for the NLKG yet. [START_REF] Hörmander | Lectures on nonlinear hyperbolic differential equations[END_REF] Therefore we shall from now on study an asymptotic behavior near its certain singular solution. While we will later see that this approach is sufficient within our purpose. Since EOM should have a singular solution of 0   as explained above, let us take an asymptotic form of the solution near vev point to be connected smoothly to it as follows. (Where though we will find another singular solution that 3v

  by factorization of eq.( 5), we shall abandon it because of its inconsistency with Lorentz invariance of Lagrangian (eq.( 2)).) , : cf. eq,( 3)' v , 0 , : constants as So we may expect that (0) 0, (0) 0   , and having a finite value at infinitely far point. (7) Thus eq.( 6) asymptotically satisfies eq.( 5) at world origin. Then by expanding near

    3 0 0 ( ) 1
0 s  ,     32 01 0 0 0 ( ) 1 1 s s s s avs s s a vs         2 2 2 1 1 0 0 0 , where , , 0< 1. v a s a as s       (8)
Hence we can take an asymptotic form near singular solution ( 0

  ) as 2 , ( 0) v     (9)
By inserting eq.( 9) into eq.( 5) and using Higgs mass definition, etc. of eq.( 3), we are able to have a Higgs mass formula without  as

          2 22 2 2 2 0 2 (Z ) , where 2 2 m W W g G                       (10)
Here we understood that eq.( 10) expresses one of elliptic curves with coordinates of

 

, Z WW    . It is noteworthy that   still has a finite and larger value than 2  at very near vev point ( 2 0  ), because the power of   is always a half power of 2  (that is, equals to 1), describing a micro elliptic mass curve even by very near this point. Further, if we introduce an infinitesimal Grassmann number (   ) for  by putting     , then eqs.( 9),(10) are elegantly represented as

( ) 0,    (9')         2 22 2 2 2 2 2 (Z ) 2 , from nilpotent property that 0. m W W g G                  (10')
After all, we now could have an asymptotic behavior of eq.(10') for eq.(9'), as shown in Fig. B1. Since dimensions of two terms in right-side of eq.(10') are to be both square of mass, we can put as

    2 2 2 0 2 , WW c W W g m      (11)   2 22 0 (Z ) 2
, where , : constant

Z Z W Z c G m c c    (
        2 22 2 0 0 2 2 (Z ) 2 m W W g G               2 2 2 2 2 2 2 2 2 1 2
, where we put as

W Z W Z W Z WZ m m k m m c c c c k          , (13)
where  : constant. Furthermore we can write as 7)

    22 11 , ; , , 11 W W W W Z Z Z Z WZ m M m M u c u c          (
      22 2 2 W W Z Z m k M M             2 2 2 2 2 , W W Z Z Z W W Z H H k M M M              (16)
To maintain Lorentz invariance of m  in eq.( 16), it should be that;

.

WZ     (17)
Then we can write as,

    00 2 2 2 2 2 2 2 2 HH 2 M , where M 2 W Z W Z m k M M k M M             (18), (19)
Since the value of M H 0 above is supposed to be in the range of 'intermediate mass' from the results of LEP, let us consider H 0 production scheme of W/Z-fusion processes which is most expected to meet with above mass formula. 8),9) As these processes are described by Feynman diagram as shown in Fig. 1, we here study the case of that w n -W fusions and z n -Z fusions are simultaneously occurred.

Then wz (N n n ) H 0 's may be produced after these graphs. On the other hand, from eq.( 19);

      0 2 2 2 H 1 M 2 WZ k M k M      (20)
Where eq.(20) forms 'mass triangle'. We will see that two Feynman diagrams (Fig. 1) give each factor for three sides of mass triangle (Fig. 2) by recognizing that we can apply eq.( 20) onto two fusion diagrams, as explained in APPENDIX -A. So, by comparing Figs. Also, for H 0 particle, from above discussion and mass triangle it must be kept that

k k k                 
                                (25)
As shown in Fig. 3, where the value of M H 0 ()  runs from

  20 W M   to   2 Z
M   via the value of stationary point. Here  is physically interpreted as a parameter related to effective rate of W and Z boson masses into Higgs boson mass in VBF process. Therefore, we will find  -value at stationary point to differentiate eq.( 25) regarding  , and to put zero; 3),( 18) and (28).

Respective W  -and Z  -gauge boson fields are computed at a point of the singular solution 0  , and the potential V is described with scalar Higgs  and  fields, in APPENDICES-B and -C. Finally, canonical quantization and renormalizability of Higgs field are shortly touched on in APPENDIX-D.

CONCLUDING REMARKS

So far, we have derived a formula and shown a value of mass of SM Higgs boson via an asymptotic behavior of a singular solution for the Euler-Lagrange equation, which extracts a proper information without  from EOM. The result is to be strongly expected to examine under the forthcoming experiments. And, SM Higgs Mass form (eq.( 27); symmetrical between W and Z with a factor of 2 (twice)) appears to show the possibility that the Higgs particle is to be composite, as supposed, which is now proceeded to study, and will be discussed elsewhere.

APPENDICES -A: Relation between Mass triangle and VBF triangles

Since we have seen that W, Z and H 0 particles should have both equal  to maintain Lorentz invariance as eq.( 17), we also should consider the case in which they still have both equal  in VBF process. Therefore we shall hereafter discuss with their rest masses, dropping out  's from their relativistic masses, in VBF triangles below, etc. with remembering eqs.( 14) and (15).

Because numbers of consuming W and Z particles are 2n WW M and 2n ZZ M in Feynman diagram (Fig. 1) at each event time and all related 's are both Fig. A1 From Fig. 2, it will be understood that each mass quantity of W and Z boson sides, to which cos and sin are multiplied respectively, contributes to Higgs mass. Therefore we can generally write VBF process for producing one Higgs mass as, with referring to Fig. A1(a), 0

H M (2 ) cos (2 )sin ,

W W Z Z MM       (1) (1) (1) (1)
Z W W kM kM               (A2)
as 'VBF angle', then we will get always an equal value: 120.611 GeV/c 2 for M H 0 regarding all- 

(1) (1)
, WZ  values from eq.(A1).

-B: Calculation of W  -and Z  -boson fields

We now understand that W  -and Z  -boson field each takes such an asymptotic coordinate value at this singular solution for  as follows. From eqs.( 11)-(15), 0 () , 2

W W W g W W mM k         (B1) 0 (Z ) 2 Z Z Z G mM k       (B2) Therefore,       22 12 0 0 1 () 2 W W W W       2 , WW W M k k v g                12 00 Hence, W W W k v       (B3), (B4) 3 0 22 0 2 Similarly, (Z ) , ZZ Z gW g B M k k v G gg                         (B5) where 12 1 2 2 1 2 kk                      (B6) Finally W  ,
Z  above should be equal (=  ) so as eq.( 17) at this singular solution for  . Hence the mass formula of eq.(10') is described as Fig. B1, in which W and Z bosons should have energies of kv    and kv    respectively at the condition of eq.( 13).

-C: Description of potential V with scalar Higgs ( 1  and  ) fields

We here describe the potential V with scalar Higgs fields in which the position of singular solution

2 ( 0) v   
is shown. We consider an isospinor scalar field (the Higgs field)  as 11)

    34 0 12 1 2 , 1 2 i i                   (C1) Then †  in potential V of eq.(1) is † * 0 * 0 ( ) ( ) ( ) ( )         2 2 2 2 1 2 3 4 1 () 2         (C2)
               ( )) x  
, and by describing the Hamiltonian (H) with the Lagrangian density (L). After setting the same-time canonical commutation relation between () x  and () x  , we are able to calculate the Heisenberg's equation of motion Renormalizability (and unitary property) of massive vector field under gauge symmetry   SU(2) U( 1)  has been firstly confirmed by t' Hooft. 13) Later, Fujikawa et al. 14) developed R  -gauge theory. The renormalizability of Higgs field under BRS-symmetry 15) was shown, using the R  -gauge, also by Fujikawa. 16)

      00 ( ), ( ), ( ) , ( ) x H

  12) at certain point ( designated by 0 ) in respective gauge field. Mathematically, eq.(10') is one of the elliptic curves with   So eqs.(11) and (12) are understood as they fix a coordinate of certain point on the micro elliptic mass curve in the first quadrant. Thus we rewrite eq.(10');

  2 and Fig.A1(b) of APPENDIX -A,

(

  Fig.1H 0 production scheme of W/Z fusion process.

