Table Of ContentThe magnetic mass of transverse gluon, the B-meson
weak decay vertex and the triality symmetry of octonion
2
1
0
Sadataka Furui
2
n
a Faculty of Science and Engineering, Teikyo University
J
1
1-1 Toyosatodai, Utsunomiya, 320-8551 Japan
∗
]
h
p January 4, 2012
-
p
e
h
[
2
Abstract
v
7
5
With an assumption that in the Yang-Mills Lagrangian, a left-handed fermion and
8
3
. a right-handed fermion both expressed as the quaternion make an octonion which
0
1
possesses the triality symmetry, I calculate the magnetic mass of the transverse self-
1
1
: dual gluon from three loop diagram, in which a heavy quark pair is created and two
v
i
X self-dual gluons are interchanged.
r
a The magnetic mass of the transverse gluon depends on the mass of the pair created
quarks, and in the case of charmed quark pair creation, the magnetic mass m
mag
becomes approximately equal to T at T = T 1.14Λ 260MeV.
c c ∼ MS ∼
A possible time-like magnetic gluon mass from two self-dual gluon exchange is
derived, and corrections in the B-meson weak decay vertices from the two self-dual
gluon exchange are also evaluated.
∗E-mail address: [email protected]
1
1 Introduction
In 1980, Linde[1] pointed out difficulties of infrared problems in the thermodynamics of
mass-less Yang-Mills gas, and a possible solution via gluon acquiring the magnetic mass. At
zero temperature, an effective gluon mass of 500 200 MeV was predicted by Cornwall[2].
±
A problem that at relatively low temperature or high density, the pressure of the QCD
thermodynamic gasbecomes negative was pointedout in[3] anddifficulties inthecalculation
of the thermodynamical potential in the g6 order are discussed in[4, 5, 6].
The ground-state energy of the finite temperature quark-gluon system was classically
derived from the Yang-Mills Lagrangian by Freedman and McLarren[7]. An extensive review
of calculations done before 2004 are given in [8].
In QCD, the gluon is screened in the plasma through gluon loops, quark loops and ghost
loops. In 1993 the non Abelian Debye screening mass was found to be sensitive to the
non perturbative magnetic mass of the gluon[9]. By adding a magnetic mass m2 to the
mag
transverse self-energy Π (K), the instability could be evaded, when m = c 3 g2N T and
T mag f32 c
c 1. m is expected to appear in the perturbative calculation in the order of g6 and
f mag
≥
higher, but no systematic calculation of it was found.
Alexanian and Nair[10] calculated the magnetic mass of the gluon in a model based
Cg2T
on Chern-Simons theory and obtained m (2.384) , where C = N for SU(N)
mag
∼ 4π
gauge theory. There is, however, other models that yield different values of the magnetic
mass[11, 12], and the situation is worrisome. The best that one could do perturbatively to
get the Debye mass is[13]
N N 1 N N
m = ( c + f)1/2gT + N g2Tlog( c + f )1/2)+c g2T +O(g3T)
D 3 6 4π c 3g2 6g2 f
where N is the number of colors and N is the number of fermion flavors, but the value of
c f
c is left open.
f
There is a systematic investigation of vacuum polarization tensor Π (k ,k) of rela-
µν 0
tivistic plasma[14]. The collective plasma effects are characterized by the frequency ω =
k
2
N N
f + cgT. The self-energy depend on four vector uα of the fluid and the momentum Kα
r 6 3
of the virtual particle via two invariants ω = Kαu and k such that KαK = ω2 k2. The
α α
−
transverse self-energy function Π (k,ω) and the longitudinal self-energy function Π (ω,k)
T L
are calculated by a pair of four vector E˜α and B˜α, each orthogonal to uα, defined by
Fαβ = uαE˜β uβE˜α +ǫαβγδB˜ u
γ δ
−
When ω > ω , plane wave solution of E˜α and B˜α exist, but when ω < ω they are screened.
p p
It was shown that the screening length for the magnetic fields diverges at ω 0, thus B˜α is
→
screened except at ω = 0.
