New relativistic Coulomb-like threshold resummation S- and L-factors in quantum chromodynamics are obtained. Consideration is performed within the framework of quasipotential approach in quantum ﬁeld theory formulated in the relativistic conﬁgurational representation in the case of two particles of unequal masses

Chernichenko, Yu. D.

BSU Digital Library

Nonlinear Phenomena in Complex Systems, vol. 17, no. 4 (2014), pp. 364 - 369
On a Physical Relativization of the Strong Interaction
within the Relativistic Quasipotential Approach
Yu. D. Chernichenko∗
International Center for Advanced Studies, Gomel State Technical University, BELARUS
(Received 05 September, 2014)
New relativistic Coulomb-like threshold resummation S- and L-factors in quantum
chromodynamics are obtained. Consideration is performed within the framework of
quasipotential approach in quantum field theory formulated in the relativistic configurational
representation in the case of two particles of unequal masses.
PACS numbers: 11.90.+t, 12.38. Lg, 12.40.-y, 14.65.-q
Keywords: quantum chromodynamics, quasipotential approach, relativistic threshold resummation
1. Introduction
In two-particle approximation the square
of the Bethe–Salpeter (BS) amplitude of two
charged particles χBS(x) at x = (r, τ) = 0
and hence at the relative time τ = 0 is an
important quantity for qq¯ systems. For example,
it appears in the expressions for the Drell ratio
R(s) [1], for leptonic widths Γ(e+e−) for 1−
states [2–8]. By considering the total cross section
for the production of fermion-antifermion (or
quark-antiquark) pairs in e+e− annihilation in
the kinematic region close to the threshold,
we can not cut oﬀ the perturbative series in
powers of the ﬁne structure constant α (i.e.,
in the number of loops), even if the expansion
parameter α is small [9]. The problem is well-
known from QED [10]. This can be seen by
considering the contributions of magnetic and
electric form factors to the total cross section
for the production of fermion-antifermion (or
quark-antiquark) pairs in e+e− annihilation in
the kinematic region close to the threshold s =
4m2 [11]. The reason consists in that the real
expansion parameter in the threshold region is
α/v where v =
√
1− 4m2/s is a quark velocity,
and m is a quark mass. Obviously, it becomes to
be singular, when the velocity v → 0. To obtain
meaningful result these threshold singularities of
∗E-mail: chern@gstu.by
the form (α/v)n have to be summarized. In the
nonrelativistic case for the Coulomb interaction
V (r) = −α/r (1)
this resummation is realized by the known S-
factor Gamov–Sommerfeld–Sakharov [12–14]
Snr =
Xnr
1− exp(−Xnr) , Xnr =
πα
vnr
. (2)
Here 2vnr is the relative velocity of two
nonrelativistic particles. In the relativistic
theory the nonrelativistic approximation needs
to be modiﬁed. For the ﬁrst time the relativistic
modiﬁcation of the S-factor (2) in QCD in
the case of two particles of equal masses
(m1 = m2 = m) was performed in [15] (see
also [16]) and it consisted in the change vnr → v.
Just the same form of the S-factor for the
interaction of two particles of equal masses was
later suggested in [11]. Another form of the
relativistic generalization of the S-factor also in
the case of two particles of equal masses was
obtained in [17]. The relativistic S-factor for two
particles of arbitrary masses (m1 6= m2) was
presented in [18]. The new method to relativistic
generalization of the S-factor in the case of
two particles of equal masses was developed
in [19]. Their method is based on the relativistic
quasipotential (RQP) approach [20] in the form
suggested in [21]. In the method developed
by them, the possibility of transformation
364
On a Physical Relativization of the Strong Interaction within the Relativistic . . . 365
of quasipotential (QP) equation from the
momentum space into relativistic conﬁgurational
