Hadron mass, Regge pole model and E-infinity theory

Chaos, Solitons and Fractals 38 (2008) 1–15

www.elsevier.com/locate/chaos

Hadron mass, Regge pole model and E-infinity theory

Yosuke Tanaka *

Department of Physics, Kyushu Kyoritsu University, Kita-kyushu 807, Japan

Accepted 3 November 2006

Abstract

With the use of Regge pole model and E-infinity theory, we have calculated the masses of the excited states inhadrons, and obtained the nice agreement with experiments. We have shown the Regge–Chew–Frautschi diagramsand discussed some properties of excited states in hadrons. We have also classified the masses of elementary particlesaccording to the mass formula mk ¼ bðkÞ‘n ðhmpiÞ‘ð/Þndk , which we have used in this paper.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Recently, with the use of E-infinity theory, El Naschie and Marek-Crnjac have calculated the mass spectrum ofelementary particles, and obtained the excellent agreement with experiments [1,2]. Using the perturation method andE-infinity theory, we have derived the mass formula which El Naschie and Marek-Crnjac have used in their papers[1,2], and estimated the mass of hadrons and leptons [3,4].

In this paper, with the use of Regge pole model and E-infinity theory, we have studied the mass of excited statesin hadrons. Using the mass formula, which we have derived in the previous papers [3,4], we have calculated themass of excited states in hadrons, such as N � and K�, and obtained the nice agreement with experiments. Somemasses of excited states in hadrons turn out to be described by the identity of E-infinity theory ð1=/Þ4 þ ð/Þ4 ¼7; N 5=2þð1680Þ, N 5=2�ð1680Þ, K5=2þð1820Þ and K7=2�ð2100Þ. We have shown the Regge–Chew–Frautschi diagram,where the squared mass ðmÞ2 is plotted against the spin J, and discussed the properties of excited states in hadrons.By using the Regge trajectory J ¼ bðmÞ2 þ c and E-infinity mass formula m ¼ mð/Þ, we have estimated the coeffi-cients b ¼ bð/Þ and c ¼ cð/Þ. We have also classified the mass of elementary particles according to the mass for-mula mk ¼ bðkÞ‘n ðhmpiÞ‘ð/Þndk , which we have used to calculate the mass of hadrons in this paper ð‘ ¼ 1; n ¼ 0;�4Þ.

In Section 2, we make a brief review of the mass formula. In Section 3, we show the numerical results of mass cal-culations. In Section 4, we give several discussions on the related problems. Summary is given in Section 5.

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.11.007

* Tel.: +81 93 693 3202; fax: +81 93 603 8186.E-mail address: [email protected]

mailto:[email protected]

2 Y. Tanaka / Chaos, Solitons and Fractals 38 (2008) 1–15

2. Mass formula and Regge pole model

Here, we make a brief review of mass formula and Regge pole model of hadrons.

2.1. Mass formula

El Naschie has derived the mass formula of elementary particles

mk ¼ m0kkkdk ;

where symbols mk , m0k , kk and dk show the mass of k-particle, the basic mass of k-particles, the scaling factor and thecorrection part, respectively [1]. We have derived the above mass formula by using the first-order perturation method[3,4]

mk ¼ hm0kidk ;

where the symbol hm0ki denotes the zeroth order value of k-particle, and is estimated to be in the following form

hm0ki ¼ m0kkk :

Thus, using E-infinity theory, we obtain the mass formula of elementary particles

mk ¼ m0kð/Þkkð/Þdkð/Þ;

where the symbol / shows the golden ratio / ¼ ðffiffiffi5p� 1Þ=2.

El Naschie and Marek-Crnjac have shown that many of basic masses are given by the expression [1,2]

hmpi ¼ �a0 ¼ 201

/

� �4

;

where symbols hmpi and �a0 denote the averaged pion mass and the inversed fine structure constant, respectively. Forexample, the neutron mass is given by

mn ¼ hmpi 1/

� �4

¼ 939:574 ½MeV�:

An another example is the mass of g meson

mg0 ¼ 7hmpi 1� 1100ð/Þ3

n o¼ 957:308 ½MeV�:

In general, the zeroth order value hm0ki and the correction Dk may be described by the E-infinity formula as follows:

hm0ki ¼ bðkÞn

1

/

� �n

;

and

Dk ¼ cðkÞ‘ ð/Þ‘:

Thus, we have the mass formula

mk ¼ hm0kið1þ DkÞ ¼ bðkÞn

1

/

� �n

1þ cðkÞ‘ ð/Þ‘

n o:

This formula may be applied not only to the mass spectrum but also to other quantities, such as static and transitionmoments of elementary particles.