  angle. Hence we now have a form of stationary value as an expected Higgs mass solution from eqs.(25),(26) and then we get the value of rest mass for SM Higgs boson by inserting experimental values 2) , of excluded by LEP's latest preferred value for SM Higgs boson mass 2) , and is in accordance with the simulation results for VBF of ATLAS and CMS which appear to show around 120 GeV/c 2 the best significance both in Higgs boson decays into -pairs and into  -pairs respectively, in the low-mass region:110 < M H 0 < 140 GeV/c 2 .4),5)Then, from eq.(27) and the value of G F in v

Fig. 3

 3 Fig.3 Higgs mass

  Fig.A1(b), the Mass triangle (Fig.2) and VBF triangle should be equivalent since these right-angled triangles both should have an equal angle    to make same rate of contribution to Higgs mass, and have an equal side 0 H (M ) . Thus the remainder sides should also be equal. In fact, once we later see that

Fig.C1 Potential V with 1 

 1 Fig.C1 PotentialV with 1  and  fields.

Fig. B1

 B1 Fig.B1 Micro elliptic mass curve in

  VBF triangles. equal, we can write next formulae after N n n

				
				WZ
			events;
	-consuming mass: 2n W	; 2n MM , W Z Z	-producing mass:	0 WZ . 0 0 H H H n M n M N M

  W  -and Z  -field. -sectional radius of potential V at each point of 1  . Thus we see that the point of e.g.

	energy state,
	1  	, v	† as ( ) 	Lowest Energy State	22 22 v   	(C3)
	Therefore we can write for () x  , under local symmetry, at every point as
		() x		1 2	0 vx ()     	(C4)
	After all, we may describe potential	V with 1  and  fields as Fig.C1, in which
	44 2 ( 3 ) , 9 44 r v v vv VV  0( ) r v         	(C5)
	where r is crossv   has twice of radius for the Lowest Energy Point (   . 0)
	-D: Canonical quantization and renormalizability of Higgs field
	Canonical quantization of Higgs scalar field ( ( )) x 	which derives the NLKG above can be furnished by
	computing the canonically conjugate momentum	( ( ) x	( , ) jj ( , ) t x L t x	L	(	00 ( )) x	( ) x

Z 0 approximation for high energy collision", Phys. Lett., Vol.B148, (1984)


Kazuyoshi Kitazawa

On a Singular Solution in Higgs Field (Ⅱ) -The Structure of SM Higgs Boson

The structure of SM Higgs boson is discussed by the set of top quark decay processes in electroweak and quark sectors with newly enlarged equation of motion (Non-Linear Klein-Gordon eq.). Their asymptotic behaviors are mathematically described on an infinitesimal hyperboloid of one sheet in multi dimensional space. Then mass value of top quark is calculated as 171.26(6) GeV/c 2 which is consistent with CDF/D0's result by minimizing the theoretical top quark mass. And from the difference between the value by assuming that SM Higgs boson is a virtual bound state of top quark-pair (( tt )*) itself with the mass formula obtained by requirement of minimal mass production as 171.26(6) /2 = 121.10(3) GeV/c 2 , and the recently obtained theoretical mass value of SM Higgs boson (120.611 GeV/c 2 ), it is expected that SM Higgs boson is to be a composite scalar meson after emitting one photon from the ( tt )* which will be a vector meson through radiative decay. Finally, a structure of SM Higgs boson mass which is composed of all spin 0 mesons' masses, is proposed. Where the truncated-Octahedron mass structure is recursively (doubly) seen.

INTRODUCTION

The mass of top quark and the structure of SM Higgs boson are still open problems which have not been fully clarified theoretically yet, although they must be related each other from the viewpoint of that they have (or seem to have) both huge masses. So until now several efforts [START_REF] Nambu | BCS Mechanism, Quasi-Supersymmetry and Fermion Mass[END_REF] have been done. In preceding paper 2)

, the author developed the Mass triangle method by which the mass formula of SM Higgs boson and its mass value were calculated, and also it was expected that SM Higgs particle would be a composite boson. Therefore we shall hereafter try to solve the next problems above by applying the technique of our method.

FORMULATION AND THE RESULT

2.1 LAGLANGIAN DENSITY OF GAUGE FIELD AND EOM OF HIGGS FIELD Since we later treat top quark decay processes in electroweak and quark sectors to which weak boson and quark are produced, we start with the Lagrangian 3) of (2) (1)

SU

U  invariant gauge field after spontaneous symmetry breaking and with unitary condition to make a gauge invariant formulation of the theory: -

  3 3 2 L L L i ig W S W S W S ig B Y                     L   2 2 + . . , R R R L D R L U R Higgs i ig B Y M D M U h c v                   L (1)
where,   0 1 2 3 5 ,5 03 : Dirac's γ matrix 1 1 ; ; , 2

LR i U D                 4) 2 3 0 1 0 0 1 0 1 1 1 , , , , ; 0 0 1 0 0 1 2 22 ud U c D s S S S tb                                              1 6 0 2 3 0 11 , 0 1 6 0 1 3 22 LR YY              
; where 0 3 3 zero matrix, 1 3 3 unit matrix , , ; , , : quark field

u c t d s b     2 2 † † 22 11 ( ) ( ) ( + ) ( ) , 2 4 4 a a ab Higgs c b c D D W W gf W W B B                                  L (2)
where, 2; =0 3.

aa D igW T ig B Y           ~

 

, : gauge fields which belong to (2), ( 1) respectively

     2 1 2 2 2
2), ( 1) respectively

2 , : self-coupling constant of 1 ( ), 2 2 abc U Y f SU v W W iW m v               22 ,

G g g 

Then mass term in eq.( 1) of 3rd term is written as

      1 . . 1 . . 1 . . , Mass L D R L U R L D R L U R d d b u c t D M D U M U h c D D U U h c vv m dd m ss m bb m uu m cc m tt h c v                                               L (3) where , . 0 0 00 du D s U c bt mm mm mm                        
Then after we apply Euler-Lagrange equation onto eq.( 1):

() 0 d dx x              LL , (4)
we get the EOM of the field as a nonlinear Klein-Gordon equation regarding  :

        2 2 3 2 2 2 2 2 2 2 11 3 24 1 1 1 0. 2 i i i W Z d u s i i i c t b v m g W W G Z t gM W W GM Z m bb m cc m tt m dd m uu m ss vv                                           (5) 2.2 DERIVATION OF TOP QUARK MASS FORMULA 2.2.

HIGGS MASS FORMULA IN ELECTROWEAK AND QUARK SECTORS

According to I 2) , we also take an asymptotic solution via its Lorentz invariant form near singular solution of the field at 'vacuum expectation value' (vev: 0   ) where Higgs scalar field has been extended, to extract an information about SM Higgs boson mass from eq.(5) (NLKG) 5) so that, 2 , ( 0) v     .

(6) By inserting eq.( 6) into eq.( 5) and using Higgs mass definition, etc. of eq.( 2), we are able to have a Higgs mass formula in these sectors:  because of its nil-potent property. And, , .

                  2 2 2 2 22 22 2 2 2 () () 2 2 2 2 2 + (Z ) 2 2 2 2 2 2 2 2 2 , i bsd bsd F cu cu F F t i bsd i cu q q W W g q q G G q G m tt G m q m m                       (
bsd bsd cu cu q bb ss dd cc qq uu q    Therefore eq.( 6) is elegantly rewritten as ( ) 0.

  

. ( 8) After all, we now could have an asymptotic behavior of eq.( 7) for eq.( 8).

