A model of mesons in finite extra-dimension

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by  Jugal Lahkar

A model of mesons in finite extra-dimension
Jugal Lahkar1,a, D. K. Choudhury1,2,,S.Roy3and N.S. Bordoloi4
1Department of Physics, Gauhati University, Guwahati-781 014, India.
2Center of theoretical studies,Pandu College,Guwahati-,India.
3Karimganj College,Karimganj-788710,Assam,India.
4Department of Physics,Cotton College,Guwahati-781001,India.
aemail:[email protected]
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1
0
2 Abstract

n
a Recently,problemofstabilityofH-atomhasbeenreportedinextra-finitedimension,and
J

4 found out that it is stable in extra-finite dimension of size,R a0,where,a is the Bohr
2 ≤ 4 0

radius.Assuming that,the heavy flavoured mesons have also such stability controlled by
]
h
p the scale of coupling constant,we obtain corresponding QCD Bohr radius and it is found
-
p
to be well within the present theoretical and experimental limit of higher dimension.We
e
h
[ then study its consequences in their masses using effective string inspired potential model

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in higher dimension pursued by us.Within the uncertainty of masses of known Heavy
v
7
Flavoured mesons the allowed range of extra dimension is L 10 16m,which is well
3 −
≤
7
6 below the present theoretical and experimental limit,and far above the Planck length
0
. 1.5 10 35 m.
1 −
≃ ×
0
7
1
: Keywords: Luscher term,compact extra dimension,QCD.
v
i
X PACS Nos. :12.39.Jh,03.65.Ge,12.39.Pn.
r
a
1 Introduction
There has been considerable discussion on the possibility of extra spatial dimension since long
ago.The idea was born in 1920’s when Kaluza and Klein[1] for the first time introduced one
additional dimension.Later different theories such as ADD[2],RS[3],UED[4]etc were introduced
supporting the idea.In recent years,equally interesting research is going on what happens to the
ordinary H-atom,when there is more than 3 dimension.The problem of stability of H-atom in a
compact space was a subject of research in[6]by Bures and Seigl.They reported that H-atom is
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stable in extra-finite dimension of size,R a0,where,a is the Bohr radius.Assuming that,the
≤ 4 0
heavy flavoured mesons have also such stability controlled by the scale of coupling constant,we
obtain corresponding QCD Bohr radius,RQCD ≤ a0|Q4CD,where a0|QCD = 16µ3αs,αs is the strong
coupling constant and µ is the reduced mass of mesons.
Recently, a string inspired potential model[5] of mesons has been reported in higher dimen-
sion of infinite extent,assumed by us. In this work we assume the stability of mesons within
their QCD Bohr radii in extra-dimension,and we study the masses of heavy flavoured mesons
within this model. The additional assumption is that the modification of Coulomb potential
in finite extra-dimension in a plausible way. Plausible relationship between confinement effect
and modified coulomb potential in finite extra-dimension is also suggested.
In section.2 we outline the formalism,in section.3 we calculate the masses of Heavy Flavoured
mesons in this model without confinement,while section.4 is devoted to numerical analysis.The
last section.5 includes conclusion and discussion.
2 Formalism:
2.1 The string inspired potential model and its limitations:
Recently a String inspired potential model[5] is reported which assumes Luscher term[9],[5] to
be the higher dimensional correction to Cornell potential[ 5],[7],
γ
v(r) = − +br +c (1)
r
where b is standard confinement parameter and
π(d 2)
γ = − (2)
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At d=3,
4α
s
γ = = α = 0.0981 (3)
s
3 ⇒
2
This coupling constant does not correspond to its numerical value for d=3.Following MS-bar
scheme using standard equation of coupling constant,
4π
α (Q2) = (4)
s (11 2nf)(ln Q2 )
− 3 Λ2(QCD)
The value given in equation(3) is much smaller than the corresponding value α = 0.39 at c-
s
scale and α = 0.22 at b-scale.Forn = 6,Λ = 0.382GeV,the value of α(Q)2 of equation(3)
s f QCD
is reached at Q2 = 1.2 1013GeV2.Further for n = 16,Λ = 0.382GeV we get theoretically
f QCD
×
allowed highest value of flavour Q2 5.7 10(20)Gev2.So higher dimensional extension of
≃ ×
Coulomb term does not correspond to standard QCD in present energy regime and hence ruled
out.
