Self-energy of Heavy Quark

19 Pages·0.15 MB·

by  M. A. Ivanov

Self-energy of Heavy Quark
M. A. Ivanov and T. Mizutani
∗
5
9
Department of Physics
9
1 Virginia Polytechnic Institute and State University

n
Blacksburg, VA 24061
a
J

4 February 7, 2008
2

1
v
3
6
3
1
0 Abstract
5
9 We demonstrate that to calculate the self-energy of a heavy quark in the heavy
/
h quark limit (or the heavy fermion limit in what is called the Baryon Chiral Pertur-
p bation Theory), the use of standard dimensional regularization provides only the
-
p quantum limit: opposite to the heavy quark (or classical) limit that one wishes to
e obtain. We thus devised a standard ultraviolet/infrared regularization procedure
h
in calculating the one- and two-loop contributions to the heavy quark self-energy
:
v
in this limit. Then the one-loop result is shown to provide the standard classical
i
X Coulomb self-energy of a static colour source that is linearly proportional to the
r untraviolet cutoff Λ. All the two-loop contributions are found to be proportional
a
to Λln(Λ/λ) where λ is the infrared cutoff. Often only the contribution from the
bubble (light quarks, gluon and ghost) insertion to the gluon propagator has been
considered as the O(α ) correction to the Coulomb energy to this order. Our result
s
shows that other contributions are of the same magnitude, thus have to be taken
into account.
∗
Permanent address: Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Re-
search, 141980 Dubna (Moscow region), Russia
0
1 Introduction
Recently, manydiscussions have beendevotedtothedefinitionoftheofheavy quarkmass.
This quantity is the basic parameter in the heavy quark physics. Generally (not only for
the heavy quarks in the heavy quark limit in which we are interested, in the following) one
of the most popular choices within the perturbative scheme is the pole mass. In reference
[1] it has been shown that the pole mass is gauge independent and infrared finite at the
two-loop level. It was checked at the three-loop level in [2] where the relation between
the mass of the modified minimal subtraction scheme and the scheme-independent pole
mass was also obtained. Later authors of work [3] argue that precise definition of the
pole mass in the heavy quark limit may not be given once nonperturbative effects are
included (a more analytical approach leading to the identical conclusion has been given
independently in [4]). To demonstrate that, they adopted the standard effective running
constant (as a typical non-perturbative quantity) obtained from the summation of leading
logarithms to calculate the Coulomb energy of a static (or infinitely heavy) colour source.
This has produced an auxiliary singularity for small momentum (the so-called, infrared
renormalon) which, they claimed, leads to the uncertainty of order Λ /m in the
QCD Q
definition of the pole mass. An important observation which served as one of the bases
to this demonstrative calculation is that the widely used dimensional regularization may
not be used to analize self-energy diagrams in the heavy quark limit, in particular, for
separating the infrared and ultraviolet domains. The best example, as we shall also
present in the following section, is that this regularization method leaves out the linear
divergence that defines the self-energy of a static quark in the lowest order: the Coulomb
energy. Also it should be emphasized here that the Heavy Quark Limit does make sense
only after introducing a scale parameter to the theory with which the quark mass may
be compared, and there is something awkward about it in the dimensional regularization
method.
It may be useful to point out here that a quite analogous situation to the heavy quark
limit can be found in the problem of the classical limit h¯ 0 for the self-energy of an
→
electroninQED[5]. Itwasfoundthatinthelimith¯ 0,ormorepreciselyfortheelectron
→
Compton wave length h¯/mc r , where parameter r regularizes small distances, the
0 0
≪
energy of the Coulomb field by a change of radius r is reproduced in the second order of
0
1
perturabative theory whereas the higher order contributions were shown to vanish in this
limit.