In this paper, we use the following form of mass formula for the ground and excited states in hadrons

mk ¼ bðkÞn hmpið/Þndk ðn ¼ 0;�1;�2; . . .Þ;

which is derived from the above general formula: m0k ¼ hmpi, kk ¼ bðkÞn ð/Þn and dk ¼ 1þ cðkÞ‘ ð/Þ

‘. This mass formulareminds us the hierarchy dimension formula of El Naschie [1]

Gm ¼1

2�a0ð/Þm ðm ¼ 0; 1; 2; . . .Þ;

where the elementary particle number NSM ¼ �a0=2 is scaled by factor ð/Þm. The elementary particle number N SM hasbeen discussed by El Naschie et al. with the use of various kinds of models [5–10].

Y. Tanaka / Chaos, Solitons and Fractals 38 (2008) 1–15 3

2.2. Regge pole model

In 1959, Regge proposed the Regge pole model to study the scattering in high energy physics [11,12]. According tothe Regge pole model, the squared mass ðmÞ2 of hadron excited states is described by the linear function of the hadronspin J

J ¼ bðmÞ2 þ c:

This comes from the assumption that the Regge pole anðtÞ is the linear function of the squared mass ðmÞ2 ¼ t

J ¼ Refang:

Phenomenologically, this relation is plotted in the ðmÞ2 � J diagram, which we call ‘‘the Regge–Chew–Frautschi dia-gram’’ in this paper. This diagram reminds us ‘‘the Schmidt diagram’’, where nuclear moments l are plotted againstthe nuclear spin J [13]. As is well known, the Regge–Chew–Frautschi diagram is very useful to classify the excited statesin hadrons.

In this paper, using the Regge–Chew–Frautschi diagram, we classify the excited states in hadrons and calculate thehadron mass of excited states within the framework of E-infinity theory of El Naschie [1]. We have estimated the coef-ficients b and c in terms of the golden ratio /. They are given in the following forms

b ¼ 1000

iþ jð/Þ8;

and

c ¼ ‘þ nð/Þ8

iþ jð/Þ8;

where we use the mass formula

mk ¼ bðkÞhmpidk ¼ bðkÞhmpif1þ cðkÞð/Þ8g:

The factors i, j, ‘ and n are determined to reproduce experimental data.

3. Numerical results

With the use of E-infinity mass formula [1–4], we have calculated the mass of excited states in hadrons and obtainedthe nice agreement with experiments [14,15]. Here, we show the numerical results of hadron mass calculations and theRegge–Chew–Frautschi diagrams of these hadrons.

3.1. The masses of excited states in baryons

Using the Regge trajectories of baryons, we can classify the excited states in baryons: a-, b-, c- and d-trajectories.

(1) The a-trajectories of baryons.In Table 1, we show the mass of baryons on the a-trajectories: N a, Ka and Ra. The mass expressions are given in the

third column. The theoretical and experimental values are shown in the fourth and fifth columns, respectively. We haveobtained the nice agreement with experiments [14,15]. It is interesting to note that the mass of Nð1680Þ and Kð1820Þ aredescribed by the identity ð1=/Þ4 þ ð/Þ4 ¼ 7.

In Fig. 1, we show the Regge–Chew–Frautschi diagram of these baryons (the a-trajectories), where the squared massðmÞ2 is plotted against the spin J; J ¼ bðmÞ2 þ c. The squared mass ðmÞ2 of N baryons are almost on the straight line.

Using the expressions of Nð1680Þ and Nð2220Þ, we have obtained the following coefficients b and c for N a baryons:

b ¼ 1000

1053� 45ð/Þ8¼ 0:950 ½GeV��2

;

and

c ¼ � 379þ 225ð/Þ8

2f1053� 45ð/Þ8g¼ �0:182:

Table 1The masses of the ground and excited states in baryons (the a trajectories)a

Particle state J P E-infinity theory Exp. (MeV)

Expression Value (MeV)

Nð939Þ 1=2þ 20 1/

� �8 939.574 939.565

Nð1680Þ 5=2þ 240 1/

� �4n1þ ð/Þ8

o1680.0 1680

Nð2220Þ 9=2þ 320 1/

� �41þ 30

53 ð/Þ8

n o2219.738 2220

Nð2700Þ 13=2þ 400 1/

� �41� 37

53 ð/Þ8

n o2700.899 2700

Kð1116Þ 1=2þ 160 1/

� �41þ 44

53 ð/Þ8

n o1116.035 1116

Kð1820Þ 5=2þ 260 1/

� �41þ ð/Þ8n o

1820.0 1820

Kð2350Þ 9=2þ 340 1/

� �41þ 21

53 ð/Þ8

n o2350.049 2350

hRið1193Þ 1=2þ 20 1/

� �81þ 3ð/Þ5n o

1193.738 1193.28

Rð1915Þ 5=2þ 280 1/

� �41� 5

53 ð/Þ8

n o1915.294 1915

hNið1318Þ 1=2þ 200 1/

� �41� 17

40 ð/Þ5

n o1318.287 1318.1

a The experimental data are taken from Refs. [14,15].