2.2 Improved potential model in finite extra-dimension:
We consider the known 3 dimensions to be in the range 0 to α and the extra dimension to be
finite within the range 0 to L[3,4,6]. Thus,
r2 = r2 +r2 +r2 +y2 (5)
d 1 2 3
= r2 +y2 (6)
where r2 = r2 +r2 +r2 , y is the size of finite extra dimension.Hence,
1 2 3
y2
r r + (7)
d
≃ 2r
Now,we consider the Coulomb potential in d-dimension,
A
V(r ) = (8)
d
−r
d
where,A = 4αs,at d=3 And,with finite extra-dimension we modified it to,
3
4α 4α
s se µLy (9)
−
3 −→ 3
3
4α
s
= (1 µ y) (10)
L
3 −
for µ y 1. Atd = 3,y = 0,and we get back the standard 3-dimensional QCD coupling
L
≪
constant.The mass parameter (µ ) occurred in equation(10)will be determined from the corre-
L
spondence with the H-atom [13].
2.3 Wave function(D-Dimensional,with only Coulomb term in finite
extra dimension):
The D-dimensional Schrodinger equation is[10],[14],
d2 d 1 d l(l +d 2) 2µ
L
[ + − − + (E V )]R(r ) = 0 (11)
dr2 r dr − r2 ~2 − 0 d
d d
for l=0,taking~ = 1,we get
d 1 A
R¨(r )+ − R˙(r )+2µ (E + )R(r ) = 0 (12)
d d L d
r r
d d
Let,R(rd) = F(rd)e−µLArd[10],[15], Now putting R(rd) in equation (12) we get,
d 1 d 1 2µ A
F¨(r )+( − 2µ A)F˙(r )+(µ2A2 − µ A+2µ E + L )F(r ) = 0 (13)
d r − L d L − r L L r d
d d d
Now,we consider theseries expansion of F(r ) as, F(r ) = α a rnf(r ,d), such that f(r ) =
d d n=0 n d d d
P
σ(d−3)
1 at d = 3.Let us consider,f(r ) = r [15],which satisfies this condition.Then the radial wave
d 2
function can be expressed as, R(rd) = αn=0anrdn+σ(d2−3)e−µArd. For ground state,n = 0,we get
P
the unperturbed wave function,
ψ(rd) = Nd(r2 +y2)σ(d2−3)e−µLA(r+y2r2) (14)
Now,at d = 3,y = 0 and we get from above equation(14),
ψ(r) = Ne µLAr (15)
−
4
which is consistent with standard H-atom wave function[13] at d=3,with A = 1 .Also,for the
µα
consistency of the wave function with H-atom wave function µ corresponds to reduced mass
L
µ of H-atom.
2.4 Normalization: with 3 non-compact and one compact extra di-
mension
The normalization condition[16] is,
α L
Z Z dCd(r2 +y2)σ(d2−3)|ψ(r,y)|2drdy = 1 (16)
0 0
d
where,C = (π)2 . The normalization constant obtained from the above equation has got
d Γ(d+1)
2
singularity at d = 3,but for d = 3,L = 0,and it is free of singularity. At d = 3,in analogy with
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H-atom[13],N =(µ3α3)2,standard QCD normalization constant is,
H π
1
(µ4α )3 2
N = [ 3 s ] (17)
s
π
Now,putting equation (14) in equation (16) we get(neglecting higher orders of L),
(2µA)(2σ+3)
2 1
Nd = [dC ( π )0.5Γ(2σ + 3)]2 (18)
d µA 2
At d=3,A = 4αs,putting (17) in (18) we get,
3
[(µ43αs)3]21 = [ 22σ+1.5 ][(4α µ)2σ+2]12 (19)
π 4π2Γ(2σ + 3) 3 s
2
. To determine σ we need specific values of µ = mQmQ and α . Unlike H atom reduced
mQ+mQ s 2
masses of mesons are flavour dependent and coupling constant is Q2 dependent. Hence,the
correct wave function in compact extra dimension with only Coulomb term in the potential,
ψ(rd) = Nd(rd)σf(d2−3)e−µA(r+y22r) (20)
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Thisisanimprovement overtheearlierresult[5],[7],[10],whereσ isassumedtobe1.Todetermine
σ we need specific numerical values of µ and α .Unlike H atom reduced mass of mesons are
s
flavour dependent and coupling constant is Q2 dependent.The values of σ for some Heavy
flavoured mesons are shown in table(3).