TOP QUARK MASS FORMULA

It will be understood that first and second terms in right-hand side of eq.( 7) are both related to decay of top quark to weak boson and to other quark. While third term corresponds to decay to another sector of top quark. Then we shall hereafter focus on former decay in which its behavior is described on an infinitesimal hyperboloid of one sheet in multi dimensional space with  

, , , ; , , bb ss dd W W Z Z c u c u       coordi-
nates as shown in Fig. A1 of APPENDIX-A. Since dimensions of two terms in right-side of eq.( 7) are both square of mass, we can put as

                            2 2 2 2 22 00 22 2 2 22 0 0 2 2 2 2 22 0 0 2 2 2 0 2 , 2 2 2 , 2 2 2 , 2 2 2 , 2 , 2 2 2 , 2 2 2 W W b F b s F s d F d Z Z c F c u F u i i ii i u b s d c c W W g m c G m c G m c G m bb m ss m dd m m c Z Z g cc m m c G m cm uu G                              (9)
at certain point (designated by 0) in respective field. So eq.( 9) are understood as it fixes a coordinate of certain point respectively on the hyperboloid curve in the first quadrant in    

, bb WW

11 0 0 0 0 ( ) 0 0 ( ) 11 0 0 0 ( ) 0 ( ) (1 ) (1 ) (1 )
(1 ) ( 1) ,

k n k k n k k n k n k t n k W bsd n k Z cu k n k n k W bsd Z cu C r r m C r r m C r r m C r r r m r m                  ( ) 0 ( ) 0 / / (1 ), t W bsd Z cu m m r m r     ( 10
)
where we assumed that the probability of t m -decay process obeys binominal distribution of being k-times in n-trials (-particles), producing masses of particles in virtual states with () W bsd m and ()

Z cu m :         2 2 2 2 2 2 2 2 2 2 ( ) 1 22 2 2 2 2 2 2 ( ) 2 2 2 2 2 2 2 , 22 2 W b s d W bsd bsd W bsd b s d W bsd cu Z Z cu cu Z cu c u Z cu m m m m m c k m m m m cc mm m m c k m m m cc                 (11)
from eqs.( 7),( 9); and we let,

    2 2 2 2 1 1 2 2 1 , 1 W bsd bsd bsd Z cu cu cu c c c c k c c c c k       . ( 12
)
It is important that we should group each mass of weak boson together with its relevant quarks' masses in closed form, since we here assumed the decaying virtual states such as

    22 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 1 1 ; , 1 1 W bsd W bsd W bsd W bsd W bsd Z cu Z cu Z cu Z cu Z cu m M u c m M u c           ( ) ( ) ( ) ( )
where : velocity of light; , : velocity of produced-mass , : rest produced-mass from top quar k.

W bsd Z cu W bsd Z cu c u u MM (13)
By inserting eqs.( 11),( 12) and ( 13) to eq.( 10), and to maintain Lorentz invariance, we are able to write as

      ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 00 00 (1 ) (1 )
, where and ,

// // t W bsd W bsd Z cu Z cu Z cu W bsd Z cu t t t W bsd Z cu t m M M M M M rr rr                  (14)         2 2 2 2 2 2 2 2 2 2 2 ( ) ( ) 2 2 , 2 . W bsd bsd W bsd b s d Z cu cu Z cu c u M k M M M M M k M M M          (
  1 2 2 , 1 ; 1 2, 1 2. bsd bsd cu cu kk        (16)
Therefore we finally obtain top quark mass value by minimizing the theoretical mass value through differentiation regarding decay-mode parameter 0 r on eq.( 14),

  2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 0 0 2 4 ( )( ) 0, W b s d W b s d Z c u Z c u W b s d t M M M M M M M M M M M M M M M M M d dr M M r                    (17)
and inserting eq.( 17) to eq.( 14) with eqs.( 15) and ( 16) to get stationary mass formula;

      2 2 2 2 2 2 2 2 2 11 44 1 2 171.26(6) GeV/c W b s d Z c u t M M M M M M M M               ( 18
)
which is consistent with CDF/D0's experimental result (171.2±2.1 GeV/c 2 ) 7) . Where we used the value of 4.68 2 GeV/c

  for b-quark mass. 7) It is noteworthy that eq.( 10) is symmetrical between r because two terms in   of eq.( 18) have an equal power form (1/4).

THE STRUCTURE OF SM HIGGS BOSON

Let us look again at eq.( 7) of which the first and second terms in right-hand side both appear also to describe processes of producing top quark with different point of view. So we shall write eq.( 7) as

        2 2 ' 2 2 2 2 2 2 . ii t F t m n t t n tt m G m       (19)
Then if we let;

          22 22 2 2 2 00 1 ' 2 2 2 2 , 2 2 2 2 , () ii Ft t tF t t t t G m m n tt cc G n m mm         ( 20
)
we can rewrite eq.( 19) ,  

M            (23)
and

      2 , 2 1 2 1 . tt k k k             (24)
Then by inserting eq.( 24) to eq.( 22),

  0 2 1 . 1 t H MM        (25)
And by differentiating eq.( 25) regarding  to compute stationary value, we finally get

  0 0 0, 1 2 . H t H dM MM d           ( 26
)
So we will have the mass of a virtual bound top quark-pair (( tt )*) by assuming that the virtual bound top quark-pair is to be SM Higgs boson. 8) Actually, we can see that right-hand side of eq.( 21) such as 2 '(1 1 ) '  in square root describes a binding process of two top quarks, namely, one top quark-pair when 1   . From eqs.( 18) and ( 26 GeV/c

H t tt MM M       ( 27
)
Then there is a mass difference because theoretical SM Higgs mass value 2) will be 120.611

2 GeV/c ; * 0 2 () 0.49(2) GeV/c . H tt M M M        ( 28
)
This value is a little smaller than masses of ,0 K  mesons, and is smaller than mass of 0  meson.

Therefore it is expected that SM Higgs boson is to be a composite scalar meson after emitting one photon from the virtual bound top quark-pair: ( tt )* which will be a vector meson and has zero charge. Namely, *0

(tt) H .

  

(29) As shown in Fig. 2, our theoretical mass value of bound top quark-pair looks to be on the line of experimental mass values of other bound quark-pairs of spin 1 (while π meson's spin is 0). In addition from eq.( 27), the mass of ( tt )* is asymmetry regarding M W and M Z while the mass of H 0 is symmetry. 2) And a short discussion for the physical interpretation of eqs.( 27) ,( 28),( 29) is given in APPENDIX-C.

FOR THE BASIC STRUCTURE OF SM HIGGS BOSON MASS

The spin of SM Higgs boson is, as known, 0 (zero). So let us consider that SM Higgs boson mass is constructed basically by heavy mesons' masses of all spin 0, such as

    00 . S S C C S S B B B B DD     ( 30
)
Then it is expected that they will form a polyhedron which is composed of planes of hexagon, in space. And, 'effective' (non-overlapped) number of planes of hexagon should be

                                    (31)
We will see that eq.( 31) is satisfied, in all polyhedron, only by truncated-Octahedron 9), 10) (tr-O) shown in Fig. 3, on which two hexagons and one square gather at each twenty-four vertexes.

And we think that the residue of mass from eq.( 31

The developed outer shell ('basic structure') is shown in Fig. 4. Moreover we will see that the central mesons are both surrounded by cloud of π mesons 11) and by shell of heavy mesons 12) , and similar mass structure (tr-O) is seen again in the b  and the threec  's themselves. Because their masses are described with same manner of eqs.( 32),( 33),(34), in accordance between experimental 7) and our theoretical values as,

        2 0 0 0 0 0 (theor.) 0 15 2 MeV/c 2 , 9300.3 , b i M KK KK M                                           (35) 2 (exper.
c i M KK KK M                                             (37)   2 3 (exper.)
where, MeV/c 3 2980.3 1.2 8940.9 3.6

c M       (38)
Here each central meson is 0 2 or 0 3 . Therefore, we understand that antiparticle SM Higgs boson is also SM Higgs boson itself. Developed inner shell (inner -'basic structure') is shown in Fig. 5. And in Fig. 6 we describe a conceptual model for spin and parity of SM Higgs boson from above discussion.

CONCLUDING REMARKS

So far, we have derived a formula and shown the mass value of top quark and of bound top quark-pair, then explained relevant structure of SM Higgs particle that it is to be a composite scalar meson whose mass composes a recursive truncated-octahedron structure of all spin 0, pseudo-scalar meson (parity: -1) 's masses in two phases. The parity of SM Higgs boson (+1) will be come from its recursive (double) structure. The result is to be expected to examine under the forthcoming experiments. And, next problem would be to explain Higgs Mechanism itself from the structure (geometry) above, which is now proceeded to study.

APPENDICES -A: Infinitesimal Hyperboloid in Multi-dimensional Space

We have seen that top quark to weak boson and to other quark decay was described on an infinitesimal hyperboloid of one sheet in multi dimensional space with ( )-coordinates there is a micro elliptic mass curve which has been shown and is equivalent one in ref. I, around origin (0).