+213

PJ

+29

+25

+21

αΛ

αΣ

αΞ

αN

2 4 6 8

( ) [ ]22 GeVm

0

Fig. 1. The Regge–Chew–Frautschi diagram of baryon masses (the a trajectories).


Also, using the expressions of Kð1820Þ and Kð2350Þ, we have obtained the following coefficients b and c for Ka baryons:

b ¼ 1000

1106� 49ð/Þ8¼ 0:905 ½GeV��2;

and

c ¼ � 1094þ 245ð/Þ8

2f1106� 49ð/Þ8g¼ �0:497:


(2) The b-trajectories of baryonsIn Table 2, we show the mass of baryons on the b-trajectories: Nb, Kb and Rb. The mass expressions are given in the

third column. The theoretical and experimental values are shown in the fourth and fifth columns, respectively. We haveobtained the nice agreement with experiments [14,15]. It is interesting to note the mass of Nð1680Þ is reproduced by theidentity ð1=/Þ4 þ ð/Þ4 ¼ 7.

In Fig. 2, we show the Regge–Chew–Frautschi diagram of these baryons (the b-trajectories).Using the expressions of Nð1680Þ and Nð2220Þ, we have obtained the following coefficients b and c for Nb baryons:

TableThe m

Particl

Nð1680

Nð2200

Kð1830

Rð1775

a Th

b ¼ 1000

1011� 50ð/Þ8¼ 0:990 ½GeV��2

;

and

c ¼ � 589þ 250ð/Þ8

2 1011� 50ð/Þ8n o ¼ �0:294:

(3) The c-trajectories of baryonsIn Table 3, we show the mass of baryons on the c-trajectories: N c and Kc. The mass expressions are given in the third

column. The theoretical and experimental values are shown in the fourth and fifth columns, respectively. We have

2asses of the excited states in baryons (the b trajectories)a

e state JP E-infinity theory Exp. (MeV)


Þ 5=2� 240 1/

� �4n1þ ð/Þ8

o1680.0 1680

Þ 9=2� 320 1/

� �41þ 8

53 ð/Þ8

n o2200.359 2220

Þ 5=2� 260 1/

� �41þ 67

53 ð/Þ8

n o1830.019 1830

Þ 5=2� 260 1/

� �41� 9

53 ð/Þ8

n o1775.624 1775

e experimental data are taken from Refs. [14,15].

−213

PJ

−29

−25

−21

βΛ

βΣβN

2 4 6 8

( ) [ ]22 GeVm

0

Fig. 2. The Regge–Chew–Frautschi diagram of baryon masses (the b trajectories).

Table 3The masses of the excited states in baryons (the c trajectories)a

Particle state J P E-infinity theory Exp. (MeV)


Nð1520Þ 3=2�220 1

/

� �41þ 20

53 ð/Þ8

n o 1520.014 1520

Nð2190Þ 7=2� 320 1/

� �41� 4

53 ð/Þ8

n o2189.788 2190

Nð2650Þ 11=2� 380 1/

� �41þ 43

53 ð/Þ8

n o2649.538 2650

Nð3030Þ 15=2� 440 1/

� �41þ 12

53 ð/Þ8

n o3030.339 3030

Nð3340Þ 19=2� 480 1/

� �41þ 38

53 ð/Þ8

n o3340.178 3340

Nð3660Þ 23=2� 540 1/

� �41� 27

53 ð/Þ8

n o3661.079 3660

Nð3960Þ 27=2� 580 1/

� �41� 9

53 ð/Þ8

n o3961.009 3960

K 1520ð Þ 3=2� 220 1/

� �41þ 20

53 ð/Þ8

n o1520.014 1520

Kð2100Þ 7=2� 300 1/

� �41þ ð/Þ8n o

2100.0 2100



obtained the nice agreement with experiments [14,15]. It is interesting to note that the mass of Kð2100Þ is described bythe identity ð1=/Þ4 þ ð/Þ4 ¼ 7.

In Fig. 3, we show the Regge–Chew–Frautschi diagram of these baryons (the c-trajectories). The squared mass ðmÞ2of N baryons are almost on the straight line.

PJ

2 4 6 8

−215

−211

−27

−23

( ) [ ]22 GeVm

81614121 0 0201

−219

−227

−223

γN

γΛ

Fig. 3. The Regge–Chew–Frautschi diagram of baryon masses (the c trajectories).