Table 1: Variation of σ with reduced mass µ
µ(Gev) Meson σ
0.2099 D(cs) 0.071
0.23727 B(sb) 0.083
1.02254 B(bc) 0.3489
2.33 τ(bb) 0.796
3 Masses of heavy flavoured mesons in compact extra-
dimension without cofinement:
3.0.1 Mass only with coulomb term in compact extra dimension:
As a application of the formalism developed in 2, we calculate the masses of Heavy flavoured
mesons. Pseudoscalar meson mass can be computed from the following relation[12],[17]:
M = m +m +∆E (21)
p Q Q
where,∆E = H . In D-spatial dimension,the Hamiltonian operator H has the form[5],[18]:
h i
2
H = ∇d +V(r ) (22)
d
−2µ
where,µ = is the reduced mass of the meson with m and m are the quark and anti-quark
Q Q
masses;V(r ) corresponds to the coulomb plus confinement term with plausible modification to
d
compact extra dimension given as
A
V(r ) = +br (23)
d d
−r
d
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In this work we first neglect the confinement term and see the effect of coulomb term with
modification in compact extra dimension,and 2 is the Laplace’s operator in D dimension[19],
∇d
which at l = 0 is given by,
d2 d 1 d
2 + − (24)
∇d ≡ dr2 r dr
Now, H can be expressed as(with only Coulomb term in the potential in compact extra di-
h i
mension),
2 A
H = ∇d + − (25)
h i h−2µi h r i
d
2 1 δ2 A
H = ∇3 + + = H + H + H (26)
h i h−2µi h−2µδy2i h−r i h 1i h 2i h 3i
d
Where first two terms are the kinematic contribution and third term corresponds to poten-
tial contribution.Now putting correct D-dimensional wave function as given in(20) and using
coupling constant from equation (10) in equation (26) we get,
2 9 µπ 2
H = ∇3 = ( ) (27)
h 1i h−2µi 32µ 12
For the only compact extra dimension,ψ = N e µAy,and with it we get,
d −
1 δ2 N2A
H = = d (1+2µAL) (28)
h 2i h−2µδy2i 4
And the potential term,
A 1 √π Γ(5σ + 1) 1 Γ(5σ)
H = = N2AdC [ 2 + √πL ] (29)
h 3i h−rdi d d 2√µA(2µA)(5σ+12) 2 (2µA)5σ
neglecting higher orders of L,since L is very small.Then we get the final result,
9 µπ 2 N2A 1 √π Γ(5σ + 1) 1 Γ(5σ)
H = H + H + H = ( ) + d (1+2µAL)+N2AdC [ 2 + √πL ]
h i h 1i h 2i h 3i 32µ 12 4 d d 2√µA(2µA)(5σ+12) 2 (2µA)5σ
(30)
We can write,
H = F(µ)+G(µL) (31)
h i
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where,F(µ) iS L independent with F(µ) = 9 (µπ)2and
32µ 12
G(µL) = Nd2A(1+2µAL)+N2AdC [1 √π Γ(5σ+12) + 1√πL Γ(5σ) ],is L dependent.
4 d d 2√µA(2µA)(5σ+21) 2 (2µA)5σ
Since G(µL) is not explicitly a linear function of L,equation(31) shows that variation of mass
with size of extra-dimension is not linear.
4 Results:
4.1 Estimation of QCD Bohr radii of heavy flavoured mesons and
comparision with varius theory and experimental models:
In table(2),we show that QCD Bohr radii of various heavy flavoured quark anti-quark systems
in ’Fermi(fm)’ and thencompare with the experimental andtheoretical limit of extra-dimension
in table(3).As the obtained QCD Bohr radii are always well within the corresponding limits
of table(3),it is therefore,tempting to assume that physical mesons too might be stable within
such QCD Bohr radii and see its consequences in their masses.
