A note on the anomalous magnetic moment of the muon

0.1 MB·

by  Davor Palle

A note on the anomalous magnetic moment of the muon
6
1
0 Davor Palle
2

n Zavod za teorijsku fiziku, Institut Rugjer Boˇskovi´c
a
Bijeniˇcka cesta 54, 10000 Zagreb, Croatia
J

7
2

] The anomalousmagnetic moment of the muonis an importantobserv-
h ablethattestsradiativecorrectionsofallthreeobservedlocalgaugeforces:
p
electromagnetic, weak and strong interactions. High precision measure-
-
n ments reveal some discrepancy with the most accurate theoretical evalua-
e tions of the anomalous magnetic moment. We show in this note that the
g UV finite theory cannot resolve this discrepancy. We believe that more
.
s reliableestimateofthenonperturbativehadroniccontributionandthenew
c measurements can resolve the problem.
i
s
y
PACS numbers: 12.60.-i; 12.15.-y;11.15.Ex
h
p
[ 1. Introduction and motivation

1
Since the most reliable Standard Model (SM) evaluations of the anoma-
v
lous magnetic moment of the muon differs three to four standard deviations
1
8 from the measurements [1], it might be useful to examine predictions for
7 this observable of the theories beyond the SM.
7
It is well known that the SM has three major obstacles: (1) massless
0
. neutrinos, (2) absence of a dark matter particle, and (3) absence of the
1
lepton and baryon number violations. We show in ref. [2] that the UV
0
6 nonsingular theory, free of the SU(2) global anomaly (called the BY theory
1 in[2]),isfreeoftheSMdeficiencies. BesidesthreeheavyMajorananeutrinos
:
v ascolddarkmatterparticle(s), theBYtheorycontainsthreelightMajorana
i neutrinoswitharelationonthethreemixingangles, alowingtheinclusionof
X
theleptonCPviolatingphase. TheeffectsoftheuniversalUVspacelikecut-
r
a off are studied in various strong and electroweak processes in refs. [3, 4, 5].
Thecosmological consequencesofthenonsingularEinstein-Cartantheoryof
gravity can be envisaged in ref. [6]. The minimal distance in the Einstein-
Cartan cosmology is compatible with the UV cut-off of the BY theory.
In this note we want to inspect the impact of the UV cut-off on the ra-
diative corrections of the anomalous magnetic moment of the muon. Next
(1)
2 muon˙arhiva printed on January 29, 2016
chapter deals with the explicit calculations, however the relevant detailed
formulae the reader can find in the Appendix. The concluding section dis-
cusses the numerical results and their consequences.
2. One loop corrections within the UV nonsingular theory
The anomalous magnetic moment of the muon is one of the most pre-
ciselymeasuredobservabletotheuncertaintyof (10 9)[1]. Itiscalculated
−
O
within the perturbative quantum field theory to the very high order.
The perturbation theory works well and it is very accurate because of
the very small fine structure constant. The anomalous magnetic moment
formula can be cast into transparent perturbation series where it is possible
tocompareandstudyanymodificationoftheStandardModel(SM)[7]. We
evaluateoneloopcorrectionswithvirtualelectroweak bosonswithintheUV
finite BY theory [2]. Although this theory contains Majorana light neutri-
nos,wecansafelyneglecttheirMajoranacharacterandtheirmassesbecause
of their smallness. The heavy Majorana neutrinos are coupled strongly to
Nambu-Goldstonescalarsattreelevel, butnottoelectroweak gaugebosons.
They are, therefore, decoupled in the evaluation of the anomalous magnetic
moment.
Since the gauge invariant physical result cannot depend on the choice of
gauge, wecanfreelyperformourcalculationsinthe’tHooft-Feynmangauge
with Nambu-Goldstone scalars instead in the unitary gauge where Nambu-
Goldstone scalars are decoupled from lepton doublets [8]. However, the
scalardoubletdoesnotcontaintheHiggsscalar[2]sincethenoncontractible