-B: Mass Triangles for Top Quark

Since we have seen that top quark and virtual-W(bsd) * , -Z(cu) * particles should have both equal t  to maintain Lorentz invariance as eq.( 14), we shall hereafter discuss with their rest masses, dropping out t  's from their relativistic masses in top quark triangles below, etc. with remembering eq.( 13) . From eq.( 15) we can describe mass triangles for virtual W(bsd) * and Z(cu) *

cu Z Z c u n n NN            (B2)
From Figs.B1(a) and (b), it will be understood that each mass quantity of virtual W(bsd) * and Z(cu) * , to which cos is multiplied respectively, and each mass quantity of other sides to which sin is multiplied respectively, contribute to hypotenuses (resultant masses). Therefore we can generally write these processes,

( ) ( ) ( ) cos sin W bsd b s d b s d W W M M M          , ( ) ( ) ( ) cos sin Z cu c u c u Z Z M M M       (

  15) Then we will get values of coefficients in eq.(15) by comparing the Mass triangles shown in Figs.B1(a) , (b) and Figs.B2(a) , (b) of APPENDIX-B as,

Fig. 1

 1 Fig.1 Top quark Mass Triangles Fig.2 Bound qq -meson mass

  ) will form, inside the tr-O, meson masses at central position and of surrounding cloud. Actually by comparing SM Higgs boson mass values between experimental 7) and theoretical 2) ones along with this model, we have two phases of SM Higgs boson mass which appear relevant according to the heavy meson ( b  or threec  's) at the central position of tr-O to be able to interact with W ± or Z gauge particle before acquiring own mass as

Fig. 3

 3 Fig.3 truncated-Octahedron (tr-O)

Fig. 5

 5 Fig.5 Developed inner shell of tr-O

Fig.

  Fig.A1 Behavior of SM Higgs and top quark in an infinitesimal hyperboloid of one sheet

  B3), (B4) by which Figs.B2(a) and (b) are described. Then each mass triangles: Figs.B1(a) and (b) should be equivalent to that of Figs.B2(a) and (b) respectively since these right-angled triangles should have an equal angle such that rates of contribution to corresponding hypotenuses (resultant masses) which should have an equal value (side) respectively. Thus the remainder sides should also be equal respectively. So with considering also eqs.(B1) and (discussion for the physical interpretation of eqs.(27),(28),(29) As shown in eqs.(27),(28),(29), the SM Higgs boson mass is to be a little smaller than virtual bound top quarkamount for emitting one photon from   * tt . On the other hand, the dynamical strong coupling theory of top quark condensation[START_REF] Nambu | BCS Mechanism, Quasi-Supersymmetry and Fermion Mass[END_REF] predicts that Higgs boson mass is twice of top quark mass, namelythat the former(eqs.(27),(28),(29)) describes a 'solidification' phenomenon around at crystallization point cryst Tfrom Quark-Gluon Plasma (QGP) state into a more stable state via a bound top quark-pair   tt while the latter would describe one into a molecular-like state of two top quarks at condensing point tt above will be discussed in next paper.

Large Anomalous Dimension and t Quark Condensate", Phys. Lett., Vol.B221, (1989), pp.177-183. Bardeen, W., Hill, C. and Lindner, M., "Minimal dynamical symmetry breaking of the standard model", Phys. Rev., Vol.D41, (1990), pp.1647-1660; Fukano, H. and Yamawaki, K., "Anatomy of Top-Mode Extended Technicolor Model", Progress of Theoretical Physics, Vol.119, (2008), pp.429-460. 2) Kitazawa, K.-Y. , "On a Singular Solution in Higgs Field", Theoretical and Applied Mechanics Japan, Vol.57, (2009), pp.217-225: refer Theory of Gauge Field, (Iwanami, 1993) No.89-08, (1989); Tanabashi, T., "Top quark and Higgs in dynamical symmetry breaking", Nagoya 1991, Proceedings, Dynamical symmetry breaking, (1991), pp.336-385. 9) Conway, J.H. and Sloane, N.J.A., "Sphere packings, Lattices and Groups", (3rd ed ), Grund. Math. Wiss. Vol.290, (Springer, 1999), P.34.; also (for the crystallographic analysis), Wells, A. F. and Sharpe, R. R., "The geometrical basis of crystal chemistry. VII. On three-dimensional polyhedra and networks", Acta Cryst., Vol.16, pp.857-871, (1963). 10) Zhang, Y., and Seeman, N.C., "The construction of a DNA truncated octahedron". J. Am. Chem. Soc., Vol.116, (1994), pp.1661-1669; http://seemanlab4.chem.nyu.edu/ . 11) Szczurek, A., "Mesons and the Nucleon Structure", Acta Phys. Pol., Vol.B27, (1996), pp.3321-3341. 12) Bali, Gunnar S.,"QCD forces and heavy quark bound states", Phys. Rep., Vol.343, (2001), pp.1-136.


Wei Du
email: wei.du@insa-lyon.fr

Jean-Marie Gorce
email: jean-marie.gorce@insa-lyon.fr

Tanguy Risset
email: tanguy.risset@insa-lyon.fr

Matthieu Lauzier
email: matthieu.lauzier@hikob.com

Antoine Fraboulet
email: antoine.fraboulet@hikob.com

Compressive Data Aggregation on Mobile Wireless Sensor Networks for Sensing in Bike Races

This paper presents an efficient approach for collecting data in mobile wireless sensor networks which is specifically designed to gather real-time information of bikers in a bike race. The approach employs the recent HIKOB sensors for tracking the GPS position of each bike and the problem herein addressed is to transmit this information to a collector for visualization or other processing. Our approach exploits the inherent correlation between biker motions and aggregates GPS data at sensors using compressive sensing (CS) techniques. We enforce, instead of the standard signal sparsity, a spatial sparsity prior on biker motion because of the grouping behavior (peloton) in bike races. The spatial sparsity is modeled by a graphical model and the CS-based data aggregation problem is solved using linear programming. Our approach, integrated in a multi-round opportunistic routing protocol, is validated on data generated by a bike race simulator using trajectories of motorbikes obtained from a real race, the Paris-Tours 2013.

I. INTRODUCTION

Modern sensing technologies already enable real-time monitoring of various environments or human activities in applications such as traffic control, health care, smart homing and sports. For example, GPS sensors have been used in recent bike races including the Tour de France for collecting biker locations, and such data can be exploited to enhance live TV broadcasting or to analyze biker performance [START_REF]Dimension Data completes big data analytics and digital delivery platform for Tour de France, Dimension Data[END_REF].

In most applications of wireless sensor networks (WSNs), the efficiency of data collection is critical as sensors come with limited power while the radio communication is energy consuming. The context of bike racing adds additional constraints due to the high mobility of the bikers. Obviously, it is not possible to ensure a direct connection between each sensor and a common sink due e.g. to body shadowing and biker spreading. Therefore, routing data between sensors is necessary to ensure reliable data collection. However, data routing is complicated because the routes are time-varying and cannot be pre-defined. Additionally, the application requires a high refresh rate on the location measurement, typically about one reading per second in a race. HIKOB [START_REF] Hikob | [END_REF] provides GPS sensors that meet these requirements and designed an infrastructure for data collection in bike races, illustrated in Figure 1, in which GPS data are relayed from sensors on bikes, via intermediate sinks on motorbikes surrounding the race, to the central sink on a data truck for further processing. The This work is funded by the French government under the grant FUI-SMACS. particular challenge we address in this paper is how to route data between sensors on bikes in an energy-efficient way.

A naive but direct solution would be a multihop opportunistic routing protocol in which each sensor transmits in several rounds not only its own data but also the data received from its neighbors. Due however to the bandwidth constraint, each sensor can only select and transmit a subset of the received data. The selection of this subset is strategic to maximize the chance of receiving all data at the sink. Since it is impossible to fully coordinate the data selection process at different sensors, it is inevitable to have redundant and missing data, i.e. some data are received for multiple times, while others are not received at all. The problem is more severe with poor radio connections which result in a high packet loss rate.

To improve the efficiency of the data collection, this paper proposes to aggregate data at sensors instead of selecting a few data, based on compressive sensing (CS) techniques [START_REF] Donoho | Compressed sensing[END_REF], [START_REF] Candè | An Introduction To Compressive Sampling[END_REF]. CS is a class of methods that deal with sparse signals and has been widely used for data collection on WSNs [5]- [START_REF] Xie | Transmission-efficient clustering method for wireless sensor networks using compressive sensing[END_REF]. However, the general CS framework in previous work is designed for static WSNs with predefined routing paths and cannot deal with packet losses which occur frequently in mobile WSNs. In contrast, we design a novel CS approach for the application of sensing in bike races. Instead of the standard signal sparsity, we enforce a problem-specific sparsity prior on the motion of the bikers. The assumption is that many bikers follow the constant velocity model and show strong group behavior. Indeed, cycling is a team sport in which bikers group and draft for the aerodynamic benefit and they accelerate only occasionally to break away from a group. Thus, we incorporate the prior information that nearby bikers tend to make synchronized moves and thus keep their relative locations in the group. This spatial sparsity is modeled by a graphical model which captures the spatial relationships between bikers. We formulate and solve the CS problem with the spatial sparsity using linear programming (LP). Furthermore, a simple multi-round routing protocol based on broadcasting and aggregation is designed and integrates seamlessly the CS method. Our approach is evaluated on data generated by a bike race simulator based on real trajectories of motorbikes obtained in the race of Paris-Tours 2013, and is shown to be able to recover accurately all data even in situations when many sensors are poorly connected.

