Constraints on effective constituent quark masses from phenomenology
D, Jane1 and M, Eosina1,2
1 J. Stefan Institute, P.O. Box 3000, SI-1001 Ljubljana, Slovenia 2 Faculty of Mathematics and Physics, University of Ljubljana
June 13, 2001
From the assumption of a two-particle Hilbert space for mesons and from rather general properties of the effective quark-quark potential we constrain considerably the choice of effective constituent quark masses.
1 Nonrelativistic models
We consider the following form of the Hamiltonian for the quark-antiquark system
p2
II — - V0(r) + (ti • (T-AJjm.iihy.r).
where fi is the reduced mass of the system and mi and m2 are the quark and antiquark masses.
We make rather general assumptions about the potential:
1.	The central potential Vq (r) is flavour independent
2.	The central potential is monotonie function or r and satisfies the conditions for a positive Laplacian and concavity
and -j-7- < 0.
dr ar	arJ
3.	The spin-spin potential Vs satisfies condition that ¡jVa decreases with total mass of both quarks M = m 1 + TO2
4.	The spin-spin potential is monotonie function of r and has positive Laplacian
d_r> g dr dr
c
1950
1900
1850
1800
1750
[MeV]
1700
u > 2 8 0MeV | u > 34 0MeV
300 400 500 600 700
s [MeV]
Figure 1: Allowed mass region for strange arid charmed quarks for different choices of light and bottom quarks.
In the family of potentials which satisfy conditions 1. and 2. one can find the "QCD inspired" Couloriib-plus-liriear potential and power law potential
V0(r) = -- + fir + U0,
r
V0(r) = A + Brf\ while conditions 3. and 4. are satisfied for example by
VH(mi,m2;r) =
a e
r/ro
in i //)•_> r
a,r0 > 0
From this assumptions one can obtain inequalities between quark masses and masses of ground state of psendoscalar and vector mesons, which to some extent restrict masses of constituent quarks as shown in Fig(l).
2 Semirelativistic models
For heavy quark Q - light (heavy) antiquark q psendoscalar mesons we use the semirelativistic model with Haniiltonian
H = y/p? + T„1 + ^p2 + m® + Z2_Z«F(„Iq) + K
m a
where we assumed that the expectation value of F(mQ)V88/mq is monotonic decreasing function of mq and that both Vss and V are flavour (mass) independent.
The Haniiltonian for all vector mesons in our model has the general form H = yV2 + rn'l + ^Jp2 + m\ + \ ~(ni\. )
s [MeV] 500
c	<	190 0MeV
1	<	185 0MeV
1	<	18 0 0MeV
1	<	175 0MeV
1	<	17 0 0MeV
u [MeV]		
260 280 300 320 340
Figure 2: Allowed mass regions for light arid strange quarks for five different choices for mass of charm quark.
where we demanded that the expectation value of \~(ii)\- n>->) is a decreasing function of the quarks masses from where it follows that
E(K*)-E(p) < m, -m, E(D*)-E(K*) < ID, - ID E(B*)-E(D*) < id,, - ID
Using this assumptions we again obtained inequalities between masses of quarks and mesons which allowed us to constrain masses of constituent quarks. In Fig(2) one can se that it is not possible to reproduce correctly the masses of ground state mesons with semirelativistic model if one takes mass of the charmed quark smaller then 1650MeV. Then the mass of bottom quark must be according to upper inequalities always larger then 4970MeV.