Using the expressions of Nð2190Þ and Nð3960Þ, we have obtained the following coefficients b and c for N c baryons:

TableThe m

Particl

Dð1232

Dð1950

Dð2420

Dð2850

Dð3230

Rð1385

Rð2030

Nð1531

Xð1672

a Th

b ¼ 1000

1090� 22ð/Þ8¼ 0:917 ½GeV��2

;

and

c ¼ � 980þ 77ð/Þ8

1090� 22ð/Þ8¼ �0:900:

Also, using the expressions of Kð1520Þ and Kð2100Þ, we have obtained the following coefficients b and c for Kc baryons:

b ¼ 1000

1049þ 21ð/Þ8¼ 0:952 ½GeV��2

;

and

c ¼ �1477þ 147ð/Þ8

2f1049þ 21ð/Þ8g¼ �0:702:

(4) The d-trajectories of baryonsIn Table 4, we show the mass of baryons on the d-trajectories: Dd, Rd, Nd and Xd. The mass expressions are given in

the third column. The theoretical and experimental values are shown in the fourth and fifth columns, respectively. Wehave obtained the nice agreement with experiments [14,15].

In Fig. 4, we show the Regge–Chew–Frautschi diagram of these baryons (the d-trajectories). The squared mass ðmÞ2of D baryons are almost on the straight line.

Using the expressions of Dð1950Þ and Dð3230Þ, we have obtained the following coefficients b and c for Dd baryons:

b ¼ 1000

1105� 5ð/Þ8¼ 0:905 ½GeV��2;

and

c ¼ 131� 35ð/Þ8

2f1105� 5ð/Þ8g¼ 0:058:

Also, using the expressions of Rð1385Þ and Rð2030Þ, we have obtained the following coefficients b and c for Rd baryons:

b ¼ 1000

1101� 13ð/Þ8¼ 0:908 ½GeV��2

;

4asses of the ground and excited states in baryons (the d trajectories)a

e state J P E-infinity theory Exp. (MeV)


Þ 3=2þ180 1

/

� �41� 1

15 ð/Þ8

n o 1231.987 1232

Þ 7=2þ 280 1/

� �41þ 40

53 ð/Þ8

n o1949.979 1950

Þ 11=2þ 360 1/

� �41� 48

53 ð/Þ8

n o2419.908 2420

Þ 15=2þ 420 1/

� �41� 25

53 ð/Þ8

n o2849.818 2850

Þ 19=2þ 480 1/

� �41� 45

53 ð/Þ8

n o3230.508 3230

Þ 3=2þ 200 1/

� �41þ 26

53 ð/Þ8

n o1385.134 1385

Þ 7=2þ 300 1/

� �41� 32

53 ð/Þ8

n o2029.803 2030

Þ 3=2þ 200 1/

� �41þ 4

5 ð/Þ4

n o1530.820 1531.80

Þ 3=2þ 200 1/

� �41þ 49

20 ð/Þ5

n o1673.657 1672.45


2 4 6 8

+215

PJ

+211

+27

+23

( ) [ ]22 GeVm

21 0 10

+219

δΔ

δΣ

δΞ

δΩ

Fig. 4. The Regge–Chew–Frautschi diagram of baryon masses (the d trajectories).


and

c ¼ � 533þ 39ð/Þ8

2f1101� 13ð/Þ8g¼ �0:242:

3.2. The masses of excited states in mesons

Using the Regge trajectories of mesons, we can also classify the excited states in mesons: K�-, q-, x- and /-mesontrajectories.

(1) The trajectory of K� mesonsIn Table 5, we show the mass of mesons on the K� trajectory: K�, K�2, K�3 and K�4. The mass expressions are given in

the third column. The calculated and experimental values are shown in the fourth and fifth columns, respectively. Wehave obtained the nice agreement with experiments [14,15].

In Fig. 5, we show the Regge–Chew–Frautschi diagram of K� mesons. The squared mass ðmÞ2 of K� mesons arealmost on the straight line. Using the expressions of K�2ð1430Þ and K�4ð2060Þ, we have obtained the following coefficientsb and c for K� mesons:

b ¼ 1000

1107� 97ð/Þ8¼ 0:905 ½GeV��2

;

and

c ¼ 180� 194ð/Þ8

1107� 97ð/Þ8¼ 0:159:

(2) The trajectory of q mesons

( ) [ ]22 GeVm

1.0 2.0

PJ

−1

0.50.40.3 0

+2

−3

+4

∗Kωρ φ

'2

*2 f,K

3*3 ,K φ

*4K

22 ,f

,ω3 3ρ

44 ,f

Fig. 5. The Regge–Chew–Frautschi diagram of meson masses.