Our technical contributions include the incorporation of a spatial prior on biker motion and the resolution of the CS problem by LP. Our approach falls in the category of modal-based CS [START_REF] Baraniuk | Model-based compressive sensing[END_REF] and CS with structured sparsity [START_REF] Bach | Structured sparsity through convex optimization[END_REF] which enforces additional structures or priors on the sparsity. Our spatial prior is a local constraint that enforces similarities on the motion of nearby bikers which is specific to our problem. In addition, we study the integration of the CS method with our multi-round opportunistic routing protocol. The rest of the paper is organized as follows. Section II introduces the idea of CS and its applications on WSNs. Section III describes our CS method with the spatial prior and the routing protocol. Section IV illustrates some results on simulated data. Conclusions and future work are given in section V.

II. CS FOR WSNS

A. General CS and its applications on WSNs

The general idea of CS is as follows [START_REF] Donoho | Compressed sensing[END_REF], [START_REF] Candè | An Introduction To Compressive Sampling[END_REF]. Let X be a signal of size n, X = [x 1 , . . . , x n ] T , and X is K-sparse in the space φ, i.e. φX contains at most K significant coefficients. φ is of size n × n and degenerates to the identity matrix if X is sparse itself. Let Y be a measurement vector of size k, Y = [y 1 , . . . , y k ] T , k n, and Y is formed by the projection of the K-sparse coefficients φX on the random matrix A of size k × n, i.e. Y = AφX. The entries of A are drawn i.i.d. from either Gaussian or Bernoulli distribution. The CS theory asserts that one can recover X from Y , if k O(Klog(n/K), by solving for the noiseless case

min ||φX|| 1 s.t. Y = AφX, (1)
or for the noisy case

min ||φX|| 1 s.t. ||Y -AφX|| 2 2 . (2)
The above optimization can be solved easily by e.g. lasso [START_REF] Tibshirani | Regression shrinkage and selection via the lasso: a retrospective[END_REF].

For applications of WSNs, CS is promising as it can reduce the number of data transmission from n to k. Previous work on CS for WSNs [START_REF] Luo | Compressive data gathering for large-scale wireless sensor networks[END_REF]- [START_REF] Xie | Transmission-efficient clustering method for wireless sensor networks using compressive sensing[END_REF] employed a similar framework, illustrated in Figure 2, in which the compressive data collection is done on a routing tree, i.e. each sensor aggregates in one packet its own data and all data received from the children and transmits the aggregated data to its parent.

There are two issues with the framework in Figure 2. First, it works only for static WSNs where the routing path for each sensor is fixed and known by the sink. For our application, due to the high mobility of the bikers, we can only perform CS with no predefined routing. Second, it does not handle packet losses due e.g. to poor radio connections which are more

Y x 4 x 1 ψ 1 x 1 x 2 ψ 2 x 2 x 3 ψ 3 x 3 4 i=1 ψ i x i . . . Y = n i=1 ψ i x i = ψX
Fig. 2. CS for data collection on a routing tree of a WSN. Define ψ = Aφ and its ith column ψ i is stored at sensor i, ψ = [ψ 1 , . . . , ψn]. Sensor i transmits a vector of size k which is the combination of its own data ψ i x i and of all data from the offspring, and the sink receives Y = ψX = AφX.

severe for mobile WSNs. Packet losses destroy the sparsity pattern of the data since the lost data can only be treated as 0, illustrated in Figure 3(a). Only when X is directly sparse, i.e. φ is an identity matrix, packet losses do not cause big problems, which is barely true for most types of data including the GPS data in our application.

B. CS with Packet Losses

We first provide a general CS method which handles packet losses naturally. Instead of aggregating φX, we aggregate X directly and randomly without doing the transform φ while assuming the sparsity of φX , i.e.

min ||φX|| 1 s.t. Y = AX. (3)
The problem of packet losses is introduced by the fact that φ is predefined for transforming X which is of size n. With packet losses, we are in fact dealing with a signal of smaller size. Let X be the vector of all received x i s. Clearly, eq. 3 can deal with packet losses because the sink receives Y = A X , A being the remaining of A with the columns corresponding to the lost x i removed, and finds the right φ for transforming X . Eq. 3 cannot be solved by standard lasso, but can be converted to a problem of linear programming (LP) as follows,

min 1 T δ. (4) s.t. Y = AX -δ φX δ δ 0
Note that δ is a vector of length n which defines the bound on each entry of |φX|. A comparison of the reconstruction results is shown in Figure 3(b), which illustrates the better performance of eq. 3. While X is not directly sparse, eq. 3 is intuitively solvable as eq. 1 and 2, because it is equivalent to the following formulation,

min ||C|| 1 s.t. Y = Aφ -1 C, C = φX, ( 5
)
where C is the sparse coefficients of X in the space φ.

III. CS FOR SENSING IN BIKE RACES

The keys to the design of a proper CS method for an application of WSNs include the integration of CS with the routing protocol and the choice of the sparse space φ which is generally specific to the application. This section describes our efforts to solve these issues.

A. Sensor Data

The data of interest is GPS readings from each sensor which are in the form of (longitude, latitude, altitude). In this paper, we convert each GPS reading into a quantity of relative location with respect to a starting point using the road map from which we further compute the motion of the bikers over time. Thus, we aim to collect from each sensor the motion data instead of the location data. In particular, for a race of n bikers, let x i (t) be the instant velocity of biker i at time t, i = 1, . . . , n, and x i (t) can be directly computed at each sensor from its GPS readings. Without causing confusions, we drop the time index for simplifying the notations as we consider independent data collection at each time. Note that our CS method can easily include the sparsity in the temporal domain, with little modification.

B. Spatial Sparsity

To perform CS, we need to know in which space φ the data of interest is sparse. For our data of biker motion, we assume that the bikers in a race follow the constant velocity model and show strong group behavior in order to save energy. Thus, we define a spatial sparsity which enforces synchronized motion between nearby bikers. The motivation comes from the flocking behavior which, while looking complicated, is controlled by three simple rules [START_REF] Reynolds | Flocks, herds and schools: A distributed behavioral model[END_REF]:

• Separation: avoid crowding neighbors.

• Alignment: steer towards average heading of neighbors.

• Cohesion: steer towards average position of neighbors. The grouping behavior of bikers resembles in some ways the flocking behavior of birds both of which aim at the aerodynamic benefit.

To model the spatial sparsity, we first construct a graph modeling the spatial relationships between the bikers. Here, we simply connect each biker with her/his k-nearest neighbors using their positions in the previous time instant. Then, we penalize the differences between the velocities of neighboring bikers under the constraint of Y = AX. That is, our CS formulation based on the spatial sparsity is defined as

min ij∈E,i<j ||x i -x j || 1 s.t. Y = AX, (6)
where E is the set of edges in the k-nearest neighbor graph. The spatial sparsity is specific to our application and related to the graph Laplacian regularization for graph signal processing [START_REF] Shuman | The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains[END_REF]. To see this, a Laplacian of a graph is defined as a matrix L with entries of

L ij =      degree(i) i = j -1 ij ∈ E, i = j 0 otherwise . (7)
Thus, the spatial sparsity ij∈E,i<j ||x i -x j || 1 is in nature similar to the graph Laplacian regularization ||LX|| 1 where L corresponds to the sparse space φ. The difference is that ||LX|| 1 = i || ij∈E,∀j (x i -x j )|| 1 enforces the total difference between each biker and the neighbors to be zero or as small as possible, whereas ij∈E,i<j ||x i -x j || 1 enforces the similarity between each pair of neighboring bikers which is a stronger prior on X.

Similarly, eq. 6 can be converted to a standard LP problem, given by

min ij∈E,i<j δ ij , (8) s
.t. Y = AX -δ ij x i -x j δ ij , ∀ij ∈ E, i < j δ ij 0, ∀ij ∈ E, i < j
which can be solved easily by any standard LP solver.

C. Integration of CS with Routing

Based on the CS formulation in eq. 6, we design a simple routing algorithm for collecting data in the application of bike races. We assume that each sensor broadcasts messages which can only be received by sensors in a certain range and the radio transmission is under certain packet loss rates. As mentioned earlier, we can not define a priori routing paths for each sensor due to the high mobility of the bikers. Thus, the routing is basically performed by broadcasting and aggregation. In particular, the data collection procedure is divided into time slots of equal size within each of which sensor data are relayed to the sink by a multi-round routing algorithm. In each round in a time slot, each sensor aggregates messages heard in the previous round using CS and broadcasts the aggregated data.