representation in the case of two particles of
equal masses [22] has also been used. Moreover,
it is important that the potential (1) they used
possesses the QCD-like behaviour [23]. This
approach gives the following expression for the
relativistic S-factor:
S(χ) =
X(χ)
1− exp [−X(χ)] (3)
where
X(χ) =
πα
sinhχ
,
and χ is the rapidity related to the total
c. m. energy of interacting particles,
√
s, by
2m coshχ =
√
s. The function X(χ) in Eq. (3)
can be expressed in terms of v as X(χ) =
πα
√
1− v2/v. The method proposed by them
in [19] has been generalized in [24] successfully
to get the following expression for the relativistic
L-factor (ℓ ≥ 1) in the case of m1 = m2 = m:
L(χ) =
ℓ∏
n=1
[
1 +
(
α
2n sinhχ
)2]
S(χ). (4)
We should like to remind that the QP wave
functions in the momentum space, Ψq(p), and
in the relativistic conﬁgurational representation,
ψq(ρ), are deﬁned by the relation
χBS(x = 0) =
1
(2π)3
∫
dΩpΨq(p) = ψq(ρ)|ρ=i
where dΩp = (mdp)/Ep is the relativistic three-
dimensional volume element in Lobachevsky
space realized on the hyperboloid E2p − p2 = m2.
Our aim here is to generalize the method
proposed in [19] to obtain the relativistic L-
factor (ℓ ≥ 0) in the case of two particles of
arbitrary masses m1 and m2. Consideration is
performed on the basis of covariant Hamiltonian
formulation of quantum ﬁeld theory [21, 25] by
transition to the three-dimensional relativistic
conﬁgurational representation for the case of
interaction of two relativistic spinless particles
having arbitrary masses m1,m2 [26, 27].
2. Relativistic threshold
resummation S- and L-factors
The basis of our consideration is the integral
form of the relativistic Schro¨dinger equation
in the conﬁgurational representation with the
quasipotential V (r;Eq′) (we use the system of
units c = ~ = 1)
1
(2π)3
∫
dΩp′
(
2Eq′ − 2Ep′
)
ξ(p′, r) (5)
×
∫
dr′ξ∗(p′, r′)ψq′(r
′) =
2µ
m′
V (r;Eq′)ψq′(r)
where µ = m1m2/(m1 + m2) is the usual
reduced mass, and dΩp′ = m
′dp′/Ep′ is the
relativistic three-dimensional volume element in
Lobachevsky space, which is realized on the upper
half of the mass hyperboloid E2p′−p′2 = m′2 of an
eﬀective relativistic particle having the massm′ =√
m1m2 and relative 3-momentum p
′, emerging
instead of the system of two particles and carrying
the total c. m. energy of the interacting particles,√
s, is proportional to the energy Ep′ of one
eﬀective relativistic particle of mass m′ [26, 27]:√
s =
√
m12 + p′2 +
√
m22 + p′2 = (m
′/µ)Ep′ .
The proper Lorentz transformations means a
translation in Lobachevsky space. The role of the
plane waves corresponding to these translations
are played by the functions
ξ(p′, r) =
(
Ep′ − p′ · n
m′
)−1−irm′
. (6)
By using the expansions
ξ(p,ρ) =
∞∑
ℓ=0
(2ℓ+ 1)iℓpℓ(ρ, coshχp)Pℓ
(
p · ρ
pρ
)
,
ψq(ρ) =
∞∑
ℓ=0
(2ℓ+ 1)iℓ
ϕℓ(ρ, χq)
ρ
Pℓ
(
q · ρ
qρ
)
,
Nonlinear Phenomena in Complex Systems Vol. 17, no. 4, 2014
366 Yu. D. Chernichenko
and also formula [22]
pℓ(ρ, coshχ) =
(−1)ℓ(sinhχ)ℓ
ρ(ℓ+1)
×
(
d
d coshχ
)ℓ( sin ρχ
sinhχ
)
, (−ρ)(ℓ) = iℓΓ(ℓ+ iρ)
Γ(iρ)
,
Eq. (5) can be transformed to the form
2
π
∞∫
0
dχ′
(sinhχ′)2ℓ+2(−1)ℓ+1
ρ(ℓ+1)
(
2 coshχ− 2 coshχ′)( d
d coshχ′
)ℓ( sin ρχ′
sinhχ′
)
×
(
d
d coshχ′
)ℓ 1
sinhχ′
∞∫
0
dρ′
ρ′ sin ρ′χ′
(−ρ′)(ℓ+1)ϕℓ(ρ
′, χ) =
V (ρ;Eq)ϕℓ(ρ, χ)
ρ
where we introduced the notation:
p
′ = m′p,p = sinh(χ′)np, |np| = 1,ρ′ = m′r′,
ρ′ = |ρ′|, r′ = |r′|, dr′ = m′−3dρ′,
dΩp′ = m
′3dΩp, dΩp =
dp
Ep
, Ep′ = m
′Ep,
Ep =
√
1 + p2 = coshχ′, V (r;Eq′) = m
′V (ρ;Eq),
Eq′ = m
′Eq, Eq =
√
1 + q2 = coshχ,
ξ(p,ρ) = (Ep − p · n)−1−iρ , ψq′(r) = ψq(ρ).
We will seek for a solution of RQP equation (2)
with the potential (1) in the form
ϕℓ(ρ, χ) =
(−ρ)(ℓ+1)
ρ
α+∫
α−
dζeiρζRℓ(ζ, χ) (7)
where ζ-integration is performed in the complex
plane over a contour with end points α± =
−R ± iε, R → +∞, ε → +0 (see Fig. 1). The
resulting solution Eq. (2) at arbitrary ℓ ≥ 0 can
