Electromagnetic mass differences oft-flavored hadrons in modified bag model

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PRAMANA © Printed in India Vol. 43, No. 2,

_ _ journal of August 1994

physics pp. 155-163

Electromagnetic mass differences of t-flavored hadrons in modified bag model

BISHWARANJAN DASH l, SRIPATI KUMAR NAYAK 2 and LAMBODAR P SINGH 3

~ Physics Department, Banki College, Banki, Cuttack, India 2 Physics Department, S.V.M College, Jagatsingpur, Cuttack, India

3physics Department, Utkal University, Vanivihar, Bhubaneswar 751 004, India MS received 16 February 1994; revised 6 June 1994

Abstract. Mass difference of t-flavored hadrons is calculated using bag model modified for considering heavy quarks inside the bag. Both electric and magnetic contributions to mass differences are evaluated without the assumption of degenerate intermediate state. Mass differences between up and down quarks inside the bag is taken to be a constant in the absence of a dynamical calculations for the same.

Keywords. Bag model; t-flavoured hadrons; electromagnetic mass differences.

PACS Nos 12.40; 13.40

1. Introduction

A new era was started in the theoretical understanding of particle properties when Gell-mann and Zweig [ 1] introduced quarks as fundamental constituents of hadrons.

Basing on the symmetry group global SU(3), three quark-model with fractionally charged quarks was proposed. These are commonly known as u, d, s quarks and are said to represent three quark flavors. Discovery [2] of mesons like ~ and Y required the existence of additional flavors like c and b. There are also two independent evidences for the existence of the heaviest top quark t, coming from forward-backward asymmetry in e ÷ e- ~ bb- [3] and the absence of flavor changing neutral current decay

b - ~ s , d [43. Besides, aesthetics of quark-lepton symmetry and the resulting three

quark family structure have led many to believe in the existense of sixth quark t.

The precision of electroweak data has made it possible to estimate indirectly the masses of the top quark via their contributions to the radiative corrections. A global

~___ l ~ ; f l + 2 3

fit to all data of standard model yield, m t . . . . 26 + 16 GeV [5] where the central value is for Higgs mass m u = 250 GeV. A new update precision LEP data are analysed to predict m, = 126_2sGeV [6] for Higgs mass unconstrained. If these predictions ÷26 are correct, the top quark should be discovered during the new run of Fermilab p/~

collider.

Thus, on the eve of a possible of discovery of t-quark, assumed to live long enough to hadronise, a calculation [7] of masses of t-flavored hadrons was undertaken to facilitate comparison with the experimental results when the t-flavored hadrons are eventually discovered and shed light on the understanding of the confinement of heavy quarks to form hadrons drawn from the above comparison. The purpose of

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the present work is to carry the above work to completion with the estimation of the electromagnetic mass differences between t-flavored mesons and baryons carrying spin 0, 1 and ½, 3 respectively.

The paper is organized as follows. In § 2 we have described the model used for our calculation. In § 3 we present our numerical results and in §4 we conclude with a discussion.

2. The model

It is long identified that the strong interaction creates complications in calculating mass differences among hadrons in the same isospin multiplet. These problems, one assumes, may perhaps be avoided in a quark model, where the mass shift due to electromagnetic interaction has two parts; one from electric and magnetic interactions between different quarks and a second from the self energies of quarks themselves.

The latter results in a mass difference between up and down quarks for which a thorough dynamical calculation has not been carried out so far [8]. The electric and the magnetic interactions on the other hand can be explicitly calculated [9] using, say, MIT bag model which provides us an explicit wavefunction for quarks constituting a hadron.

The MIT bag model, implementing quark and gluon confinement in a simple manner, has been very successful in description of ground state hadrons formed with light quarks [10]. But the application of this model to heavy quark states or nonspectroscopic calculation have been very limited, reflecting a lack of proper understanding of its two most important parameters like the bag pressure B and zero point energy Zo. This has also resulted in poor agreement of model predictions with the experiment as one goes to heavier quark systems. Heavy quarks, in all likelihood, destroy the sphericity of the bag. However, a/nonspherical bag has not yet been solved. Therefore Ponce [11] modified the spherical bag model by invoking mass-dependent series exl~ansions for co, the frequency of a quark inside a spherical bag and also for the zero point energy Zo. The virtue of this expansion lies in the better agreement with experimental results in the c and b sectors. We, in the following, will use these expansions for heavy t-quark.

As stated before, we write the electromagnetic mass difference in a quark model as

AM = (AM)c~ + (AM)ma, + (AM)q (1)

where (AM),L~ag are contributions to AM from electric and magnetic interactions between quarks. (AM)q is the up and down quark mass difference.