Table 2: QCD Bohr radius
Mesons Reduced mass(Gev) a = 3 (m)
0|QCD 16µαs
B0(db) 0.1733 4.9 10 15
−
×
B0(sb) 0.23727 3.59 10 15
s × −
B+(cb) 1.02254 0.83 10 15
c × −
D (cd) 0.15826 3.03 10 15
− −
×
D+(cs) 0.2099 2.29 10 15
s × −
Table 3: Different experimental and theoretical limit on the size of extra dimension
Experiment and Models Limit on the size of extra-
dimension (m)
Fermi-LAT[21] 8 10 9m (LED)
−
×
LEP-I[20] 4.5 10 14m
−
×
ADD [2] 10 3m
−
∼
Martin Bures[6] a0(0.13225 10 10)m
≤ 4 × −
ALEPH,DELPHI,OPAL[20] 6 10 18m
−
∼ ×
RS[3] 2 10 9m
−
×
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4.2 Variation of mass of mesons with size of extra-dimension :
With the expression obtained for H ,we calculate the mass of heavy flavoured meson[5],[12],
h i
B(bc). We take the value of L according to the condition R 3 ,[6]. In table(4),we
′ ′ QCD ≤ 16µαs
show the variation of mass of B(bc) meson with the size of finite extra-dimension,which is
well within the corresponding QCD Bohr radii,0.83fm(table(2)). This table(4) shows that as
size of extra-dimension increases,mass of meson also increases. Further,for L=0,at d=3,the
calculated mass is above the experimental value suggesting that effectively confinement effect
reduces mass of a meson in 3D. We therefore raise the question if such reduction of mass
due to the confinement effect can be obtained equivalently through the assumption that gluon
effect can as well propagate to extra-dimension within the QCD Bohr radii. So,in table(5), we
generalize the effect of confinement through the compact extra dimension of suitable size ,for
B(bc).This table(5) shows that mass of B(bc) meson can be obtained with the assumption of
extra-dimension of size L where the gluon can effectively propagate without any confinement
effect.Similar analysis is done for τ(bb) in table(6),at which experimental mass is obtained
without confinement effect.The input parameters for numerical calculations used are m =
b
4.66Gev,m = 1.31Gev and α values 0.39 and 0.22 for c-scale and b-scale respectively.
c s
Table 4: Variation of mass of B(bc)meson with size of extra-dimension without confinement:
L(fm) m(q)+m(q) M (Gev) Exp.mass(Gev)
P
0.01 5.97 6.4343 6.27 0.006
±
0.02 5.97 6.4474
0.03 5.97 6.4578
0.04 5.97 6.469
0.05 5.97 6.4821
Table 5: Mass of B(bc)meson without confinement in finite extra-dimension:
SizeL(fm) m(q)+m(q) M (Gev)
P
0.005 5.97 6.24
0.006 5,97 6.247
0.007 5.97 6.25
0.008 5.97 6.26
0.009 5.97 6.27
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Table 6: Mass of τ(bb)meson without confinement in finite extra-dimension:
SizeL(fm) m(q)+m(q) M (Gev)
P
0.1 9.32 9.479
0.12 9.32 9.467
0.15 9.32 9.46
0.18 9.32 9.427
0.2 9.32 9.40
5 Conclusion and Discussion:
In this paper the recent result that the H-atom can be stable in finite extra dimension if the size
of extra dimension does not exceed a0[6],where,a is the Bohr radius,has been extended to QCD
4 0
and corresponding QCD Bohr radii for quark-antiquark systems are evaluated. It is found that
the QCD Bohr radii of various Heavy Flavoured mesons are well within the present theoretical
and experimental limit of size of extra-dimension. We then study its consequences in their
masses using an effective string inspired potential model[5] .Within the uncertainty of masses
of known Heavy Flavoured mesons the allowed range of extra dimension is L 10 16m,which
−
≤
is also well below the present theoretical and experimental limit, but far above the Planck
length 1.5 10 35 m.It is interesting to note that the potential suggested by us has got
−
≃ ×
strong resemblance tothecorresponding potential indeconfined phaseinQGP[22]andindicates
plausible relationship between the confinement effect and the coulomb potential in finite extra-
dimension.Hence our result does not contradict the present notion that the extra degrees of
freedom for gluons and quarks in extra-dimension is not yet discovered.
Acknowledgement
Oneoftheauthors(D.K.C)thanksProf.MikeTepper,RudolfPeirelsCenterofTheoreticalPhysics,University
of Oxford, Prof.D.K.Srivastava of VECC,Kolkata,and Prof.P.Poulose ,IIT Guwahati,India for
useful discussions.We thank Dr. Bhaskar Jyoti Hazarika,Centre of theoretical Physics,Pandu
College,Assam,India for collaboration at the initial stage of the work. One of the authors (J.L)
acknowledges CSIR,India for financial support during the research work.
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