Scalar and pseudoscalar meson pole terms in the hadronic light-by-light contributions to $a_μ^{had}$

16 Pages·0.17 MB·English

by  E. Bartos

Scalar and pseudoscalar meson pole terms in
the hadronic light–by–light contributions to
2
0 ahad
0
µ
2

n
a
J E. Bartoˇs1, A.-Z. Dubniˇckov´a1, S. Dubniˇcka2, E. A. Kuraev3

1
and E. Zemlyanaya4
2

3
v
4 1 Dept. of Theor. Physics, Comenius Univ., Bratislava, Slovak Republic
8
0 2 Inst. of Physics, Slovak Acad. of Sci., Bratislava, Slovak Republic
6 3 Bogoliubov Lab. of Theor. Physics, JINR Dubna, 141980 Dubna, Russia
0
4 Lab. of Information Technologies, JINR Dubna, 141980 Dubna, Russia
1
0
/
h
p
- Abstract
p
e
h Third QED order hadronic light–by–light (LBL) contributions
v: aLµBL(M) to the anomalous magnetic moment of the muon ahµad from
i the pole terms of scalar σ, a (980) and pseudoscalar π0, η, η′ mesons
X 0
(M)intheframeworkofthelinearizedextendedNambu–Jona–Lasinio
r
a model are evaluated. The off–shell structure of the photon–photon–
meson vertices is taken into account by means of constituent quark
triangle loops. Themass of thequarkis taken to bem = m = m =
u d q
(280±20)MeV. The unknown strong coupling constants of π0,η,η′
anda mesons withquarksare evaluated in acomparison of thecorre-
0
sponding theoretical two–photon widths calculated in the framework
of our approach with experimental ones. The σ–meson coupling con-
stant is taken to be equal to π –meson coupling constant as it follows
0
from the linearized Nambu–Jona–Lasinio model Lagrangian. Then
oneobtainsaLBL(π )=(81.83±16.50)×10−11,aLBL(η)=(5.62±1.25)×
µ 0 µ
10−11,aLBL(η′)=(8.00±1.74)×10−11,aLBL(σ)=(11.67±2.38)×10−11
µ µ
and aLBL(a )=(0.62±0.24)×10−11. The total contribution of meson
µ 0
poles in LBL is aLBL(M)=(107.74±16.81)×10−11.
µ
1
1 Introduction
The muon is described by the Dirac equation and its magnetic moment is
related to the spin by means of the expression
e
~µ = g ~s (1)
2mµ!
where the value of the gyromagnetic ratio g is predicted (in the absence of
the Pauli term) to be exactly 2.
In fact, however, the interactions existing in nature modify g to be ex-
ceeding the value 2 because of the emission and absorption of virtual pho-
tons (electromagnetic effects), intermediate vector and Higgs bosons (weak
interaction effects) and the vacuum polarization into virtual hadronic states
(strong interaction effects).
In order to describe this modification of g theoretically, the magnetic
anomaly was introduced by the relation
g −2
a ≡ = (2)
µ
2
α α 2 α 3
a(1) + a(2)QED +a(2)had +a(2)weak +O
µ π µ µ π µ π
(cid:18) (cid:19) (cid:16) (cid:17)(cid:18) (cid:19) (cid:18) (cid:19)
where α = 1/137.03599976(50) [1] is the fine structure constant.
Here we would like to make a note that from all three charged leptons
− − −
(e , µ , τ ) the muon magnetic anomaly is the most interesting object for
theoretical investigations due to the following reasons:
i) it is one of the best measured quantities in physics [2]
aexp = (116592020±160)×10−11 (3)
µ
ii) its accurate theoretical evaluation provides an extremely clean test of
”Electroweak theory” and may give hints on possible deviations from
Standard Model (SM)
iii) the new measurement [3] in BNL is expected to be performed with a
definitive accuracy
∆aexp = ±40×10−11 (4)
µ
i.e. it is aimed at obtaining a factor 4 in a precision above that of the
last E–821 measurements (3).
2
At the aimed level of the precision (4) a sensibility will already exist to
contributions [4, 5]
a(2,3)weak = (152±4)×10−11, (5)
µ
arising from single– and two–loop weak interaction diagrams.