space is a symmetry breaking mechanism in the BY theory and theUV cut-
off is fixed attree level by theweak bosonmass Λ= π 2M , e = gsinΘ ,
√6g W W
cosΘ = MW [2]. The Higgs scalar is decoupled from other particles in
W MZ
the BRST transformations [10] and does not play essential role in the proof
of the renormalizability of the spontaneously broken gauge theories [11].
Consequently, the UV finite BY theory without the Higgs scalar is also
renormalizable.
Itisnecessarytocommenttheclaimthattherecentlydiscovered125GeV
scalar resonance is the SM Higgs scalar [12]. P. Cea [13] proposed the most
naturalexplanationofthe125GeV resonanceasamixtureoftoponiumand
gluonium. The possible new 750 GeV heavy boson resonance at the LHC
[14] might be a perfect candidate for a heavier scalar twin of the 125 GeV
boson [15].
Let us go back to the description of the electroweak one loop contribu-
tions to the anomalous magnetic moment. The SM one loop result takes
the form [1, 7, 8, 9] (m denotes the mass of the muon and s = sinΘ ):
w W
muon˙arhiva printed on January 29, 2016 3
1
a (g 2),
µ µ
≡ 2 −
1α α
aγ = + (( )2)
µ 2π O π
G m2 10 1 5
aEW(1) = F + (1 4s2)2 + (10 5) .
µ √28π2{ 3 3 − w − 3 O − }
We have to rederive these results equiped with theSMFeynman rules in
the ’t Hooft-Feynman gauge. One can straightforwardly extract the contri-
butions with one virtual photon [16] and one virtual W and Z bosons [8] in
terms of the coefficient functions of the tensor and vector Green functions
that can be deduced from the scalar Green functions-master integrals [17]
(for definitions and explicit expressions see the Appendix):
1
lim m2(C +C )(photon) = , (1)
11 21
Λ 2
→∞
1 10 1
lim (C C )(W boson)= . (2)
Λ 4 11− 21 3 16M2
→∞ W
1 1
lim [ (1 2s2)2+s4](2C +3C +C )+s2(1 2s2)
Λ { 2 4 − w w 0 11 21 w − w
→∞
1
(C +C ) (Z boson)= [(3 4c2)2 5], (3)
× 0 11 } 48M2 − w −
Z
The deviation between the SM and the BY theory could be found in
the scalar master integrals. The UV cut-off Λ in the spacelike domain of
the Minkowski spacetime is introduced as a Lorentz and gauge invariant
quantity. The analytical continuation to the timelike domain is performed
on the Riemann’s sheets, when necessary. By the symmetrization of the
externalmomentaofthemasterintegralswiththeUVcut-offweinsuretheir
translational invariance. Consequently, all the master integrals of the BY
theory have a correct limit Λ of the standard QFT master integrals.
→ ∞
We leftourpresentation oftheexplicitresultsandcommentstothenext
section.
4 muon˙arhiva printed on January 29, 2016
3. Results and conclusions
The one loop results for the BY theory follow from Eqs. (1-3) and the
expansion formulas for scalar one, two and three point functions in the
SM and the BY theory presented in the Appendix. Besides the difference
between the SM and BY in the cut-off (Λ = or Λ = π 2M ), one has
∞ √6g W
to exclude the Higgs boson contribution for the BY theory (M )[18]:
H
→ ∞
1 1m2 5 m6 α
aγ(BY)(1) = ( + ) , (4)
µ 2 − 4 Λ2 12 Λ6 π
G m2 10 41M2 99M4
aW(BY)(1) = F W + W , (5)
µ √28π2{ 3 − 6 Λ2 8 Λ4 }
G m2 1 5 3 M2
aZ(BY)(1) = F (1 4s2)2 +( +2s2 4s4) Z
µ √28π2{3 − w − 3 2 w − w Λ2
1 M4
+ (43 332s2 +664s4) Z . (6)
48 − w w Λ4 }
We evaluate numerically the difference with the following set of the well
established parameters (see also the figure for the cut-off dependence):
m(muon)= 105.658 MeV, M = 91.188 GeV, G = 1.166 10 5 GeV 2,
Z F − −
×
α= 1/137.036, M = 80.385 GeV, s2 = 0.22295, Λ = 326.2 GeV,
W w
[aBY aSM]γ,1 loop = 6.09 10 11,
µ − µ − × −
[aBY aSM]W,1 loop = 4.307 10 10,
µ − µ − × −
[aBY aSM]Z,1 loop = +1.595 10 10,
µ − µ × −
[aBY aSM]γ+W+Z,1 loop = 3.321 10 10,
µ − µ − × −
muon˙arhiva printed on January 29, 2016 5
Cut-off dependence
0
"Deviation"
-1e-10