More specifically, assume that there are L rounds in each time slot. In the first round, sensor i broadcasts a message of the format (i, x i ) and receives a list of such messages from the 1-hop neighbors. In the following rounds, sensor i computes and broadcasts a random combination of all received data from the previous round, given by

x l i = n j=1 a l ij x l-1 j , l 2, ( 9
)
where x 1 i = x i and a l ij is a random Bernoulli coefficient and takes on the value of 1, -1 or 0. a l ij = 0 means that no data from sensor j is received by sensor i in the previous round. The use of the Bernoulli random numbers allows the efficient coding of the coefficients with very few bits. Furthermore, define A l i = [a l i1 , . . . , a l in ] which is a row vector, A l = [A l 1 ; . . . ; A l n ]. Define X l = [x l 1 , . . . , x l n ] t . Thus, we have

X l = A l X l-1 = A l A l-1 X l-2 = . . . = l r=2 A r X 1 , (10)
where X 1 = X and X l is in fact the aggregated sensor data in the lth round. Define B l = l r=2 A r , and we have X l = B l X. Let B l i be the ith row of B l , B l = [B l 1 ; . . . ; B l n ], and we have

x l i = B l i X.
For the consistency of the notation, we let B 1 be an identity matrix of n × n.

Thus, we store at sensor i [A 2 i , . . . , A L i ] which consumes n × (L -1) bits. At the lth round, l = 1, . . . , L, sensor i transmits a message of the form (B l i , x l i ). Note that both B l i and x l i can be computed recursively at sensor i using received messages in the previous round by B l i = j a l ij B l-1 j and x l i = j a l ij x l-1 j . As each entry of B l , b l ij , is the sum of a number of products of a number of random Bernoulli numbers, b l ij is actually a Binomial random number. If we control L to be small and A l i to be sparse, we can also encode b l ij with few bits. For example, if we limit each message to aggregate at most m data, b l ij can be encoded by log 2 (m) bits. As a result, the length of the payload of the message in the lth round is bit(B l i , x l i ) = nlog 2 (m) + bit(a real number).

The sink nodes receives messages in each round from sensors within the 1-hop range and every message of (B l i , x l i ) provides a linear equation x l i = B l i X which is used to construct the linear system Y = AX. When the linear system is overdetermined, i.e. more than n messages are received, we obtain X by the analytical solution X = (A T A) -1 A T Y . When it is underdetermined, we solve it with the CS formulation in eq. 6 using a LP solver. Our multi-round routing algorithm is illustrated in Figure 4. A limitation of such routing is that the A in the resulted linear system Y = AX is no more the random matrix required by the CS theory. However, we will show in the experiment section that empirically, we can still solve the system even under severe packet losses.

IV. EXPERIMENTS

We evaluated our approach on simulated data generated from real trajectories of motorbikes surrounding the peloton in the race of Paris-Tours 2013. In particular, we developed a bike race simulator based on a mobility model derived from the GPS recordings of the motorbikes. In the simulation, we control the speed of the group by the speed of the motorbikes and program individual behaviors of each biker by taking into account a set of parameters including fatigue level and objective. Thus, the simulator produces realistic trajectories

sink i j k (B l i , x l i ) (B l-1 j , x l-1 j ) ( B l -1 k , x l -1 k ) x l i = j a l ij x l-1 j B l i = j a l ij B l-1 j
Fig. 4. Multi-round routing. Sensor i stores [A 2 i , . . . , A L i ] and broadcasts in the lth round a message of (B l i , x l i ) both of which are computed recursively by

B l i = i a l ij B l-1 j and x l i = j a l ij x l-1 j
. Each message of (B l i , x l i ) received by the sink provides a linear equation x l i = B l i X which is used to construct the linear system Y = AX. Messages from sensors out of the range of the sink such as j are aggregated and relayed by other sensors. of 130 bikers for a duration of 780 seconds, shown in Figure 5. Figure 5(a) shows the motion (i.e. instant velocity) of the bikers calculated as the changes of locations over time, Figure 5(b) shows the changes of the motion of each biker over time, and Figure 5(c) shows the differences between the motion of nearby bikers calculated as the total difference between each biker and its 10 nearest neighbors in the space.

The performance or reconstruction accuracy is evaluated on the metric of stress, defined as

stress(X, X) = i (x i -xi ) 2 i x 2 i . (11)
The smaller the stress is, the better the reconstruction is.

In the first experiment, we assume that the sink receives Y = AX, with A being a random Bernoulli matrix of size k × n and its entries taking on either 1 or -1 with equal probability. This is the simplest case which is as good as collecting data via a routing tree on a static WSN with the architecture in Figure 2. We experiment with different ks in order to evaluate the impact of k to the reconstruction accuracy, illustrated in Figure 6. It can be seen that the accuracy improves with the Fig. 6. Impact of k on the accuracy of the reconstruction by using the CS method in eq. 6 solved by a LP solver. We aggregate and reconstruct the data and compute the stress in each time of the simulation. The average stress in the entire simulation period is 0.4%, 0.16% and 0.08% for k = 20, 60 and 90 respectively. For k = 60, the reconstruction is the least accurate at time 374, with a stress of 0.91%, when many bikers attack each other by accelerating irregularly, which breaks the assumption of the spatial sparsity.

increase of k which is intuitive. Overall, the reconstruction is fairly accurate for k = 60 as the stress is reduced to less than 1% which means that, for example, for the average velocity of 10 meter/second, the average error in the reconstruction is less than 1 meter/second. In the second experiment, we simulated the data collection procedure with the multi-round routing algorithm. In particular, we put two sink nodes in the front and at the back of the biker group respectively. We assume that the transmission range for each sensor is 50 meters and that the number of rounds, L, in the multi-round routing algorithm is determined dynamically at each time so that every sensor can reach a sink node within L rounds, L 3. While such control of L is impossible in practice, in doing so, we make sure that each sensor data is aggregated and relayed to the sink with certain probability, which allows the evaluation of the reconstruction accuracy. In such a configuration, the experiment shows that, with the current location data, we always get overdetermined systems if there are no packet losses. Thus, we tune the packet loss rate for each connection so that each packet is dropped from a link randomly and independently with a probability p. We performed 10 simulations of the data routing for p = 0.25, 0.5 and 0.75 respectively and found that even for p = 0.75, i.e. on average 75% packets are dropped, we still get overdetermined systems in about 30% times due to the dense formation of the bikers in those times, and the underdetermined systems contain in the worst case about 20 linear equations. Figure 7 shows the stresses over time under different ps. Overall, the reconstruction accuracy with the multi-round routing is comparable to that with the routing tree but looks more noisy. One important difference is that, while the multi-round routing often gathers more equations in Y = AX even under severe packet loss rates, the combination matrix A is not purely random as required by the CS theory which should have an effect on the accuracy. We are interested in a more thorough study on the impact of the routing.

V. CONCLUSIONS AND FUTURE WORK

This paper presents a CS-based approach to compressive data collection on mobile wireless sensor networks which is designed for the application of sensing in bike races. The Fig. 7. Impact of packet loss rate p on the accuracy of the reconstruction with the multi-round routing algorithm. The average stress in the entire simulation period is 0.48%, 0.14% and 0.04% for p = 0.25, 0.5 and 0.75 respectively. For p = 0.5, the reconstruction is the least accurate at time 365, with a stress of 0.92%. method incorporates the problem-specific spatial sparsity and is integrated in a multi-round routing protocol based simply on broadcasting and aggregation. On motion data generated by a bike race simulator, our method achieved a fairly accurate reconstruction, with a stress of less than 1%, under a severe packet loss rate of 50%. Note that we did not compare our method with others simply because none of the previous work that we found are applicable in the situation of opportunistic routing in highly mobile and dynamic WSNs whereby each sensor has no knowledge of the next-hop sensors.

In the near future, we are interested in performing a test on real data from major bike races such as the Tour de France with which we intend to learn a better model of bike motions. Alternative to CS, networking coding is a technique which is also widely used for efficient data collection on WSNs. Thus, we also intend to investigate the feasibility of network coding in our application and compare it with CS-based approaches.

Fig. 1 .

 1 Fig. 1. The HIKOB infrastructure for data collection in bike races.

  no packet losses DCT coefficients with packet losses (a) Impact of packet losses to the sparsity pattern. by removing lost data and linear programming reconstruction by taking lost data as 0 and lasso (b) Comparison of the reconstruction accuracy.