However, as we have mentioned above, the muon magnetic anomaly may
also contain contributions from a new physics, which can be revealed, we
hope, in a comparison of aexp with an accurate theoretical evaluation of ath.
µ µ
While QED and weak interaction contributions to ath seem to be estimated
µ
very reliably, there is still opened a door for improvements in hadronic con-
tributions.
As the most critical from all hadronic contributions are the light–by–light
(LBL) meson pole terms [6], we recalculate the third–order hadronic LBL
contributions to the anomalous magnetic moment of the muon ahad from the
µ
pole terms of the scalar σ, a and pseudoscalar π0, η, η′ mesons (M) in
0
the framework of the linearized extended Nambu–Jona–Lasinio model. The
reason for the latter are predictions of series of recent papers
aLBL = (−52±18)×10−11 [7]
µ
aLBL = (−92±32)×10−11 [8]
µ
aLBL = (−79.2±15.4)×10−11 [9] (6)
µ
aLBL = (+83±12)×10−11 [10]
µ
aLBL(π ) = (+58±10)×10−11 [10,11]
µ 0
which differ not only in the magnitude, but even in the sign. Moreover, in
these papers only the pseudoscalar pole contributions were considered. In
our paper we include the scalar meson (σ,a ) pole contributions as well.
0
∗ ∗
The crucial point is description of transition form factor γ → Mγ .
Current methods used for this aim are the ChPT and the vector–meson–
dominance (VMD) model.
In this paper the corresponding transition form factors by the constituent
quark triangle loops with colourless and flavourless quarks with charge equal
to the electron one are represented. (An application of a similar modified
constituent quark triangle loop model for a prediction of the pion electro-
magnetic form factor behavior can be found in [12], where also a comparison
with the naive VMD model prediction is carried out.) The mass of the quark
in the triangle loop is always taken to be m = m = m = (280±20)MeV
u d q
[13], which was determined in the framework of the chiral quark model of
the Nambu–Jona–Lasinio type by exploiting the experimental values of the
pion decay constant, the ρ–meson decay into two–pions constant, the masses
3
′
of pion and kaon and the mass difference of η and η mesons. The unknown
strong coupling constants of π0,η,η′ and a mesons with quarks are evalu-
0
atedinacomparisonofthecorrespondingtheoreticaltwo-photonwidthswith
experimental ones. The σ–meson coupling constant is taken to be equal to
π0–meson coupling constant as it follows from the corresponding Lagrangian.
The σ–meson mass is taken to be m =(496±47)MeV as an average of the
σ
values recently obtained experimentally from the decay D+ → π−π+π+ [14]
and excited Υ decay [15] processes.
As a result we present explicit formulas for aLBL(M) (M = π0,η,η′,σ,a )
µ 0
in terms of Feynman parametric integrals of 10–dimensional order, which
subsequently are calculated by MIKOR method [16].
The paper is organized as follows. In the next section all definitions and
a derivation of basic relations is presented. More detail can be found in
Appendix A and in Appendix B. The last section to numerical results and
discussion is devoted.
2 Meson pole terms in light–by–light contri-
butions to ahad
µ
The third QED order hadronic LBL scattering contributions to ahad are gen-
µ
erally represented by the diagram in Fig. 1A , which contains a class of
pseudoscalar mesons(π±, K± andalsoK0 )squareloopdiagrams(Fig. 1B),
S,L
a class of quark square loop diagrams (Fig. 1C) and scalar σ, a (980) and
0
pseudoscalar π0, η, η′ meson polediagrams(Fig. 2), where theoff-shellstruc-
ture of the photon–photon–meson vertices is taken into account by means of
flavourless and colourless constituent quark triangle loops. Here the interac-
tion of mesons (M) with quarks is described by the linearized Nambu–Jona–
Lasinio type Lagrangian
L = g q¯(x)[σ(x)+iπ(x)γ ]q(x), (7)
qq¯M M 5
with unknown strong coupling constant g . The latter for π0, η, η′ and
M
a mesons are evaluated by a comparison of the corresponding theoretical
0
two-photon widths with the experimentally determined values [1].