be represented in terms of the hypergeometrical
function as
ϕℓ(ρ, χ)=Nℓ(χ)(−ρ)(ℓ+1)eiρχ+iAχ+iπ(ℓ+1) (8)
×F (ℓ+ 1− iA, ℓ+ 1− iρ; 2ℓ+ 2; 1− e−2χ)
- +4 i
- -4 i
- +2 i
- -2 i
-4 i
-2 i
+4 i
+2 i
-
-plane
FIG. 1. Contour of integration in Eq. (7) and
singularities of the function Rℓ(ζ, χ)in the complex ζ-
plane.
where A = αµ/(m′ sinhχ) and constant of
normalization Nℓ(χ) is found from the condition
lim
α→0
ϕℓ(ρ, χ) (9)
= ρpℓ(ρ, coshχ) −−−→
ρ→∞
sin(ρχ− πℓ/2)
sinhχ
.
Нелинейные явления в сложных системах Т. 17, № 4, 2014
On a Physical Relativization of the Strong Interaction within the Relativistic . . . 367
The relativistic L-factor is connected with the
RQP partial wave function ϕℓ(ρ, χ) as follows:
Luneq(χ) (10)
= lim
ρ→i
∣∣∣∣∣ Γ(2ℓ+ 2)(2 sinhχ)ℓΓ2(ℓ+ 1)(∆∗)ℓ
[
ϕℓ(ρ , χ)
ρ
] ∣∣∣∣∣
2
,
∆∗ =
1
i
[
exp
(
i
∂
∂ρ
)
− 1
]
.
By using Eqs. (8–10), we ﬁnally ﬁnd the following
expression for the relativistic L-factor:
Luneq(χ) (11)
=
ℓ∏
n=1
[
1 +
(
αµ
m′n sinhχ
)2]
Suneq(χ),
Suneq(χ) = lim
ρ→i
∣∣∣∣ϕ0(ρ, χ)ρ
∣∣∣∣
2
(12)
=
Xuneq(χ)
1− exp [−Xuneq(χ)] , Xuneq(χ) =
2παµ
m′ sinhχ
.
The function sinhχ in Eqs. (11) and (12) can be
expressed in terms of the relative velocity of an
eﬀective relativistic particle with mass m′, u′rel,
determined by the relation
u′rel =
2u√
1− u2 , (13)
u =
√
1− 4m
′2
s− (m1 −m2)2 ,
in the form sinhχ = µu′rel/m
′.
Thus, in terms of relative velocity of an
eﬀective relativistic particle (13) the factors (11)
and (12) are given by expressions
Luneq(u
′
rel)=
ℓ∏
n=1
[
1+
(
α
nu′rel
)2]
Suneq(u
′
rel), (14)
Suneq(u
′
rel) =
Xuneq(u
′
rel)
1− exp [−Xuneq(u′rel)] , (15)
Xuneq(u
′
rel) =
2πα
u′rel
.
The S-factor in Eq. (15) only formally has the
same form, as the nonrelativistic S-factor (2).
However, the S-factor in Eq. (15) has an obviously
relativistic nature since as the argument r in the
Coulomb potential (1) and the relative velocity of
(13), both are relativistic invariants.
The relativistic factors (11) and (12) [or (14),
(15)] have the following important properties:
• In the nonrelativistic limit, u ≪ 1, they
reproduce the well-known nonrelativistic result.
• In the relativistic limit, u→ 1, the factors
(11) and (12) [or (14), (15)] go to unity.
• In the case of equal masses they coincide
with S-factor (3) and L-factor (4).
• In the ultrarelativistic limit, as it
was argued in [28, 29], the bound state
spectrum vanishes since m′ → 0. This feature
reﬂects an essential diﬀerence between potential
models and quantum ﬁeld theory where an
additional dimensional parameter appears. One
can conclude that within a potential model, the S-
and L-factors which correspond to the continuous
spectrum should go to unity in the limit m′ → 0.
Thus, in contrast to the nonrelativistic case, the
relativistic factors (11) and (12) [or (14), (15)]
reproduce both the known nonrelativistic and the
expected ultrarelativistic limits.
To illustrate the diﬀerences between
the nonrelativistic S-factor (2) and the new
relativistic S-factor in Eq. (15) in more detail,
in Fig. 2 we plot the behavior of these factors as
functions of u at diﬀerent values of the parameter
α (the numbers at the curves). From this ﬁgure
one can see that in the region of nonrelativistic
values of u, u ≤ 0.2 where the inﬂuence of the S-
factor is high, the diﬀerence between (2) and (15)
is practically absent. However, when α increases,
the expression (2) gives a less suitable result in
the region of large values u, in particular, as
u→ 1.
Thus, the above performed analysis
demonstrates that the factor in Eq. (15)
coincides in form with the factor (2). However,
the relative velocity of an eﬀective relativistic
particle (13) emerging instead of the system
of two particles, now plays the role of the
Nonlinear Phenomena in Complex Systems Vol. 17, no. 4, 2014
368 Yu. D. Chernichenko
0.0 0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
6
 