In [15, 16, 17], I proposed a calculation of the Domain Wall Fermion(DWF) propagator
in quaternion basis and expressed vector fields in terms of the Plu¨cker coordinate, following
E´. Cartan[18]. In this framework, the spin structure of the quark pair in the self-energy
diagram with two self-dual gluon exchange is uniquely defined when the color and spin of
the incoming gluon arefixed. The process is similar to the one investigated in the technicolor
theory[20].
In order to calculate B˜α at ω = 0, I adopt a model, in which a quark pair is created, and
two self-dual gluons are exchanged and the pair is annihilated[16, 17]. From a left-handed
quark described by a quaternion and a right-handed quark described by another quaternion,
one can construct an octonion. An octonion posesses the triality symmetry[18, 19] and the
spin structure of the magnetic coupling of a self-dual gluon to the quark loops can be fixed
when the exchanged internal gluons are restricted to be self-dual. There is no direct evidence
that the triality symmetry plays a role in the nature, but the difference of about a factor of 3
intheeffective flavor number foropeningtheconformalwindowintheSchro¨dinger functional
scheme[21]v.s. inthemomentumsubtraction(MOM)scheme[22]couldbeunderstood, ifthe
Schro¨dinger functional does not select a triality sector but MOM scheme selects one triality
sector. Using the Banks-Zaks expansion[23], Grunberg[25] showed that the non-perturbative
effect modifies the critical flavor number of the perturbative QCD which was around 8[21]
3
to 4. Infrared stable QCD running coupling was observed also in holographic QCD[27], and
in the polarized electron nucleon scattering[26].
In order to check the relevance of self-dual two gluon exchange in the coupling ver-
tex, I calculate the vertex of a B meson weak decay vertex into lepton and anti-neutrino.
The decay propability of a B meson into a lepton and neutrino is measured by the Babar
collaboration[31] and by the Belle collaboration[32]. Possible deviation from the standard
model via Penguin diagram is discussed in [33] and I compare with the possible correction
from two self-dual gluon exchange diagrams.
The structure of this paper is as follows. In the sect.2, I summarize the magentic mass
problem and present the calculation of the self-energy using the quaternion bases. In sect.3,
the B-meson weak decay vertex including the two self-dual magnetic gluon exchange is
investigated. Conclusion and discussion are given in the sect.4.
2 The self-energy of a gluon via exchange of self-dual
gluons between heavy quark and heavy anti quark
In 1993 the non abelian Debye screening mass was found to be sensitive to the non per-
turbative magnetic mass of of the gluon[9]. The importance of the magnetic mass was
recognized in [3], as he calculated the pressure of finite temperature plasma, which is de-
fined as p = T logZ/V where Z is the partition function. The pressure derived at a
−
high-temperature region
8 1 7
p = π2T4 +N ( T2µ2 + π2T4)
f
45 2 60
16π2 1 5
[ T4 +N ( T4 +T2µ2/4π2)] (1)
−(22 4N )log(T/Λ ) 6 f 72
− 3 f QCD
becomes negative when extrapolated to low temperature T or to high chemical potential µ.
The negative pressure implies that the convergence of the perturbation series breaks down,
and a QCD phase transition at this point was discussed.
4
Ξ12 Ξ4 Ξ31 Ξ23 Ξ4 Ξ12
x1 x3 x2 x1 x2 x1 x3 x2
Ξ314 Ξ23 Ξ124 Ξ124 Ξ31 Ξ234
Figure 1: The self-energy diagram of trans- Figure 2: The self-energy diagram of trans-
verse polarized gluon through exchange of verse polarized gluon through exchange of
self-dual gauge fields. Π self-dual gauge fields. Π
11 22
In the infrared, however, the ring sum term log(1 + Π (K)/K2) was found to contain
T
infrared singular term, indepent of the choice of the gauge. In order to evade the equation
of the one loop transverse gluon propagator near T = 0
8+(ξ +1)2
k2 +Π (k = 0,k) = k2 g2N T k = 0
T 0 c
− 64
be satisfied for positive k, a modification Π (K) Π (K)+m2 was proposed[6]. With
T → T mag
an arbitrary parameter c 1, m = c 3 g2N T c NcT was expected for α (k) = 0.3
f ≥ mag f32 c ∼ f 3 s
at k 1GeV. The magnetic mass contributes to the pressure of the quark-gluon plasma in
∼
the order of g6T4. In lattice simulations, the magnetic gluon propagator does not show a
peak at the zero momentum and the pole structure of the magnetic mass was, excluded[35],
although how to detect the timelike pole on the lattice remains an open question[34].