For the on–mass shell scalar (S) and pseudoscalar (P) mesons decays one
can write the following matrix elements
iαg
S
M(S(p) → γ(k )+γ(k )) = − K(m /m )(k k g − (8)
1 2 S q 1 2 µν
πm
q
−k k )e (k )e (k ),
1ν 2µ 1ν 1 2µ 2
4
(cid:1) (cid:2) (cid:3)
A B C
Figure 1: Third order hadronic light–by–light scattering contribution to ahad
µ
(A) and class of pseudoscalar meson square loop diagrams (B) and quark
square loop diagrams (C) contributing to (A).
αg
P
M(P(p) → γ(k )+γ(k )) = J(m /m )(k k e e ), (9)
1 2 P q 1 2 1 2
πm
q
with
1 1−x 1
1−4xy 2 dx
K(z) = 2 dx dy, J(z) = ln[1−z2x(1−x)] (10)
1−xyz2 z2 x
Z Z Z
0 0 0
and
p2 = m2 , k2 = k2 = 0, (abcd) = ǫ aαbβcγdδ.
M 1 2 αβγδ
They lead to the theoretical two–photon widths (for more detail see Ap-
pendix A)
α2m3g2
Γγγ = S SK2(m /m ), (11)
S 64π3m2 S q
q
α2m3g2
Γγγ = P PJ2(m /m ). (12)
P 64π3m2 P q
q
dependent on the scalar g and pseudoscalar g meson coupling constants
S P
with quarks, respectively.
Taking the meson masses with errors and the following world averaged
values of the two–photon widths from the last Review of Particle Physics [1],
Γγγ = (0.008±0)keV, Γγγ = (0.464±0.044)keV, Γγγ = (4.282±0.339)keV
π0 η η′
and Γγγ = (0.24 ± 0.08)keV, the quark mass value m = (280 ± 20)MeV,
a0 q
one finds the meson coupling constants with quarks as follows
g2 = 9.120±1.305,g2 = 2.220±0.378, (13)
π0 η
g2 = 6.708±1.096,g2 = 0.757±0.277.
η′ a0
5
q q
M M
k1 k2 k2 k1
p1 p2 p1 p2
(cid:1) (cid:2)
A B
q
q
M
(cid:17) q
k1
k2 q
p1 p2
(cid:1) (cid:2) (cid:3)
C
Figure2: Meson(M) polediagramsinthethirdorderhadroniclight–by–light
scattering contributions to ahad.
µ
Since there is no experimental result on the σ-meson decay into two–
photons up to now, we identify g2 with g2 as it follows from the Lagrangian
σ π0
(7).
Now we are ready to evaluate LBL meson pole terms contributions to
ahad.
µ
Usually the fermion–photon vertex function is written in the form
ˆ
Γ = u¯(p )V (q)u(p ), (14)
µ 2 µ 1
Vˆ (q) = γ F (q2)+ 1 [qˆ,γ ]F (q2), q = p −p ,
µ µ 1 4m µ 2 2 1
with on mass–shell muons, i.e. p2 = p2 = m2 (m is the muon mass). Then
1 2
the expression for the muon anomalous magnetic moment is
(g −2)
µ
a = = F (0) (15)
µ 2
2
3 1
= − lim Sp(pˆ +m)Vˆ (q)(pˆ +m)P (q),
16m2 q2→0 q2 2 µ 1 µ
with a projection operator
m
P (q) = q2γ + [qˆ,γ ].
µ µ µ
3
6
The gauge invariant set of Feynman diagrams containing the LBL scat-
tering block with meson pole intermediate states is drawn in Fig. 2.