-2e-10
-3e-10
n
o
ati
vi
e
D -4e-10
-5e-10
-6e-10
-7e-10
200 400 600 800 1000 1200
Cut-off (GeV)
Fig. 1: Cut-off (Λ) dependence of the deviation [aBY aSM](γ+
µ − µ
W +Z).
It is evident from the recent comparison between the experimental and
the theoretical SM prediction in ref. [1, 7]:
a1+2 loops(Higgs) = + (10 11),
µ O −
aexp aSM = 2.7 10 9,
µ − µ × −
that the loop corrections of the BY theory cannot explain the deviation
from the experimental value. It seems that more reliable estimates of the
hadronic contributions is necessary. They should be studied by the non-
perturbative methods supplemented by the experiments with hadrons [1].
The new experiments to measure the anomalous magnetic moment ot the
muon are planned in the U.S.A. and Japan with a good potential to further
reduce the experimental error.
6 muon˙arhiva printed on January 29, 2016
Appendix
The appendix is devoted to the exposure of the explicit formulas for
scalar, vector and tensor functions, as well as for one, two and three point
scalar Green functions.
Let us start with definitions and conventions as in [17]:
ı d4k 1
A(m2) = ,
16π2 (2π)4 k2 m2
Z −
ı d4k 1;k
B (q2;m2,m2)= µ ,
16π2 0;µ 1 2 (2π)4[k2 m2][(k+q)2 m2]
Z − 1 − 2
B = q B ,
µ µ 1
ı d4k 1;k ;k k
C (p2,p2,p2;m2,m2,m2)= µ µ ν ,
16π2 0;µ;µν 1 2 1 2 3 (2π)4 [k2 m2][(k+p )2 m2][(k+p +p )2 m2]
Z − 1 1 − 2 1 2 − 3
Cµ = pµC +pµC , p = p p ,
1 11 2 12 − 1− 2
Cµν =gµνC +pµpνC +pµpνC +(pµpν +pµpν)C .
24 1 1 21 2 2 22 1 2 2 1 23
The standard Green function can be found in ref. [17], while the Green
functions in the noncontractible space (Λ < ) have the form [3]:
∞
1
BΛ(p2;m ,m )= [ B˜Λ(p2;m ,m )+ B˜Λ(p2;m ,m )],
ℜ 0 1 2 2 ℜ 0 1 2 ℜ 0 2 1
Λ2 0 1
B˜Λ(p2;m ,m ) = ( dyK(p2,y)+θ(p2 m2) dy∆K(p2,y)) ,
ℜ 0 1 2 Z0 − 2 Z−(√p2−m2)2 y+m21
2y
K(p2,y) = ,
p2+y+m2+ ( p2+y+m2)2+4p2y
− 2 − 2
q
( p2+y+m2)2+4p2y
∆K(p2,y) = − 2 ,
q p2
muon˙arhiva printed on January 29, 2016 7
1
CΛ(p2,p2,p2;m2,m2,m2) = [ C˜Λ(p2,p2,p2;m2,m2,m2)
ℜ 0 1 2 3 1 2 3 3 ℜ 0 1 2 3 1 2 3
+ C˜Λ(p2,p2,p2;m2,m2,m2)+ C˜Λ(p2,p2,p2;m2,m2,m2],
ℜ 0 2 3 1 2 3 1 ℜ 0 3 1 2 3 1 2
Λ2
CΛ(p ,m ) = dq2Φ(q2,p ,m )+ dq2Ξ(q2,p ,m ),
ℜ 0 i j i j i j
Z0 ZTD
CΛ(p ,m )= ReC (p ,m ) ∞dq2Φ(q2,p ,m ),
ℜ 0 i j 0∞ i j −ZΛ2 i j
Φ function derived by the angular integration after Wick s rotation,
′
≡
C standard t Hooft Veltman scalar function,
0∞ ≡ ′ −
TD timelike domain of integration.
≡
We list the Green functions (real parts only) necessary for the one loop
contributions with a virtual W boson (photon and Z boson contributions
proceed similarly):
C = C (m2,0,m2;0,M2 ,M2 ),
0 0 W W
1 dB (m2;0,M2 ) dC
lim C = [ 0 W (m2 = 0) C (m2 = 0)+M2 0(m2 = 0)],
m2 0 11 2 dm2 − 0 Wdm2
→
C (m2) 1 M2 dC
lim C = 24 (m2 = 0) C (m2 = 0)+ W 11(m2 = 0)
m2 0 21 − dm2 − 2 11 2 dm2
→
1dB (m2;0,M2 )
+ 1 W (m2 = 0),
2 dm2
dC (m2) 1 dC
24 (m2 = 0) = [C (m2 = 0) M2 11(m2 = 0)],
dm2 4 11 − W dm2
1
C (m2) = [B (m2;0,M2 ) B (0;M2 ,M2 ) (m2 M2 )C ],
11 2m2 0 W − 0 W W − − W 0
1
B (m2;0,M2 )= [ A(M2 )+A(0)+(M2 m2)B (m2;0,M2 )],
1 W 2m2 − W W − 0 W
Λ2+M2
BΛ(0;0,M2 ) = ln W,
0 W M2
W
dBΛ(m2;0,M2 ) 1 1 1 M2
0 W (m2 = 0) = W ,
dm2 2M2 − 4(Λ2+M2 )2
W W
d2BΛ(m2;0,M2 ) 1 1 1 M2 1 M4
0 W (m2 = 0) = + W W ,
d(m2)2 3M4 3(Λ2+M2 )3 − 2(Λ2+M2 )4
W W W
8 muon˙arhiva printed on January 29, 2016
1 M2 m2
C0Λ(m2) = C0∞(m2)+∆C0Λ, C0∞(m2)= m2 ln WM−2 ,
W
1 1/Λ 1 m2y2 4m2y2
∆C0Λ(m2) = 3m2[Z0 dyy−1(1+−MW2 y2)2(s1+ (1−m2y2)2 −1)
1/Λ y2(M2 m2)+1
+2 dyy−1(1 W − )],
Z0 − (1+y2(MW2 −m2))2+4m2y2
q
1 M2
CΛ(m2 = 0) = ( 1+x(1+x) 1), x = W,
0 M2 − − Λ2
W
dCΛ 1 1 2
0 (m2 = 0) = [ + ( (1+x) 2+(1+x) 3)],
dm2 M4 −2 3 − − −
W
d2CΛ 1 2 4 37
0 (m2 = 0) = [ + ( 2(1+x) 5+8(1+x) 4 (1+x) 3
d(m2)2 M6 −3 3 − − − − 3 −
W
1
+9(1+x) 2 3(1+x) 1+ )].
− −
− 3
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