Here we first study the quantum and classical limits for the lowest order contributions
to the quark self-energy and quark-gluon vertex, and show that the standard dimen-
sional regularization method only reproduces the quantum limit. Since one may clearly
identify the classical limit to be equal to the currently fashonable heavy quark limit, we
could conclude that the dimensional regularization method is not suited for studying this
case. Then in the following section we calculate the two-loop contributions to the quark
self-energy in the heavy quark limit, exploiting the conventional ultraviolet and infrared
regularization method. We have found that each contribution to this order is all propor-
tionaltoΛln(Λ/λ)whereΛandλaretheultraviolet andinfraredcutoffs, respectively and
that, unlike in QED, these contributions do not mutually cancel out. We point out that
the gluonic bubble diagrams (Fig.4d) (implicitly) taken often as the majour contribution
in some processes is of the same magnitude as those from vertex and quark self-energy
diagrams (Figs. 4a-4c) for the self-energy in the heavy quark limit. In the last section
some comments will be given regarding the alleged infrared renormalon and the Baryon
(or heavy fermion) Chiral Perturbation Theory.
2 One-loop diagram
The self-energy ofquark with mass m inthe second order of perturbative theory is defined
by the diagram in Fig.1,
d4k 1 1
δm = Σ (m) = C g2 γµ γ (1)
2 2 F s (2π)4i( k2) m k p µ|6p=m
Z − − 6 − 6
where C = (N2 1)/2N , for a gauge group SU(N ). First, we note that this expression
F c − c c
coincides with the electromagnetic self-energy except for the trivial factor C g2. While
F s
the integral has no infrared divergence, it develops the linear ultraviolet divergence. We
are interested in this quantity in the heavy quark limit: m . As was mentioned in [3],
→ ∞
the dimensional regularization cannot serve this purpose. To make this point more clear,
we calculate the quark self-energy by way of introducing an ultraviolet regularization,
2
1 F( k2/Λ2)
− . (2)
( k2) → ( k2)
− −
Here, the function F( k2/Λ2) is assumed to decrease rapidly in the Euclidean region
−
k2 = k2 + providing the ultraviolet convergence of the Feynman integrals. Now,
E − → ∞
the heavy quark limit (or classical limit) means the case where m Λ in comparison
≫
with the quantum limit: m Λ. The both limits may be realized using the Mellin
≪
representation for the form factor F( k2/Λ2) under calculation of the integral in eq. (1).
−
We can write in the Euclidean region
−δ−i∞
∞ 1 dζ
F(k2) = c ( )n(k2)n = c(ζ)(k2)ζ, (3)
E n − E 2i sinπζ E
n=0 Z
X −δ+i∞
where c(n) c for n being positive integers, and 0 < δ < 0.5.
n
≡
Finally, we have
−δ−i∞
1 dζ 1 d4k 4m 2(k+ p)
δm = C g2 c(ζ) − 6 6
2 F s2i sinπζ Λ2ζ (2π)4i( k2)1−ζ(m2 (k +p)2)
−δ+Zi∞ Z − −
−δ−i∞ 1
1 dζ 1 Γ(2 ζ) d4k 4 2(1 α)
= C g2m c(ζ) − dα(1 α)−ζ − −
F s 2i sinπζ Λ2ζ Γ(1 ζ) − (2π)4i(α2m2 k2)2−ζ
−δ+Zi∞ − Z0 Z −
−δ−i∞
3α 1 dζ Γ(1+2ζ) m2 ζΓ(1 ζ)(1+ζ)
s
= C m c(ζ) − . (4)
2π F 2i sinπζ ( ζ) Λ2 Γ(3+ζ)
−δ+Zi∞ − (cid:18) (cid:19)
To get the leading term in m/Λ in the quantum limit, we move the integration counter
to the right and take into account of the first double pole at ζ = 0. Assuming that
F(0) = 1, we have
3α Λ2
δmqu = sC m ln +O(1) . (5)
2 4π F m2
(cid:26) (cid:27)
To get the leading term in Λ/m in the classical limit, we move the integration counter
to the left and take into account the first pole at ζ = 0.5. From the standard relation
−
between the Mellin and its inverse transformations
3
∞
c( 0.5) = 2 dtF(t2),
−
Z
0
we have
∞
α α
δmcl = sC Λc( 0.5) = sC Λ dtF(t2). (6)
2 2π F − π F
Z
0
If we compare theabove result withthe calculationof theexpression inEq.(1) devising
the dimensional regularization,
3α 5 m2
s
δm = C m N + ln (7)
2 4π F ǫ 3 − µ2
(cid:26) (cid:27)
where N = 1/ǫ + ln4π γ, (γ: the Euler-Mascheroni constant) and µ is the subtrac-
ǫ
−
tion point, one can see that it only reproduces the quantum limit Eq.(5) for the quark
self-energy by identifying µ as the ultraviolet cutoff. Thus, the naive dimensional regu-
larization does not allow to see the heavy quark limit.