Their contributions to ahad are
µ
aLBL(S)+aLBL(P) = (16)
µ µ
3 1
− lim Sp(pˆ +m)VˆS+P(pˆ +m)P (q),
16m2 q2→0 q2 2 µ 1 µ
with
α d4k d4k T(P) T(S) 1
VˆS+P = 1 2Oνλσ µνλσ + µνλσ , (17)
µ 26π3 (iπ2)2 k2 −m2 k2 −m2  k2k2(k −k )2
Z 1 P 1 S 1 2 1 2
 O = 2O +O ,
1 2
ˆ ˆ
pˆ −k +m pˆ −k +m
Oνλσ = γσ 2 2 γλ 1 1 γν,
1 (p −k )2 −m2 (p −k )2 −m2
2 2 1 1
ˆ ˆ ˆ
pˆ −k +m pˆ +k −k +m
Oνλσ = γσ 2 2 γν 1 1 2 γλ
2 (p −k )2 −m2 (p +k −k )2 −m2
2 2 1 1 2
where
4α2g2
T(P) = P(µνk q)(λσk k )f (k2)f (k ,k ), (18)
µνλσ π2m2 1 2 1 1 1 2 1 2
q
with
1 m2 1 1−x1m2
f (k2) = q dx, f (k ,k ) = 2 dx qdx , (19)
1 1 m2 −x(1−x)k2 2 1 2 1 d 2
Z q 1 Z Z
0 0 0
d = m2 −k2x (1−x )−k2x (1−x )+2k k x x ,
q 2 3 3 1 2 2 1 2 2 3
x = 1−x −x .
3 1 2
Here we would like to note that the functions f (k2), f (k ,k ) at the limit
1 1 2 1 2
of an infinite value of the quark mass, m → ∞, tend to one.
q
The expression for T(S) has a similar form
4α2g2
T(S) = − S(g k q −k q )f˜(k2)f (k ,k ), (20)
µνλσ π2m2 µν 1 1µ ν 1 1 λσ 1 2
q
1 m2(1−2x(1−x))dx
f˜(k2) = q ,
1 1 m2 −x(1−x)k2
Z q 1
0
1 1−x1m2
q
f (k ,k ) = 2 dx {(1−4x x )[g k (k −k )−
λσ 1 2 1 1 2 λσ 2 1 2
d
Z Z
0 0
k (k −k ) ]+2(k −k )2x (2x −1)g dx .
2λ 1 2 σ 1 2 1 1 λσ 2
o
7
Usual procedure of performing the loop–momenta integration using the
Feynman parameters method (see Appendix B) leads to
aLBL(M) = −R F (m ,m ) (21)
µ M M M q
where
3m2α2g2
R = M (22)
M
64π5m2
q
and
F (m ,m ) = 2M +M , F (m ,m ) = 2N +N .
P P q a b S S q a b
The M ,N are given by the expressions
i i
1 AP
M = dΓ a +BP ;
a Z azD12 "D1 a #
1 AP
M = dΓ b +BP ; (23)
b Z bσ2D22 "D2 b #
1 AS
N = dΓ a +BS ;
a Z azD12 "D1 a#
1 AS
N = dΓ b +BS ,
b Z bσ2D22 "D2 b#
with dΓ the 10–dimensional integrals on Feynman parameters
a,b
1 1 x¯1 1 z¯1 1
dx dx
dΓ = τ4 dx 3 dz dz α α2d3αdyρρ¯2dρ, (24)
a xx¯ 1 x x¯ 1 2 2 3
Z Z Z Z 3 3 Z Z Z
0 0 0 0 0 0
1 1 x¯1 1 z¯1 1
dx dx
dΓ = τ4 dx 3 dz dz α α2α3d4αρρ¯2dρ,
b xx¯ 1 x x¯ 1 2 2 3 4
Z Z Z Z 3 3 Z Z Z
0 0 0 0 0 0
m
x¯ = 1−x,z¯ = 1−z ,τ = q,d3α = dα dα dα ,d4α = dα dα dα dα .