Table 5The masses of the ground and excited states in K mesonsa

Particle state JP E-infinity theory Exp. (MeV)


K�ð892Þ 1� 120 1/

� �4n1þ 4ð/Þ8

o892.522 892.1

K�2ð1430Þ 2þ 200 1/

� �41þ 40

21 ð/Þ8

n o1426.399 1426

K�3 1780ð Þ 3� 260 1/

� �41� 1

21 ð/Þ8

n o1780.259 1780

K�4ð2060Þ 4þ 300 1/

� �41þ 2

21 ð/Þ8

n o2060.398 2060



In Table 6, we show the mass of mesons on the q trajectory: q, a2, q3, and a4. The mass expressions are given in thethird column. The calculated and experimental values are shown in the fourth and fifth column, respectively. We haveobtained the nice agreement with experiments [14,15].

In Fig. 5, we show the Regge–Chew–Frautschi diagram of q mesons. The squared mass ðmÞ2 of q mesons are almoston the straight line.

Using the expressions of a2ð1320Þ and q3ð1690Þ, we have obtained the following coefficients b and c for q mesons:

b ¼ 1000

1120� 53ð/Þ8¼ 0:893 ½GeV��2

;

and

c ¼ 503� 106ð/Þ8

1120� 53ð/Þ8¼ 0:447:

Table 6The masses of the ground and excited states in q mesonsa



qð770Þ 1�100 1

/

� �4n1þ 1

5 ð/Þo

770.131 770

a2ð1320Þ 2þ 200 1/

� �41� 38

21 ð/Þ8

n o1318.019 1318

q3ð1690Þ 3� 240 1/

� �41þ 9

7 ð/Þ8

n o1690.003 1691

a4ð2040Þ 4þ 300 1/

� �41� 20

53 ð/Þ8

n o2039.713 2040



(3) The trajectory of x mesonsIn Table 7, we show the mass of mesons on the x trajectory: x, f2, x3 and f4. The mass expressions are given in the

third column. The calculated and experimental values are shown in the fourth and fifth columns, respectively. We haveobtained the nice agreement with experiments [14,15].

In Fig. 5, we show the Regge–Chew–Frautschi diagram of x mesons. The squared mass ðmÞ2 of x mesons are almoston the straight line.

Using the expressions of f2ð1270Þ and x3ð1670Þ, we have obtained the following coefficients b and c for x mesons:

TableThe m

Particl

xð783

f2ð127

x3ð167

f4ð203

a Th

b ¼ 1000

1162� 75ð/Þ8¼ 0:861 ½GeV��2

;

and

c ¼ 702� 150ð/Þ8

1162� 75ð/Þ8¼ 0:602:

(4) The trajectory of / mesonsIn Table 8, we show the mass of mesons on the / trajectory: /, f 02 and /3. The mass expressions are given in the third

column. The calculated and experimental values are shown in the fourth and fifth columns, respectively. We haveobtained the nice agreement with experiments [14,15].

In Fig. 5, we show the Regge–Chew–Frautschi diagram of / mesons. The squared mass ðmÞ2 of / mesons are almoston the straight line.

Using the expressions of f 02ð1525Þ and /3ð1850Þ, we have obtained the following coefficients b and c for / mesons:

b ¼ 1000

1105þ 162ð/Þ8¼ 0:902 ½GeV��2;

and

c ¼ �114þ 324ð/Þ8

1105þ 162ð/Þ8¼ �0:096:

7asses of the ground and excited states in x mesons a

e state JP E-infinity theory Exp. (MeV)


Þ 1� 100 1/

� �41þ 39

40 ð/Þ4

n o782.910 782.6

0Þ 2þ 180 1/

� �41þ 32

21 ð/Þ8

n o1273.755 1274

0Þ 3� 240 1/

� �41þ 2

3 ð/Þ8

n o1668.327 1668

0Þ 4þ 300 1/

� �41� 2

3 ð/Þ8

n o2027.051 2026


Table 8The masses of the ground and excited states in / mesons a



/ð1020Þ 1� 160 1/

� �41� 14

29 ð/Þ4

n o1019.414 1019.5

f 02ð1525Þ 2þ 220 1/

� �41þ 11

21 ð/Þ8

n o1524.715 1525

/3ð1850Þ 3� 280 1/

� �41� 34

21 ð/Þ8

n o1853.008 1853



3.3. The coefficients b and c in the Regge-trajectory: J ¼ bðmÞ2 þ c

As stated in the previous sections, by using the expressions of the Regge trajectory J ¼ bðmÞ2 þ c [11,15] and themass formula of El Naschie m ¼ mð/Þ [1], we have obtained the coefficients b and c in terms of the golden ratio /;b ¼ bð/Þ and c ¼ cð/Þ. The calculated results are shown in Sections 3.1 and 3.2.