Ordinarily, studies of heavy quark limit may be carried out by using the expansion of
quark propagator in the inverse quark mass:
M + k/2m
0
S(p+ k) = i 6
6 6 |6p=m [1+i/2m+f(~k2/m2)][k +2mif(~k2/m2)]
4
iM 1
0
= +O( ) (8)
k +iǫ m
4
Here, k = ik , M = (1/2)(I +γ0), and f(x) = (1/2)(√1+x 1). It is readily seen
4 0 0
− −
that the Dirac spinor u(p~) for quark may be chosen as
I
u(p~) =
0
!
so that u¯(p~)u(p~)=u¯(p~)γ0u(p~)=1, and u¯(p~)~γu(p~)=0.
Inthesecondorder: Eq.(1), uponintroducingtheultravioletregulatortotheotherwise
linearly divergent expression, the leading contribution reads
d3~k F(~k2)
δmcl = Σ (m) = 2πα C Λ (9)
2 2 s F (2π)3 ~k2
Z
4
which obviously coincides with the Eq.(6). Onecan see that this expression is the classical
Coulomb energy of a static color source. To give more physical meaning to Eq.(9), one
may introduce the charge density defined as
d3~k
ρ(~r2) = ei~k~r F(~k2) (10)
(2π)3
Z q
supposing that F(~k2) > 0, and the electron (quark) ”radius” r = 1/Λ.
0
Then, we have
g2 1 1
δmcl = s C d~r d~r ρ(~r2) ρ(~r2) (11)
2 2r0 F ZZ 1 2 1 "4π|~r1 −~r2|# 2
in accordance with the classical result.
As was mentioned above, there is no infrared divergence in the second order, but it
plays the crucial role in higher orders in perturbative expansion.
To understand its role, we compute the quark-gluon vertex induced by both quark-
gluon and self-gluon interactions: Figs.2-3. For simplicity we omit all the common coef-
ficients and color factors.
(a) Quark-gluon interaction (diagram in Fig.2):
By introducing the gluon mass λ serving to regularize the infrared singularity we get
d4k 1 γµ(m+ k+ p)γν(m+ k+ p)γ
Λν(p,p) = 6 6 6 6 µ (12)
1 (2π)4iλ2 k2 2 |6p=m
Z − m2 (k +p)2
−
(cid:18) (cid:19)
Using the form factor as in Eq. (2) one obtains
γν 1 2ln Λ2 ln Λ2 +O(1) if Λ m
(4π)2 λ2 − m2 ≫
Λν(p,p) =  (cid:26) (cid:27)
1  γν 1 2ln Λ2 +O(1) if Λ m
(4π)2 λ2 ≪
One can see that the infrared singu(cid:26)larity defines th(cid:27)e behavior of the vertex in the heavy
quark limit. Again result obtained from the dimensional regularization for ultraviolet
divergence corresponds only to the quantum case:
1 Λ2 m2
Λν(p,p) = γν 2ln N +ln +O(1) . (13)
1 (4π)2 λ2 − ǫ µ2
(cid:26) (cid:27)
5
Note that if we do not introduce the infrared cutoff within the dimensional regularization
scheme, we find the auxiliary 1/ǫ pole in the expression for the vertex in Eq.(12) leading
to
1 m2
Λν(p,p) = γν 2NIR NUV +3ln +O(1) .