Electromagnetic self energy of a particle appropriate to a bag model can be written as [9]

e2 f

f0 ~ dk s i n k t x - y l

~ bag d3xd3y

k+AE Ix-yl

x <PIJ.(x,0)IN> <NIJ (y, 0)IP>

(2) where AE is the mass of the intermediate state N minus the external state P, the k integration is the residual of photon propagator integral. The quark current J~ is given by

J~,(x) = ~ Q~l,(X)y~,q~(x)

(3)

~t

156 Pramana- d. Phys., Vol. 43, No. 2, August 1994

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Electromagnetic mass differences of t-flavored hadrons

where Q, and q, are the charge and field operators of the ~tth quark respectively.

When (3) is substituted into (2) two types of terms originate; one where a virtual photon is emitted and absorbed by the same quark and another in which virtual photon is exchanged between t w o different quarks. The former gives self energy of a quark and finally leads to the mass difference between the up and down quark when confined inside the bag. But its explicit estimation needs more work [8]. We thus concentrate on evaluation of the electric and magnetic interaction between a pair of quarks using bag model wavefunctions.

The contribution to mass difference due to electric interaction involves J0 and this being a diagonal operator the intermediate state (N) can only be same as the external state (P) and one gets

(AM)~, = ~ C,~[I~p(R) +

&,(R)] (4)

~,#;~>#

where R is radius of bag,

c ~ = Q,Q# (5)

is the product of charges of the two quarks and

(e2~N2N2f~ydyf(-w#-~#m#)[jo(~-~)]2+(w'-~m'~

\ w e /

4x 2 t . \ w, / \ x= \ w, /

x [ 2 y + R s i n ( ~ ) _ 4 R2sin2(x~y/R) ]

₍₆₎

x 2 y

J ,

where N~ is the normalization constant for the wavefunctions of the quark of type

~( and is given by

(N~)_ ~ = RSj~(x ) 2w,(w, -

(I/R)) +

(m,/R)

(7) w~(w~- m,)

and j's are spherical Bessel functions, m~ and w~ are mass and frequency of the atth quark in the lowest mode, respectively, w~ is given by

[X 2 + (m R)2] 1/2

w, = (8)

R

where x, is the root of transcendental equation, generated by the linear boundary condition of the bag model.

tanx~ - X~ (9)

1 - m R -

[ ( m R ) 2 + x 2 ]

1/2"

The magnetic interaction on the other hand involves the spatial part of the quark current which is not diagonal. Thus one can write the contribution to the mass-shift due to magnetic interaction as sum of contributions from the possible intermediate Pramana - J . Phys., Vol. 43, No. 2, August 1994 157

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Bishwaranjan Dash et al

states as

with

(AM)mas = E

(AM]I'N

rmag

(lO)

(AM] PN = - _-Irlag ~

CP~[Jp(AE, R)+

Jp,(AE, R)] (ll)

~ , # ; ~ > #

where C~e~ v are co-efficients which are appropriate differences of the product of quark charges and spin operators and

e 2 8 [ ( w 2 _ m 2 ] ( w 2 _ m ~ ) ] l / 2 R 2

~o k

J p(AE, R ) = - - - - N 2N 2L' , ~" ~ - - J o dk

4n 3n ~ w~w~ x 3~ k + AE

x(f'x"/"dzSin2Zsin(kRz~-jl(ky)sin2(x--~RY-))

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\ J o z \ x= / In the limit AE---,0 eq. (12) reduces to the form,

m )(wp - mp)] _ J p(0, R) = ~ \ - , N~ N~ w~ wp

x dyjo Jl x3

¢1

x 4 y - - - sin + 2ycos . (13)

For the heavy t-quark, as stated earlier, we will use the following expansion for w~, as suggested by Ponce [11]

7~ 2 T~ 2

wt = Xt + 2X---~ - 2X 2 + " " (14)

where Xt =

mtR.

The corresponding xt will be obtained from

It is also worth mentioning here that in evaluating the magnetic interaction we have considered the contributions of the intermediate states whose wavefunctions are totally symmetric in flavor and spin. To be specific, we have omitted the contributions of T~ hadron intermediate state with flavor structure a s

t(ud)antisy m

tO J~p and J~, for T~ and T *÷ hadrons since the co-efficients C,p will have only one non-zero value for heavy quark-light quark combinations. The contribution of Tff intermediate state towards (AM)mag for T~ and T~' + will not, thus, significantly alter the values estimated here.

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Electromagnetic mass differences of t-flavored hadrons

3. Numerical results

In tables I and 2 w e h a v e presented magnetic co-efficients for mesons and baryons repectively, which are appropriate difference of the product of charges and spin

Table 1. The magnetic co-eflicients of the mesons.