In Table 9, the numerical values of b and c are summarized for baryons. We have obtained the following results forbaryons:

TableThe co

Traject

Na

Ka

Nb

N c

Kc

Dd

Rd

a Th

TableThe co

Traject

K�

qx/

a Th

0:905 ½GeV��26 b 6 0:990 ½GeV��2

;

and

�0:900 6 c 6 0:058:

In Table 10, the numerical values of b and c are also summarized for mesons. We have obtained the following results formesons:

0:861 ½GeV��26 b 6 0:905 ½GeV��2;

and

�0:096 6 c 6 0:602:

9efficients b and c in the Regge trajectory of baryonsa; J ¼ bðmÞ2 þ c

ory b ðGeVÞ�2 c

0.950 �0.1820.905 �0.497

0.990 �0.294

0.917 �0.9000.952 �0.702

0.905 0.0580.908 �0.242

e expressions b ¼ bð/Þ and c ¼ cð/Þ are given in the text.

10efficients b and c in the Regge trajectory of mesonsa ; J ¼ bðmÞ2 þ c

ory b ðGeVÞ�2 c

0.905 0.1590.893 0.4470.861 0.6020.902 �0.096

e expressions b ¼ bð/Þ and c ¼ cð/Þ are given in the text.


Parker discussed the Regge trajectories of hadrons in the Encyclopedia of Physics [15].

(i) In the Regge–Chew–Frautschi diagram of D� baryons, he obtained the derivative b;0:9 ½GeV��2 and discussedthe rotation-like spectra of D� baryons ðJ ¼ 3=2þ; 7=2þ; 11=2þ; . . .Þ.

(ii) In the Regge–Chew–Frautschi diagram of q� and K� mesons, he obtained also the derivative b;0:9 ½GeV��2 anddiscussed the J ¼ Lþ 1 states of mesons ðL ¼ 0; 1; 2; 3; . . .Þ. As shown in Tables 9 and 10, we have obtained thealmost same results as Parker’s values [15].

In this paper, as stated before, we have estimated the coefficients b and c with the use of the fractal theory of ElNaschie [1], and discussed the Regge trajectories of hadrons.

4. Discussions

Here, we give some discussions on the related problems.

4.1. The precise estimation of baryon masses

In this work, we have found that we can estimate high precisely the mass of baryons in some cases. Some of baryonmasses turn out to be reproduced by the identity of E-infinity theory

1

/

� �4

þ ð/Þ4 ¼ 7:

Here, we show some examples.(1) Masses of the excited states in N a and Nb baryons: Nð1680; 5=2þ and 5=2�Þ. With the use of the above identity,

we have the expression

mNð1680Þ ¼ 2401

/

� �4

f1þ ð/Þ8g:

This expression gives the theoretical value mNð1680Þ ¼ 1680 ½MeV�, which is just equal to the experimental value(1680 [MeV]), as shown in Tables 1 and 2. These two states ð5=2þ and 5=2�Þ are well known as the members of theeven–odd doublet in the Regge trajectories.

(2) Mass of the excited stated in Ka baryon: Kð1820; 5=2þÞ. By using the above identity, we have the followingexpression

mKð1820Þ ¼ 2601

/

� �4

f1þ ð/Þ8g:

This gives the theoretical value mKð1820Þ ¼ 1820 ½MeV�, which is just equal to the experimental value (1820 [MeV]), asshown in Table 1.

(3) Mass of the excited state in Kc baryon: Kð2100; 7=2�Þ. Also, with the use of the above identity, we obtain theexpression

mKð2100Þ ¼ 3001

/

� �4

f1þ ð/Þ8g;

which gives the theoretical value mKð2100Þ ¼ 2100 ½MeV�. This is just equal to the experimental value ð2100 ½MeV�Þ, asshown in Table 3.

(4) We summarize these results with the use of the simple formula

mk ¼ f ðkÞ4 fDð26Þ � Dð6Þg 1

/

� �4

f1þ ð/Þ8g ¼ 140f ðkÞ4 ½MeV�;

where symbols mk ; fðkÞ4 and DðnÞ denote the mass of k-particle, the numerical factor and the hierarchy dimension,

respectively.

(1) f ðkÞ4 ¼ 11; F 1ð1540Þ:


(2) f ðkÞ4 ¼ 12; N að1680Þ;Nbð1680Þ; gð1680Þ:(3) f ðkÞ4 ¼ 13; Kað1820Þ;Nð1820Þ:(4) f ðkÞ4 ¼ 14; Dð1960Þ:(5) f ðkÞ4 ¼ 15; Nð2100Þ;Kcð2100Þ;Rð2100Þ:

More details on this problem will be discussed elsewhere.These facts show the powerful and beautiful aspects of E-infinity theory of El Naschie [1]. In future, more examples

of highly precise estimations will be found in the mass analyses of the excited states in baryons and mesons.