 
0.1
0.2
0.4
u
S
FIG. 2. Behavior of the S-factor at different values
of the parameter α (the numbers at the curves). The
solid lines correspond to the new relativistic S-factor
(15) and the dashed lines to the nonrelativistic S-
factor (2).
parameter of velocity, but not the relativistic
relative velocity of interacting particles, v.
These new relativistic threshold S- and L-factors
could have a signiﬁcant impact in interpreting
strong-interaction physics.
3. Conclusion
The new relativistic threshold resummation
factors (14) and (15) for the interaction of
two relativistic particles of unequal masses
were obtained. To reach this aim the
relativistic quasipotential equation in relativistic
conﬁguration representation [26] with the
Coulomb potential for the interaction of two
relativistic particles of unequal masses was used.
The new relativistic factors obtained here
reproduce both the known nonrelativistic and
expected ultrarelativistic limits and correspond
to the QCD-like Coulomb potential. The new S-
factor coincides in form with the nonrelativistic
(2); however, the role of the parameter of velocity
is played not by the relative velocity of interacting
particles, v, but by the relative velocity (13) of
an eﬀective relativistic particle emerging instead
of the system of two particles.
It was shown that there is a diﬀerence (see
Fig. 2) between the expression (15) obtained here
and the nonrelativistic (2). As the new relativistic
factors (14) and (15) were obtained within the
framework of completely covariant method, one
can expect that these factors takes into account
more adequately relativistic nature of interaction.
Acknowledgment
The authors would like to thank Profs. Yu.
S. Vernov and R. N. Faustov for interest in this
work and valuable discussions, as well as Yu. A.
Kurochkin, A. E. Dorokhov, I. S. Satsunkevich,
V. V. Skalozub and A. V. Kiselev for their
commentary, discussion of obtained result and
debates.
This work has been support in part by the
International Program of Cooperation between
Republic of Belarus and JINR, as well as the State
Program of Scientiﬁc Research “Konvergentsiya”.
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Nonlinear Phenomena in Complex Systems Vol. 17, no. 4, 2014


On a Physical Relativization of the Strong Interaction within the Relativistic Quasipotential Approach

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On a Physical Relativization of the Strong Interaction within the Relativistic Quasipotential Approach

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