I do not assume a magnetic mass of g2T for a producion of a g6 term in the pressure,
but I consider a three loop diagram of order g6 including two self-dual gluon exchange, as a
building block of the stabilized quark-gluon plasma system. The model does not contradict
with the lattice simulation.
In[15,16], Idiscussed thequarkpropagatorusingquaternion[19]basis. Inthisflamework,
Iconsider tensorcoupling oftheexternal gluonfieldtoaheavy quarkinternalself-dual vector
field x ,x ,x couple to the fermion spinors.
1 2 3
I consider the self-energy or the magnetic mass of a gluon, which is polarized in the
5
transverse direction. The transverse gluon field is
F = ig(∂ Aa ∂ Aa +gfabcAbAc)λa,
µν − µ ν − ν µ µ ν
where λa is the SU(3) color basis. The tensor coupling in the case of Π are, γ (γ γ
T11 4 2 3
−
γ γ )/2 = γ γ in the vertex and in the case of Π is γ (γ γ γ γ )/2 = γ γ
3 2 1 5 T22 4 3 1 1 3 2 5
−
The coupling of F (q) to the quark is
ij
γ γ γ γ
¯ i j j i
iψ(k) − ψ(p)F (q)
ij
2
kj
= ψ∗(k)γ ǫ σiψ(p)(ǫ piA (q)+ )
5 ijn ijn j
m ···
pikj
= ψ∗(k)γ ǫ ǫ σiψ(p)A (q)
5 ijn ijn j
m
(p k)
= ψ∗(k)γ × nψ(p)(σ A(q)) (2)
5 n
m ×
Internal self-dual gluon and the heavy quark couplings in Π and Π are
11a 22a
γ (γ γ γ γ )/2 = γ , γ (γ γ γ γ )/2 = γ , and γ (γ γ γ γ )/2 = γ .
4 3 4 4 3 3 4 2 4 4 2 2 4 1 4 1 3 1
− − − − − −
In Π , I choose the quark represented by ξ to be at rest and the self-dual gluon x and
11 4 3
x have momenta k and p , respectively. The quark ξ and ξ have momenta k and
2 y z 12 31
−
p, respectively, and ξ and ξ have momenta p and k , respectively. The propagator of
314 124 z y
−
quark ξ have the numerator γ p +γ k , but γ is multiplied by γ at the junction to the
23 3 z 2 y 3 2
quark ξ , and γ is multiplied by γ at the junction to the quark ξ and effectively the
314 2 3 124
numerator is proportional to σ , as required by the assignment of ξ .
x 23
I use the Clifford products rule (a + a)(b + b) = a b + a b + b a a b + a b to
0 0 0 0 0 0
− · ×
evaluate Clifford products of the following bases.
Cφ = ξ ξ i ξ j ξ12k
1234 23 31
− − −
Cψ = ξ ξ i ξ j ξ k
123 234 314 124
− − −
φ = ξ +ξ i+ξ j +ξ34k
0 14 24
ψ = ξ +ξ i+ξ j +ξ k (3)
4 1 2 3
The x gluon coupling to quark pair with 2 self-dual gluon exchange Πa in Coulomb
1 11
gauge consists of two types i.e ( ξ ξ +ξ ξ ), or (ξ ξ ξ ξ ). The possible quark-anti
12 314 31 124 24 3 34 2
− −
6
quark state between the self-dual gluon exchange of the former is ξ ξ and the latter is
4 23
ξ ξ .