4.2. Comments on the mass calculations of hadrons

We make comments on the hadron mass calculations using the quantum theory and E-infinity theory.At this stage, within the framework of quark model, as far as using the quantum theory, it may be difficult to repro-

duce systematically the masses of excited states in hadrons, because we know scarcely the quark wave functions andquark–quark interactions in hadrons. On the other hand, with the use of E-infinity mass formula, as shown in thisand previous papers [1–4], we can reproduce precisely the experimental data of hadron masses in the ground and excitedstates. This fact shows the powerful and elegant aspects of E-infinity theory of El-Naschie [1]. In other words, we maysay that E-infinity theory is simpler and superior to the quantum theory, as far as studying the mass spectrum of ele-mentary particles [3–5].

4.3. Regge trajectory, string and E-infinity theory

Parker discussed the Regge trajectory, with the use of the dimension analysis in the string theory [15]. According toParker, the angular momentum J of hadrons is written in the following form:

J ¼ Aðc3j�1m2Þ þ Bð�hÞ;
where symbols m; c; �h and j denote the mass of hadrons, the speed of light, the Planck constant and the elasticity ofstring, respectively. The coefficients A and B show the dimensionless constant. From above equation, he obtained
J ¼ bðmÞ2 þ const:;

which is just the expression of the Regge trajectory of hadrons. As is well known, this fact suggests that hadrons consistsof quarks and strings.

In this paper, as stated before, we have estimated the coefficients b and const, with the use of E-infinity theory [1]. ElNaschie has studied the relations between E-infinity theory and string theory, such as Heterotic- and Super-string the-ory [1].

4.4. The classification of elementary particle masses

In this section, we classify the mass spectrum of elementary particles with the use of the extended mass formula

mk ¼ bðkÞ‘n ðhmpiÞ‘ð/Þndk ð‘ ¼ 1; 2; 3 . . . ; n ¼ 0;�1;�2 . . .Þ;
which we have used in this paper; ‘ ¼ 1 for hadrons. The elementary particle masses are classified in Table 11, where theprevious mass calculations are summarized [1–4].
(1) The mass of baryons.Many of baryon masses are described by the formula [1–3].

mk ¼ bðkÞn hmpidk ð‘ ¼ 1; n ¼ 0Þ;
k ¼ K;N;D;R�;N� and X: Some of baryon masses are given by the formula [1–3]
mk ¼ bðkÞn hmpið/Þ�4dk ; ð‘ ¼ 1; n ¼ �4Þ;

k ¼ N and R.

(2) The mass of mesons. At this stage, as far as concerning the nonet scalar- and vector-mesons, masses of all mesonsare described by the formula [1–3]

mk ¼ bðkÞn hmpidk ð‘ ¼ 1; n ¼ 0Þ;

Table 11The classification of elementary particle masses according to the mass formulaa; mk ¼ bðkÞ‘n ðhmpiÞ‘ð/Þndk

‘ Particles n

�4 0 2 4 5 11

26 7 8

‘ ¼ 1 Baryons N K;NR D;R�

N�;X

Mesons p;Kg; g0

p�;K�

g�; g�0

‘ ¼ 2 Quarks s d u

c

b

t

Charged leptons s l e

Weak bosons Z0 W �

Higgs bosons H�;H 0

‘ ¼ 3 Sub-quarks wi

hj

Ck

a The mass expressions are given in Refs. [1–4,17–21].


k ¼ p;K; g; g0; p�;K�; g� and g0�:

(3) The mass of quarks.The current masses of quarks are given as follows [1,3,4]:(i) ‘ ¼ 2; n ¼ 6; k ¼ u,(ii) ‘ ¼ 2; n ¼ 5; k ¼ d; c; b and t,(iii) ‘ ¼ 2; n ¼ 4; k ¼ s.

(4) The mass of charged leptons.The masses of charged leptons are classified as follows [1,2,4]:(i) ‘ ¼ 2; n ¼ 7; k ¼ e,(ii) ‘ ¼ 2; n ¼ 11=2; k ¼ l,(iii) ‘ ¼ 2; n ¼ 2; k ¼ s.

(5) The mass of weak bosons.The masses of weak bosons are given as follows [1,2,4]:(i) ‘ ¼ 2; n ¼ 8; k ¼ W �,(ii) ‘ ¼ 2; n ¼ 7; k ¼ Z0.

(6) The mass of Higgs bosons.At this stage, several expressions are proposed for the mass of Higgs bosons [16–20].(i) ‘ ¼ 2; n ¼ 0; k ¼ H�0.