Fig. 3 .

 3 Fig.3. We generate a signal which is 10-sparse in the discrete cosine transform (DCT) domain, shown as the red line in (a). We simulate 10 packet losses by setting randomly 10 data in the signal to 0, and the DCT coefficients become non-sparse, shown as the blue line in (a). In practice, packet losses close to the root of the routing tree cause much more severe problems. We compare the accuracy of the reconstruction with packet losses. It can be seen that taking lost data as 0 doesn't work well, shown as the black line in (b), due to the non-sparse pattern in (a), whereas linear programming with removing lost data produces a fairly good reconstruction, shown as the blue line in (b).

  Biker motion, x i , (change of location over time).

  Change of motion over time, x i (t) -x i (t -1). Difference of motion between nearby bikers, ij∈E ||x i -x j || 1 , ∀i.

Fig. 5 .

 5 Fig. 5. Spatial and temporal sparsity of biker motion. The values are normalized in {0, 1}, and the X axis is the time from 0 to 780 seconds and the Y axis is the biker/sensor ID indexed from 1 to n = 130.


Olivier Ferret

dir. Anne-Marie Mercier-Faivre

Joanny Moulin

Joanny, « Olivier Moulin

Anne-Marie Ferret

dir. Mercier-Faivre

privée, de l'Ancien Régime à la Restauration Sous la direction d'

Keywords:

teaching and research institutions in France or abroad, or from public or private research centers.

Cet ouvrage collectif est en partie tiré d'un colloque organisé en 2011 par l'équipe XVIII e siècle de l'UMR LIRE (Littérature, Idéologies, Représentations XVIII e -XIX e ), à l'Institut des sciences de l'homme et de l'Université Lumière-Lyon 2. Il s'agit d'un ouvrage tout à fait remarquable, fort bien édité, qui tout à la fois illustre la mise en oeuvre de méthodes nouvelles dans la pratique de la science littéraire et contribue de manière significative aux avancées de la connaissance dans le champ émergent des études biographiques. Dans sa « Présentation générale », Olivier Ferret engage le débat principalement avec les thèses de François Dosse, l'auteur du Pari biographique (Paris, La Découverte, 2005), dont à bon escient il critique respectueusement certaines des thèses, comme la classification de la biographie en trois « âges », « héroïque », « modal » et « herméneutique », qui néanmoins continuent de présider discrètement à l'ensemble de l'ouvrage. Ce texte introductif mérite en soi qu'on s'y arrête, d'abord pour la raison qui vient d'être dite, qui démontre une pensée bien affirmée sur la biographie comme relevant « d'un domaine de la connaissance, voire d'une discipline du savoir ». Il le mérite ensuite parce qu'il justifie l'àpropos de cet ouvrage par le fait qu'il participe à montrer comment la biographie moderne, ainsi définie, se constitue comme telle dans l'histoire de la littérature vers la fin du XVII e siècle, c'est-à-dire au début de ce « long XVIII e siècle » qui définit la période d'application de la présente étude. En effet, il faudra un jour prochain démontrer comment la naissance de la biographie a coïncidé avec celle du roman, « dans une période où l'histoire n'est pas encore conçue comme une science historique, et fait encore largement partie des « belles-lettres » ». Là n'est toutefois pas l'objectif de cette anthologie d'articles, qui se propose de montrer, par des études de cas bien choisies, comment la biographie, en tant que genre littéraire au sens large, participe pragmatiquement à la construction de discours politiques. « Sont ainsi à explorer les liens qu'entretient l'écriture biographique avec la construction d'imaginaires politiques, mais également avec les dispositions de construction, à l'occasion polémique, d'une légitimité, conduisant ainsi à une interrogation sur les processus de légitimation politique et sur le rôle que jouent les écrits biographiques dans cette opération ». Saluons ici une méthodologie de la recherche littéraire qui allie avec bonheur l'héritage du nouvel historicisme avec celui des théories de la réception. Une brève introduction seconde présente et commente rapidement quelques exemples bien choisis de ces illustrations qui accompagnent souvent les biographies étudiées, démontrant que la biographie déborde du champ de la littérature tout comme elle se distingue de celui de l'histoire. Cet ouvrage très bien conçu se divise en quatre parties -« Agir », « Édifier », « Justifier », « Interpréter » -, chacune d'entre elles étant présentée par un texte de cadrage d'Anne-Marie Mercier-Faivre, si bien que l'on ne perd jamais de vue la forte cohérence de l'ensemble, qui se lit de ce fait avec une aisance qui fait toute l'élégance d'une érudition distinguée. Le recueil commence fort, avec une étude que Christophe Cave consacre aux Anecdotes sur Mme la comtesse du Barry de Pidansat de Mairobert et autres ouvrages du même ordre, relevant des libelles et s'inspirant du modèle du récit libertin, érotique ou pornographique. Ce « type de biographie politique, entre libelle et effet de vérité », relève de « la méthode voltairienne en philosophie appliquée […] à la politique ». Samy Ben Messaoud étudie plusieurs exemples de vies privées aussi différents que la Vie privée de Louis XV et les Mémoires de la vie privée de Benjamin Franklin, qu'analyse ensuite plus avant l'article d'Olivier Ritz. C'est une des grandes réussites de ces articles, qu'ils se donnent pour objet de recherche des groupes d'oeuvres et d'auteurs, ne s'enfermant jamais dans le pré carré d'un seul ouvrage, mais pratiquant une forme de comparatisme au rythme enlevé. « La biographie politique émerge entre ombres et lumières, écrit Ben Messaoud, à travers ces textes excessifs certes, mais qui traduisent un besoin de transparence et une aspiration à la constitution d'un nouveau Panthéon ». L'un des textes les plus passionnants est sans aucun doute celui de Shojiro Kuwase, « Reconnaître Rousseau », dans lequel il se penche sur la tension entre l'image idéale que s'étaient construite les lecteurs de Rousseau et l'autoportrait posthume très décevant que le philosophe brosse de lui-même dans la seconde partie des Confessions, publiée en 1789, quatre ans après sa mort. Refus de lire ce livre décevant, rééditions expurgées, théorie du complot supposant que des ennemis avaient falsifié cette oeuvre en y insérant des anecdotes scandaleuses, publication de biographies de Rousseau gommant les frasques avouées de Jean-Jacques : il n'y a peut-être pas d'exemple plus probant de la différence radicale entre autobiographie et biographie, et ce texte de Kuwase fait naître le désir de lire une étude bien plus longue sur cette question-là. Après un article édifiant de Florence Boulère sur l'utilisation des biographies courtes dans les manuels d'éducation des années 1780, Simone Messina se penche sur la réhabilitation de Robespierre par Albert de Laponneraye, à travers les Mémoires de Charlotte Robespierre sur ses deux frères (1834). Autre article magistral, celui de Jean-Luc Chappey, « Usages politiques et sociaux de la biographie entre la Convention et le Directoire (1794-1799) », démontre que si la Révolution française n'invente pas la biographie, elle lui confère une portée nouvelle. Dans un premier temps, des biographies à charge dénoncent les « monstres » de la Terreur, des martyrologes établissent des listes de leurs victimes sous forme de séries de brèves notices biographiques, participent directement au déclenchement de la « Terreur blanche » : ces massacres de républicains qui se déroulèrent, particulièrement dans le Sud-Est, à partir de l'hiver 1794. Les députés de la Convention cherchent alors à canaliser ce mouvement en publiant des biographies et de petits dictionnaires biographiques faisant l'éloge de « héros nationaux », d'où émergent les types sociaux, et tout particulièrement celui du savant qui consacre sa vie au progrès du genre humain.