1234 1
Since the trace of the two types are the same, I consider the amplitude Πa of the x
11 1
gluon
k p γ k +m 1 1 1 γ p +m
J (k,p) = m4g6 z × yTrγ γ − 3 z γ γ − 2 y
a m 1 5 k2 +m2 3k2mp2 2 p2 +m2
z z y y
k p γ k +m γ p +γ k +m γ p +m
z y 3 z 2 y 3 z 2 y
γ γ × γ γ γ γ (4)
× 1 5 m k2 +m2 2 5 p2 +k2 +m2 3 5 p2 +m2
z y z y
Similarly, the self-energy from Πb becomes, by choosing the intermediate quark-anti
22
quark state ( ξ ξ +ξ ξ ), or ( ξ ξ +ξ ξ ),
23 124 12 234 14 3 34 1
− −
k p γ k +m 1 1 1 γ p +m
J (k,p) = m4g6 x × zTrγ γ − 1 x γ γ − 3 z
b m 2 5 k2 +m2 1k2 mp2 3 p2 +m2
x x z z
k p γ k +m γ p +γ k +m γ p +m
x z 1 x 3 z 1 x 3 z
γ γ × γ γ γ γ (5)
× 2 5 m k2 +m2 3 5 p2 +k2 +m2 1 5 p2 +m2
x z x z
The trace in the numerator of the Π
11
4(m6 +3m4(p2 +k2) m2p2k2) (6)
y z − y z
The numerator of the Π is
22
4(m6 +3m4(p2 +k2) m2p2k2) (7)
z x − z x
I integrate numerically
∞ ∞ m6 +3m4(k2 +p2) m2k2p2 dkdp
4m4 − (8)
Z Z (k2 +m2)2(p2 +m2)2(k2 +p2 +m2)(2π)2
0 0
where the factor 4 comes from integral on k and p from to + .
−∞ ∞
I chosse the scale m = 1 and evaluate
Λ Λ 1+3(x+y) xy dxdy
4 − (9)
Z Z (x+1)2(y +1)2(x+y +1)(2π)2
0 0
by varying Λ. Numerically, the integral (9) is approximately 7 1
(2π)2
At T = 0, when the wave function of the quark pair is available from lattice simulation,
onecanperformtheintegrationover q numerically andobtainanon-screened magneticmass.
7
1 1
At T = 0, it is necessary to replace the self-dual gluon exchange propagator in
6 k2p2
1 1
J(k,p) by and check the self-consistency.
k2 +m2 (T)p2 +m2 (T)
mag mag
In a quenched SU(2) lattice Landau gaugesimulation, the temperature dependence of the
transverse gluon and longitudinal gluon from0 temperature to twice the critical temperature
T were recently measured[28, 29]. Near the T = T , the transverse gluon propagator showed
c c
a peak near q = 0.4GeV, but the longitudinal gluon propagator did not show this behavior.
In a quenched SU(3) lattice Landau gauge simulation[30], the transverse gluon propagator,
or the magnetic gluon propagator showed smooth momentum dependence in the range from
T = 0.867T to T = 4.97T .
c c
In the present work, I ignore these temperatures and the number of color dependence,
and evaluate, using the bose Bose-Einstein distribution for the gluon, the magnetic mass at
finite temperature T as
∞ f(q) dq
m (T) = T 4πq2 (10)
mag
Z eq/T 1(2π)3
0 −
Since q is in GeV, I use k = 8.6 10−5eV/K= 8.6 10−14GeV/K as the unit, and express
B
× ×
the temperature in the unit of T = 1.14Λ [8] and Λ 230MeV.
c MS MS ∼
There are two polarization directions and two different time orders, and assuming that
the coupling of self-dual gluon and the external gluons to quarks are the same, I estimate the
7 4 28 16π
f(q) using the numerical results of T = 0 as f(q) = g(q)6 = α (q)3 × . Here,
m3(2π)2 s m3
I adopt α (q) obtained from Lattice simulation[22] which is consistent with the prediction
s
of the holographic theory[27]. The parametrization of α(s) is[26]
γn(q)
α (q) = . (11)
s
log q2+mg(q)2
Λ2
(cid:16) (cid:17)
where
γ
n(q) = π(1+[ +(bq)c]−1) (12)
log(m2/Λ2)(1+q/Λ) γ
−
and γ = 4 = 12 , m (q) = m , m = 1.024GeV, a = 3.008GeV−1, d = 0.840, b =
β0 33−8 g 1+(aq)d
1.425GeV−1, c = 0.908 and Λ = 0.349GeV.