(7) The mass of subquarks.We have used subquark model of Terazawa [21] to estimate the masses of quarks, leptons and weak bosons [4].(i) ‘ ¼ 3; n ¼ 4; k ¼ wi; hj and Ck :

These are summarized in Table 11.


5. Summary

Finally, we summarize our work as follows. In this paper, with the use of Regge pole model and E-infinity theory, wehave calculated the masses of excited states in hadrons, such as N � and K�, and obtained the nice agreement with exper-iments [14,15]. Some masses of excited states in hadrons turn out to be reproduced precisely by the identity of E-infinitytheory ð1=/Þ4 þ ð/Þ4 ¼ 7 [N 5=2þð1680Þ, N 5=2�ð1680Þ, K5=2þð1820Þ and K7=2�ð2100Þ]. We have shown the Regge–Chew–Frautschi diagrams, where the squared mass ðmÞ2 of hadrons are plotted against the spin J, and discussed several prop-erties of excited states in hadrons [14,15]. Using the Regge trajectory J ¼ bðmÞ2 þ c and E-infinity mass formulam ¼ mð/Þ, we have obtained the coefficients b ¼ bð/Þ and c ¼ cð/Þ. We have also classified the masses of elementaryparticles with the use of mass formula mk ¼ bðkÞ‘n ðhmpiÞ‘ð/Þndk , which we have used in this paper.

Acknowledgement

We would like to thank Ms. H. Shiroshita for typing the manuscript of this paper.

References

[1] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals2004;19:209–36.

[2] Marek-Crnjac L. On mass spectrum of elementary particles of the standard model using El Naschie’s golden field theory. Chaos,Solitons & Fractals 2003;15:611–8.

[3] Tanaka Y. The mass spectrum of hadrons and E-infinity theory. Chaos, Solitons & Fractals 2006;27:851–63.[4] Tanaka Y. Elementary particle mass, subquark model and E-infinity theory. Chaos, Solitons & Fractals 2006;28:290–305.[5] El Naschie MS. From Hilbert space to the number of Higgs particles via the quantum two-slit experiment. In: Weibel P, Ord G,

Rossler OE, editors. Space time physics and fractality. Wien and New York: Springer; 2005. p. 23–31.[6] El Naschie MS. On a Fuzzy Kahler-like manifold which is consistent with the two slit experiment. Int J Nonlinear Sci Numer

Simul 2005;6(2):95–8.[7] He J-H. In search of 9 hidden particles. Int J Nonlinear Sci Numer Simul 2005;6(2):93–4.[8] El Naschie MS. The unreasonable effectiveness of the electron-volt units system in high energy physics and the role played by

�a0 ¼ 137. Int J Nonlinear Sci Numer Simul 2006;7(2):119–28.[9] El Naschie MS. Super strings, entropy and the elementary particles content of the standard model. Chaos, Solitons & Fractals

2006;29:48–54.[10] El Naschie MS. String theory: what it cannot do but E-infinity could. Chaos, Solitons & Fractals 2006;29:65–8.[11] Regge T. Introduction to complex orbital momenta. IL Nuovo Cimento 1959;14:951–76.[12] Regge T. Bound states, shadow states and mandelstam representation. IL Nuovo Cimento 1960;18:947–56.[13] Wong SSM. Introductory nuclear physics. London: Prentice-Hall; 1990.[14] Eidelman S et al. The review of particle physics. Phys Lett B 2004;592:1. Also available in the following URL, http://pdg.lbl.govl.

See also Review of particle properties. Phys Lett B 1986;170:1.[15] Parker SP. McGraw-Hill encyclopedia of physics. 2nd ed. New York: McGraw Hill, Inc.; 1993.[16] Mahmoud IS. The Higgs mass using E-infinity theory. Chaos, Solitons & Fractals 2006;30:263–8.[17] Marek-Crnjac L. Different Higgs models and the number of Higgs particles. Chaos, Solitons & Fractals 2006;27:575–9.[18] El Naschie MS. Experimental and theoretical arguments for the number of the Higgs particles. Chaos, Solitons & Fractals

2005;23:1091–8.[19] El Naschie MS. Determining the mass of the Higgs and the electro weak bosons. Chaos, Solitons & Fractals 2005;24:899–905.[20] El Naschie MS. The Higgs and the expectation value of the number of elementary particles in a super symmetric extension of the

standard model of high energy physics. Chaos, Solitons & Fractals 2005;23:363–71.[21] Terazawa H. Subquark model of leptons and quarks. Phys Rev D 1980;22:184–99.

http://pdg.lbl.govl.

Hadron mass, Regge pole model and E-infinity theory

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