1 1 1 2 3 1 2 3 4
m
The explicit expressions for polynomials A, B are
8yρ
AP = 2a2 +2r a (a +b )−r2a b −8r3,
a 3z 1 1 1 1 1 1 1 1 1
h yρ i
BP = 4+12r +4a b ,
a 1 1 1 z
8ρ 16 8
AP = [a3 −r2a b (b −1)]− r2a + r3(1−2b ),
b 3σ 2 2 2 2 2 3 2 2 3 2 2
8 4ρ
BP = 4a − +r (8b −4)+ a b (b −1),
a 2 3 2 2 σ 2 2 2
8
yρ 1 1
AS = −8a3 −16r a (a b − a + b )
a z 1 1 1 1 1 3 1 3 1
(cid:20)
2 2 2
− 8r2(a b − a + b − )b d (x ,x )
1 1 1 3 1 3 1 3 1 1 1 2
(cid:21)
yρ 32
+ (16/3)a2(a +1)+ r (b −1)(a +1)a (25)
z 1 1 3 1 1 1 1
(cid:20)
16
+ r2 (a +1)(b −1)2 d (x ),
1 3 1 1 2 1
(cid:21)
16
BS = (8a + )d (x )−(4+12a )d (x ,x )
a 1 3 2 1 1 1 1 2
8yρ
− (a +1)(b −1)2d (x )
1 1 2 1
z
yρ
+ (12a b −8a +8b −8)b d (x ,x ),
1 1 1 1 1 1 1 2
z
ρ 16 32
AS = a2(a −1)+ r a (a −1)(b −1)
b σ 3 2 2 3 2 2 2 2
(cid:26)(cid:20)
16
+ r (a −1)(b −1)2 d (x )
2 2 2 2 1
3
(cid:21)
16 16
+ − a2 + r2b (b −1) d (x ,x ) ,
3 2 3 2 2 2 1 1 2
(cid:20) (cid:21) (cid:27)
8 8ρ
BS = − b (b −1) d (x ,x )
b 3 σ 2 2 1 1 2
(cid:20) (cid:21)
16 8ρ
+ 8a − − (a −1)(b −1)2 d (x )
2 2 2 2 1
3 σ
(cid:20) (cid:21)
with
a = α α¯ , b = α α γ ,a = α (α α¯ +α¯ ), (26)
1 3 2 1 2 3 1 2 4 3 2 3
b = α (α γ +α¯ ), r = ρ(y¯−c y), r = −c ρ,
2 4 3 1 3 1 1 2 2
d (x ,x ) = 1−4x x , d (x ) = 2x (2x −1).
1 1 2 1 2 2 1 1 1
Quantities c , γ , z, D , D and σ are given in Appendix B.
1,2 1 1 2
9
3 Results and discussion
The result for the meson pole contributions of LBL type has the form
aLBL(M) = aLBL(π0)+aLBL(η)+aLBL(η′)+aLBL(σ)+aLBL(a ) (27)
µ µ µ µ µ µ 0
where
aLµBL(π0) = −Rπ0Fπ0(mπ0,mq)
aLBL(η) = −R F (m ,m )
µ η η η q
aLµBL(η′) = −Rη′Fη′(mη′,mq) (28)
aLBL(σ) = −R F (m ,m )
µ σ σ σ q
aLBL(a ) = −R F (m ,m ).
µ 0 a0 a0 a0 q
Now, taking the values (13) of g2 determined in Section 2, the mass
M
of the quark m = (280 ± 20)MeV [13], the mass of the σ–meson m =
q σ
(496±47)MeV [14, 15], the mass of the muon and the corresponding scalar
and pseudoscalar mesons from the Review of Particle Physics [1] one finds
aLBL(π0) = (81.83±16.50)×10−11
µ
aLBL(η) = (5.62±1.25)×10−11
µ
aLBL(η′) = (8.00±1.74)×10−11 (29)
µ
aLBL(σ) = (11.67±2.38)×10−11
µ
aLBL(a ) = (0.62±0.24)×10−11.
µ 0
Here we would like to note that our value for the π0 is in the framework of
error bars consistent with [10, 11].
So, the total contribution of meson poles in LBL is
aLBL(M) = (107.74±16.81)×10−11, (30)
µ
where the resultant error is the addition in quadrature of all partial errors of
(30).
Togetherwiththecontributionsofthepseudoscalarmeson(π±,K±,K0 )
S,L
square loops and constituent quark square loops [7, 8] it gives
aLBL(total) = (111.20±16.81)×10−11. (31)
µ
The others 3-loop hadronic contributions derived from the hadronic vac-
uum polarizations (VP) were most recently evaluated by Krause [17]
a(3)VP = (−101±6)×10−11. (32)
µ
10