1 (4π)2 − ǫ − ǫ µ2
(cid:26) (cid:27)
(b) Self-gluon interaction (diagram in Fig.3):
This contribution may be written as
d4k 1 2k2γν 4kν k +2mkν k pγν γν p k
Λν(p,p) = − − 6 − 6 6 − 6 6 (14)
2 (2π)4i(λ2 k2)2 2 |6p=m
Z − m2 (k +p)2
−
(cid:18) (cid:19)
This integral turns out to be free from infrared singularity. Using the ultraviolet form
factor gives
γν 1 3ln Λ2 +O(1) if Λ m
(4π)2 m2 ≫
Λν(p,p) =  (cid:26) (cid:27)
2  O(Λ) if Λ m
m ≪
It is readily seen that this diagram gives no contribution in the heavy quark limit. Again
result obtained from the dimensional regularization for ultraviolet divergence corresponds
only to the quantum case:
1 m2
Λν(p,p) = γν 3N 3ln +O(1) . (15)
2 (4π)2 ǫ − µ2
(cid:26) (cid:27)
3 Two-loop Diagrams in the Heavy Quark Limit
The contributions to the quark self-energy from the two-loops (or fourth order in coupling
constant) are defined by the diagrams in Fig.4. As a consequence of our discussion in
the previous section, we shall not adopt the dimensional regularization, but merely use
the ultraviolet form factor Eq.(2) together with the infrared regularization of the gluon
propagator in the following.
6
(a) Diagram in Fig. 4a.
With Λ and λ being the ultraviolet and infrared cutoff masses, as in the previous
section, one finds
δm(a) = Σ(a)(m) = C(a)g4ΛI(a)(κ2) (16)
4 4 4 s 4
where κ2 λ2/Λ2; C(a) = 1 (for QED), and tatbtatb = C /2N (for QCD). The integral
≡ 4 − F c
I(a)(κ2) is equal to
4
d4k d4k F(k2) F(k2) 1 1 1
I(a)(κ2) = i 1 2 1 2 (17)
4 (2π)4 (2π)4k2 +κ2k2 +κ2(k +iǫ)(k +k +iǫ)(k +iǫ)
Z Z 1 2 14 14 24 24
(E) (E)
Here, to be definite, the integration is over the dimensionless Euclidean variables denoted
as (E) under the integration sign. The calculation of this integral is given in Appendix.
We have
d3~k F(~k2) α 1
δm(a) = 2πα C Λ s ln(1/κ2) . (18)
4 s F Z (2π)3 ~k2 (cid:20)−4πNc (cid:21)
(b) Diagram in Fig. 4b.
Here one obtains
δm(b) = Σ(b)(m) = C(b)g4ΛI(b)(κ2) (19)
4 4 4 s 4
where C(b) = ifabctatbtc = (N /2)C (of course there is no QED counterpart to this
4 − c F
diagram). The integral I(b)(κ2) is given as
4
d4k d4k d4k F(k2) F(k2) F((k k )2)
I(b)(κ2) = 1 2 3 1 2 1 − 2
4 (2π)4 (2π)4 (2π)4k2 +κ2k2 +κ2(k k )2 +κ2
(EZ) (EZ) (EZ) 1 2 1 − 2
N
b
(20)
·(k +iǫ)(k +iǫ)
14 24
with the numerator N equal to
b
N = (k 2 k )M γνM γν +γνM (k + k )M γν +γνM γνM (k 2 k ).
b 1 2 0 0 0 1 2 0 0 0 2 1
6 − 6 6 6 6 − 6
7
Since the integration is over Euclidean variables here, k = γ0ik ~γ~k. With this it is
4
6 −
easy to check that u¯(p~)N u(p~) = 0, which means that the contribution from the diagram
b
in the Fig.4b behaves as Λ/m for large m, thus vanishes in the heavy quark limit.
(c) Diagram in Fig. 4c.