Chappey va jusqu'à parler d'une « véritable invasion du politique par le biographique », qui avec l'émergence du héros militaire, particulièrement après le coup d'État du 18 brumaire, puis la constitution de dossiers individuels, s'épanouit en ce qu'il appelle une « biocratie ». Et Chappey de conclure en émettant l'hypothèse que le culte romantique du moi fut sans doute une réaction au « processus de saisie du biographique par l'État ». Tout aussi passionnante est l'étude de la Notice sur la vie de Sieyès par Erwan Sommerer : une autobiographie déguisée en biographie, pièce maîtresse dans la stratégie de reconquête du pouvoir par l'abbé Sieyès. Francesco Dendena analyse de façon comparable les autobiographies des Feuillants, et Marie-France Piguet les Mémoires de M. le Comte de Montlosier. Dans la dernière partie, consacrée à la façon dont la biographie opère une interprétation évolutive des figures historiques, Myrtille Méricam-Bourdet observe le renversement de perspective par lequel on accorde finalement à Oliver Cromwell le crédit de la sincérité de ses opinions religieuses, pour aboutir au Cromwell de Voltaire, qui ouvre la voie à une articulation des ressorts collectifs et individuels de l'histoire. C'est encore sur les « Vies » de Voltaire que Laurence Macé s'arrête, à la lumière de la « Poétique des Vies particulières » de Jean Sgard. Fadi El Hage montre comment les biographies de Napoléon publiées sous la Restauration participent d'une lutte pour établir la légitimité des Bourbons. Enfin, c'est fort à propos qu'Éric Gatefin ferme la marche en revenant à Sainte-Beuve, et termine en cédant la parole au maître de la critique biographique : « car enfin ce temps qui a précédé notre naissance, ce dixhuitième siècle tout entier, nous le savons, avec un peu de bonne volonté et de lecture […] ; nous pourrions entrer à toute heure dans un salon quelconque et ne pas y être trop dépaysés ». Une bibliographie générale venant s'ajouter à celles des articles, un index des noms de personnes, et des notices biographiques des auteurs apportent à ces actes de colloques la touche finale qui en fait un beau livre, démontrant à plus d'un titre que l'art du « viographe », comme disait Jacques Le Vasseur en 1633, fait aujourd'hui l'objet d'une recherche très vivace.


Célestin Matte
email: celestin.matte@insa-lyon.fr

Mathieu Cunche
email: mathieu.cunche@inria.fr

DEMO: Panoptiphone: How Unique is Your Wi-Fi Device?

Keywords: CCS Concepts, Networks → Network privacy and anonymity, •Security and privacy → Mobile and wireless security, Security, Privacy, 802.11, Information Elements, Probe Requests

, which aims to show the identifying information that can be found in the frames broadcast by a Wi-Fi-enabled device. Information is passively collected from devices that have their Wi-Fi interface enabled, even if they are not connected to an access point. Panoptiphone uses this information to create a fingerprint of the device and empirically evaluate its uniqueness among a database of fingerprints. The user is then shown how much identifying information its device is leaking through Wi-Fi and how unique it is.

INTRODUCTION

Tracking people through their mobile devices has become common despite being controversial. Tracking individuals using Wi-Fi signals emitted by their portable device is being used by surveillance [START_REF] Gellman | NSA tracking cellphone locations worldwide, Snowden documents show[END_REF] and commercial organizations [START_REF] Goodin | No, this isn't a scene from minority report. This trash can is stalking you[END_REF]. Wi-Fi tracking is possible because Wi-Fi-enabled devices routinely transmit probe requests to search for nearby networks, and these requests contain the unique MAC address of the device [START_REF] Musa | Tracking unmodified smartphones using wi-fi monitors[END_REF]. An attacker can easily capture and track these requests using off-the-shelf hardware.

In response to these privacy violations, most Operating Systems (OSs) have now implemented different variants of This work is partially funded by Région Rhône-Alpes's ARC7. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). MAC address randomization [START_REF] Gruteser | Enhancing location privacy in wireless LAN through disposable interface identifiers: A quantitative analysis[END_REF]. While it is a necessary step towards increased privacy, it has been shown that MAC address randomization may not be sufficient in itself to provide adequate privacy [START_REF] Vanhoef | Why MAC Address Randomization is not Enough: An Analysis of Wi-Fi Network Discovery Mechanisms[END_REF]. Indeed, among other things, probe requests include valuable information for device fingerprinting, in the form of Information Elements (IEs) [6, §7.2.3], also called tagged parameters, or tags. These IEs are not mandatory and are used to advertise the support of various functionalities. They are generally composed of several subfields whose size can range from one bit to several bytes. The previous paper [START_REF] Vanhoef | Why MAC Address Randomization is not Enough: An Analysis of Wi-Fi Network Discovery Mechanisms[END_REF] showed that a subset of these IEs do not change over time, and can bring up to 7 bits of entropy. With our tool, we extend this work by listing all IEs handled by libpcap and provide a user-friendly way to display them, along with a calculation of their impact on privacy.

UNIQUENESS EVALUATION

The goal of Panoptiphone is to exhibit the trackability of a device by evaluating its uniqueness. This evaluation is based on the fingerprint built using the IEs found in the probe requests sent by this device. The uniqueness is evaluated with regard to a database of fingerprints. Following the approach of Panopticlick [START_REF] Eckersley | How unique is your web browser? In Privacy Enhancing Technologies[END_REF], we consider two metrics to evaluate this uniqueness: the anonymity set size that corresponds to the number of devices that are sharing the same fingerprint, and the entropy that quantifies the amount of identifying information provided by information elements.

The entropy of an IE or a set of IEs is computed as follows:

Hi = - j∈E i fi,j * log fi,j (1)
where Ei is the domain of possible values for element i and fi,j is the frequency (i.e., probability) of the value j for the element i in the database. We consider the absence of an element as a possible value.

THE PANOPTIPHONE TOOL

The Panoptiphone tool is based on a three-step process. First, radio signals emitted by a device are captured through a Wi-Fi interface in monitor mode, then the resulting data is analyzed to evaluate the uniqueness of the device, and finally the result is displayed as a feedback to the user. The architecture of the tool is presented on figure 1.

To capture data, our tool only requires a Wi-Fi card supporting monitor mode. On a modern Linux system, this is the case for most basic off-the-shelf cards. Using an external dongle can simplify the estimation of proximity of users' The tool is composed of two scripts. The first one, panoptiphone.sh, is a small bash script configuring the Wi-Fi interface and launching tshark with appropriate options. The latter can output exhaustive information in a XML stream, which can then be parsed in real time by the second script, panoptiphone.py. The latter is a python script making the computations using and storing information in a local database, and displaying results. The display includes the list of Information Elements, which are presented using their libpcap name, along with metrics for each element.

We rely on a database of fingerprints obtained from the Sapienza dataset [START_REF] Barbera | CRAWDAD dataset sapienza/probe-requests (v. 2013-09-10[END_REF], composed of 8 millions of probe requests from 160 000 devices. Pending on the user consent, our tool can add a fingerprint of tested devices to the database.

The only information captured by our tool (in its current form) is the IEs contained in probe requests, sent by devices having an enabled Wi-Fi interface. Traffic data sent by associated devices, timing information or physical-layer information are not considered.

Once the fingerprint is captured, the privacy metrics (anonymity set size and entropy) are computed for each IE as well as for the whole fingerprint. The result of this analysis is then displayed to the user.

PRIVACY-PROTECTION MEASURES

It is often non-trivial to manipulate private data while disclosing privacy breaches. In order to guarantee the privacy of our tool's users, we keep as little necessary information as possible. In particular, we do not keep association between the different IEs, except for the global fingerprint, which is kept SHA256-hashed. Thus, the only information that can be obtained out of it is whether a full global fingerprint has already been seen. Furthermore, we encrypt elements which are direct identifiers or contain private information: MAC addresses, WPS's UUIDs, SSIDs.

In real-time mode, the tool only detects devices in a range close to the antenna (a few centimeters), to ensure only agreeing participants will have their data collected.

INTERACTION WITH PARTICIPANTS

During the demonstration, conference participants will be able to interact with Panoptiphone by testing the uniqueness of their Wi-Fi-enabled device. By bringing their device close to the antenna of Panoptiphone, they will trigger a capture event that will capture the fingerprint of their device, which will be processed by the tool to compute the uniqueness of the device. The result of this process will be displayed as a feedback to the user on a screen. Figure 2 presents an example of several commands and their output, starting with an example output of the real-time mode.

In addition, participants will be able to contribute by giving their data providing they agree to the storage of their fingerprint. The tool has several additional features such as the display of the global statistics of the fingerprints stored in the database as well as specific informations elements.

CONCLUSION

We introduce Panoptiphone, a user-friendly tool to shed light on the trackability of Wi-Fi-enabled devices, even when they are using industry-standard techniques such as MAC address randomization. We hope that this will raise awareness on the necessity to make deeper modifications on the Wi-Fi 802.11 protocol regarding information contained in probe requests that simple identifier randomization. We also aim to raise public concern on the trackability of devices carried by almost anyone at any time.

WiSec' 16

 16 July 18-22, 2016, Darmstadt, Germany c 2016 Copyright held by the owner/author(s). ACM ISBN 978-1-4503-4270-4/16/07. DOI: http://dx.doi.org/10.1145/2939918.2942417

Figure 1 :

 1 Figure 1: Architecture of the system