Particle Intermediate state Co-efficients

P ( q t q2 ) N C es

T + ( t d ) T + 0

T *+ - - ~ 2

T°(frO T O 0

T , O 4_

3

T * + ( t ~ T * + 4_

9

T + _2_

9

T,O(t~) T , o _ s_

9

T O 4

9

Table 2. The magnetic co-efticients of baryons.

Particle Intermediate state Co-efficients C eN _ap

P(ql q2q3) N qx q2 q• q3 q2q3

T + + (ttu) T ~- + 1 6 8 $

u u 27 27 27

T * + + s 16 16

27 27 27

T; (ttd) T;

16 4 4

2-5 T~ 2-3

~ . . + 8 8 8

JI d 2-5 ~~ 2-5

T ~ +(tuu ) T ? + 2 7 8 2 7 8 27 16

T , + + 16 16 8

1 27 27 27

T ~ (tud) T ~ s 4 s

27 ~ 27

T , + 16 s 4

27 27 27

T . tdd) T o T~ Y~ Y4

T , O s_ s_ 2__

27 27 27

T* ÷ + (ttu) T* + + 2~ 20 20

u 27 27 27

T + + ± s 8

u 27 27 27

T~

^{+ (ttd)}

T~

⁺ 2720 271 O 2710

T+ 4 4 4

~ ~ 2~

T* + + (tuu) T* + + 2o 2o 2o

--1 27 27 2"/

T + + s s

27 27 27

r ~ + ( t u d ) T * + 2720 271 O 271 O

T + s __4 2

27 27 27

T~O(tdd) T~O

^lO 1o s

27 27

0 4 4 1

T j 2~ 2~ 2~

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Bishwaranjan Dash et al

operators. I m p o r t a n t parameters like b a g radius and q u a r k eigenvalues x,, ct = u, d, t for each of the hadrons are given in table 3. We have calculated numerically the integrals l,p(R) and J~p(AE, R) for various t-flavored hadrons taking m, = ma = 0 and m t = 150 GeV. The results are given in tables 4, 5 and 6. The consequent values of (i) (AM)el (ii) (AM),,,, are given in table 7. Since an exact evaluation of (AM)q part of eqn. (1) in the framework of bag model has not been done as of yet, we m a k e the usual assumption of a constant up and down q u a r k mass difference and determine the same from m o + - moo. Thus we work with (Am)q = - 2.06 + 0.28 MeV determined

Table 3. Radius R and quark eigenvalues x~, (a = u, d, t) for each of the hadrons.

R

Particle (in GeV- 1) x, x, = xd

T + + 4-5045 3.1392827 2.04

T f 4.5045 3.1392827 2'04

T *+ + u 4.5096 3.1392853 2'04

T* ÷ 4.5096 3' 1392853 2-04

T~ + 5.2826 3.1396230 2.04

T~ 5.2826 3.1396230 2.04

T O 5.2826 3.1396230 2.04

T *+ + 5.2868 3.1396246 2.04 T* + 5.2868 3.1396246 2"04

T *° 5.2868 3.1396246 2.04

T O 4.4944 3' 1392775 2.04

T + 4"4944 3.1392775 2.04

T *° 4.5045 3.1392827 2.04

T* + 4-5045 3.1392827 2"04

Table 4. Evaluation ofelectric and magnetic integrals of mesons. The states within parantheses in column 3 denote the intermediate states. The integrals are defined in eqs (6), (12) and (13).

l~p + I~ J,p + Jp~

Particle (in MeV) (in MeV × 10 -3)

T 2.3665 1.0075

(r) 1-7317

(T*)

T* 2.3610 1.0030

(r*) 0'7770

(r)

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Electromagnetic mass differences of t-flavored hadrons

Table 5. Electric interaction integrals I~p + lp~ for baryons.

I~(R) + l~(R)

Integrals (in MeV)

Particle q 1 q2 q I q3 q2 q3

T + + _u 2"9680 2"3610 2"3610

Td + 2"9680 2'3610 2"3610

T~" + 2'0137 2"0137 1'7560

T~" 2"0137 2"0137 1"7560

T O 2"0137 2"0137 1"7560

T* ~ + 2"9640 2"3583 2"3583

7"* ÷ 2"9640 2"3583 2"3583

T* + + 2"0123 2"0123 1'7548

1

T~ + 2"0123 2"0123 1"7548

T *° 2"0123 2"0123 1'7548

1

Table 6. M a g n e t i c interaction integrals J~p(AE, R) + JB~(AE, R) for baryons.