This contribution may be written as
δm(c) = Σ(c)(m) = C(c)g4ΛI(c)(κ2) (21)
4 4 4 s 4
where C(c) = 1 (for QED), and tatatbtb = C2 (for QCD). The integral I(c)(κ2) is equal to
4 F 4
d4k d4k F(k2) F(k2) 1 1 1
I(c)(κ2) = i 1 2 1 2
4 (2π)4 (2π)4k2 +κ2k2 +κ2 (k +iǫ)(k +k +iǫ)(k +iǫ)
Z Z 1 2 (cid:26) 14 14 24 14
(E) (E)
1 1 1
= I(a)(κ2) (22)
−(k +iǫ)(k +iǫ)(k +iǫ) − 4
14 24 14 (cid:27)
Here, we have taken into account the mass renormalization in the second order (the dark
disk in Fig. 4c). From the above result, it is easy to see that in QED the contributions
coming from Fig. 4a and Fig. 4c cancel. This is a crucial point in solving the problem
of classical limit h¯ 0 in QED [5] since the contribution to the self-energy of a massive
→
fermion from the photon vacuum polarization is proportional to
k2
Πµν(k) (gµνk2 kµkν)ln 1+
m ∝ − m2
(cid:20) (cid:21)
and vanishes in the infinite mass limit: m .
→ ∞
For QCD one finds
d3~k F(~k2) α
δm(c) = 2πα C Λ s2C ln(1/κ2) . (23)
4 s F (2π)3 ~k2 −4π F
Z (cid:20) (cid:21)
(d) Diagram in Fig. 4d.
The contribution from the diagrams in Fig. 4d, involving the vacuum polarization by
massless quarks, gluons, and ghost is written as
d3~k F(~k2)
δm(d) = Σ(d)(m) = 2πα C Λ Πren( ~k2) (24)
4 4 s F (2π)3 ~k2 −
Z
8
In the above expression, the renormalized vacuum polarization reads (see Appendix
for a somewhat detailed calculation)
α 1 ~k2
Πren( ~k2) = s dαln 1+α(1 α) 2N 1+4α(1 α) 4n α(1 α)
− −4π − κ2 c − − f −
Z (cid:20) (cid:21)(cid:26) (cid:20) (cid:21) (cid:27)
0
α ~k2
s˜
= bln +O(1) (25)
−4π κ2
upon expanding in small parameter κ2. Here the appearance of the coefficient
˜
b = (10N 2n )/3 is due to the use of the Feynman gauge.
c f
−
Summing up all the contributions from the one- and two-loops, one finds
d3~k F(~k2) α ˜b
δm = 2πα C Λ 1 s N ln(1/κ2)+ ln(~k2/κ2) .
s F Z (2π)3 ~k2 (cid:26) − 4π(cid:20) c Nc (cid:21)(cid:27)
(26)
d3~k F(~k2) α 2 b
2πα C Λ 1 s ln(Λ2/λ2)N + .
→ s F Z (2π)3 ~k2 (cid:26) − 4π c(cid:20)3 Nc(cid:21)(cid:27)
Here, we express the final result in term of the first coefficient in the Gell-Mann-Low
function b = (11N 2n )/3. By adopting the number (flavor) of light quarks to be three:
c f
−
n = 3, we can see for N = 3 that the O(α ) corrections to the leading static Coulomb
f c s
self-energy coming fromthe bubble insertions to the gluon propagator is of the same order
of magnitude as those from other diagrams: Figs.4a-c, so the latter contribution should
be included.
In ref. [3] the self-energy of the heavy quark in the heavy quark limit was calculated
by introducing the running constant α (~k2) in the Coulomb energy expression: Eq.(9),
s
and by introducing the explicit integration cutoff in place of the ultraviolet form factor
F( k2/Λ2) (Fig.5),
−
8π d3~k α (~k2)
s
δm = (27)
3 (2π)3 ~k2
Z
|~k|<µ0
with
α (µ2)
α (~k2) = s 0 . (28)
s 1 (α (µ2)b/4π)ln(µ2/~k2)
− s 0 0
9