8
10
8
6
HLmTmag T 4
2
0
0.0 0.5 1.0 1.5 2.0
T
Tc
Figure 3: The magnetic mass of the gluon through charm quark loops, as a function of T/T
c
and a fit.
The charmed quark-anti quark pair creating 3 loop diagram gives the magnetic mass of
the order of T near T = T . The magnetic mass m (T)in this case can be parameterized
c c mag
as
m (T)/T = 0.711(T/T )2 +0.563(T/T )4 0.0628(T/T )6
mag c c c
−
The self-energy decreases as the quark mass of the loop increases. The magnetic mass
of a gluon through bottom quark loops is less than k T and the thermal response of the
B c
bottom quark and the charmed quark differ qualitatively.
When T = 0, Ievaluate theself-energy asthe 0 component of theClifford product. In the
Coulomb gauge, the exchanged self-dual gluons are assumed to be x ,x and the quark-anti
1 2
quark state that couple to x can be ψCφ (ξ ξ +ξ ξ ), or φCψ (ξ ξ +ξ ξ ).
4 1 23 2 31 14 234 24 314
→ →
The intermediate quark-anti quark state independent of the momentum could be ξ ξ in
1234 4
the former case and ξ ξ in the latter case. However, since these states don’t have virtual
123 0
momentum excitations, I consider, instead transition of ψCφ to φCψ and vice versa in the
intermediate state.
I define self-energy Πc for exchange of self-dual gluons x ,x is obtained by using eigen
44 1 2
9
states (ξ ξ +ξ ξ ) or (ξ ξ1+ξ ξ ).
14 234 24 134 23 13 2
k γ k +m 1 γ (k q )+m 1 γ (r k )q +m
J (k,p) = m4g6 xTrγ − 1 x γ γ − 3 x × y γ γ − 2 x x y
c m 1 k2 +m2 2 5q2 (k q )2 +m2 1 5r2 (r k )2q2 +m2
x y x × y x x x y
p γ p +m γ (p r )+m γ (q p )r +m
y 2 y 3 y x 1 y y x
γ γ γ × γ γ (13)
×m 2 p2 +m2 1 5(p r )2 +m2 2 5(q p )2r2 +m2
y y × x y y x
Ξ14 Ξ3 Ξ24
x4 x2 x1 x4
Ξ234 Ξ12 Ξ314
Figure 4: The self-energy diagram of time-like gluon through exchange of self-dual gauge
fields. Πc
44
When the exchanged self-dual gluons are x x , the trace in the numerator of J (k,p)
1 2 c
becomes
4m6[(p 2 +k 2)q r (1+(q r )2)k p ][m2 (k q )(p r )] (14)
y x y x y x x y x y y x
− − × ×
Numerically one could fix the momentum transfer (k r )q p and integrate over r ,k and
x x y y x x
−
p , or the momentum transfer (q p )r k and integrate over q ,p and k .
y y y x x y y x
−
When the external gluon is a spacelike photon, the infrared divergence can be regularized
andyields anomalousmagneticmoment g = 1+ α . Since regularizationof q ,r is necessary,
2 2π y x
the numerical calculation of the self-energy of the timelike gluons and photons remain as a
future problem.
3 The B-meson weak decay vertex
When the two self-dual gluon exchange is important in the heavy quark heavy anti quark
pair to a gluon coupling, similar two gluon exchange in B meson weak decay is expected to
be important. I consider B− τ−ν¯ decay, in which b quark u¯ quark couple to W boson
→
10