Particle

P(qlq2q3)

I n t e r m e d i a t e J,p(AE, R) + Jp~(AE, R)

states Integrals (in MeV)

N(qlq2q3) qlq2 qlq3 q2q3

T ++ T ++ 0'5114 x 10 -'5 1"003 x 10 -3 1"003 × 10 - 3

u u

T * ÷ + 0"5602 x 10 -5 1"135 x 10 -3 1'135 x 10 -3

T~" T~ 0"5114 x 10 - 5 1"003 x 10 - 3 1"003 x 10 - 3

T~ '+ 0"5602 x 10 -5 1"135 × 10 - 3 1"135 x 10 - 3

T~" + T 1 + 0"7295 × 10 - 3 0'7295 × 10 - 3 0'1867

T~ ++ 0"7971 X 10 -3 0"7971 X 10 -3 0"1952

T~ T + 0"7295 x 10 - 3 0"7295 × 10 -3 0"1867

TI *+ 0"7971 x 10 -3 0"7971 × 10 - 3 0"1952

T O T O 0"7295 x 10 - 3 0"7295 x 10 -3 0"1867

T *° 0"7971 x 10 - 3 0"7971 × 10 - 3 0"1952

T *+ T~ + 0"5096 x 10 -5 1'001 × 10 - 3 1"001 × 10 - 3

Td + 0"5326 x 10 -5 1"086 x 10 - 3 1"086 x 10 -3 T *++ _u T *+4 _u 0"5096 x 10 - s 1"001 × 10 -3 1"001 × 10 - 3

T ++ _u 0'5326 x 10 -5 1'086 × 10 - 3 1"086 x 10 -3

T~ '+ + T *++ 0"7283 x 10 - 3 0"7283 x 10 -3 0"1866

T~ "+ 0"7797 x 10 - 3 0"7797 x 10 - 3 0"1903

T~ + T *+ 0"7283 × 10 -3 0"7283 × 10 - 3 0"1866

T~" 0"7797 x 10 -3 0"7797 × 10 -3 0"1903

T *° T~ '° 0"7283 × 10 - 3 0"7283 × 10 -3 0"1866

T o 0"7797 x 10 - 3 0"7797 x 10 - 3 0"1903

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Bishwaranjan Dash et al

Table 7. Predicted mass difference for t-flavored hadrons.

(AM)g =

(AM)el (AM)raag

rrlu -- m,, (AM)to,,I Mass difference (in MeV) (in MeV) (in MeV) (in MeV) T +. + - T~ + 3"1.48 0.0029 - 2'06 _+ 0"28 1"0909 _+ 0"28 T~ + - T ~ 2"513 -0"2517 -2"06+0"28 0"2013+0"28 T~ - T O 0"7571 0"1274 - 2"06 + 0"28 - 1'1755 + 0"28 T * + + - T * + . 3"1444 -0"0013 -2"06+0"28_ 1"0831+0"28_

T~ + + - T* + 2'5114 - 0"2498 - 2-06 + 0 " 2 8 0-2016 _ 0"28 T~ + - T~ ° 0'7566 0"1250 - 2'06 +__ 0"28 - 1"1784 ___ 0"28 T + - T O 1"5776 0"0035 - 2"06 ___ 0"28 3"6411 _ 0"28 T* ÷ - T *° 1"574 - 0"0008 - 2"06 + 0 " 2 8 3"6332 + 0"28

from m o + - redo, the error coming from the error in D-mass difference [5]. G o o d agreement of electromagnetic mass difference for b-mesons [5], [12] based on this value of (AM)q encourages us to stick to it as a good working hypothesis also for the t-sector. Combining now three pieces of contributions (AM)e~, (AM)mos and (AM),, we estimate mass differences of hadrons and have presented it in the last column of table 7.

4. Discussion

Keeping in view a possible t-quark discovery in piO collider run at Fermilab, we have extended our previous work on t-flavored hadron mass calculation to estimate electromagnetic mass differences for t-flavored mesons of spin 0 and 1 and t-flavored b a r y o n s

1 and 3 carrying spin i 3"

The M I T bagrnodel and its modification have been used to estimate the electric and magnetic interactions of the quarks inside the hadron. The general feature of the magnetic interaction contribution staying small compared with the electric interaction is also persistent for the t-flavored hadrons. The up and down q u a r k mass difference, however, has not lent itself to a bag model calculation so far. A parametric expression such as (AM)~ = ( A / R ) + Bn s + Cn c where ns and nc are the n u m b e r of strange and charm quarks were used earlier [9]. The predictions of electromagnetic mass differences on two extreme parametrization like C = B and C = ( m d m , ) B with A and B obtained by fitting with known electromagnetic mass differences does not produce reasonable agreement with the experimental values for charmed mesons in either case [9]. Thus we have carried the assumption of a constant (AM)~ = - 2.06 + 0.28 MeV, that yielded reasonable agreement for b-flavored mesons to the t-quark sector.

The degree of agreement of the results presented here, we believe, will provide a test of the model used and shed light on the nature of confinement of heavy quarks as well.

Acknowledgements

We gratefully acknowledge some useful discussions with Prof. N Batik and D r A R Panda.

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