One-loop self-energies in the electroweak model with a
nonlinearly realized gauge group
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Bettinelli, D. , R. Ferrari, and A. Quadri. “One-loop self-energies
in the electroweak model with a nonlinearly realized gauge
group.” Physical Review D 79.12 (2009): 125028. © 2009 The
American Physical Society
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http://dx.doi.org/10.1103/PhysRevD.79.125028
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American Physical Society
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Thu May 26 20:30:06 EDT 2016
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Detailed Terms
PHYSICAL REVIEW D 79, 125028 (2009)
One-loop self-energies in the electroweak model with a nonlinearly realized gauge group
D. Bettinelli,1,* R. Ferrari,2,3,† and A. Quadri3,‡
1
2
Physikalisches Institut, Albert-Ludwigs-Universität, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
3
Dipartimento di Fisica, Università degli Studi di Milano and INFN, Sezione di Milano via Celoria 16, I-20133 Milano, Italy
(Received 9 March 2009; published 26 June 2009)
We evaluate at one loop the self-energies for the W, Z mesons in the electroweak model where the
gauge group is nonlinearly realized. In this model the Higgs boson parameters are absent, while a second
mass parameter appears together with a scale for the radiative corrections. We estimate these parameters
in a simplified fit on leptons and gauge bosons data. We check physical unitarity and the absence of
infrared divergences. Landau gauge is used. As a reference for future higher order computations, the
regularized D-dimensional amplitudes are provided. Eventually the limit D ! 4 is taken on physical
amplitudes.
DOI: 10.1103/PhysRevD.79.125028
PACS numbers: 11.10.Lm, 11.10.Gh, 11.30.Na
I. INTRODUCTION
A consistent strategy for the all-orders subtraction of the
divergences in nonlinearly realized gauge theories has
recently been developed [1–5] by extending some tools
originally devised for the nonlinear sigma model in the flat
connection formalism [6]. The approach relies on the local
functional equation for the one particle irreducible (1-PI)
vertex functional [6] (encoding the invariance of the group
Haar measure under local left transformations), the weak
power-counting theorem [7], and the pure pole subtraction
of properly normalized 1-PI amplitudes [8]. This scheme
of subtraction fulfills all the relevant symmetries of the
vertex functional. Physical unitarity is established as a
consequence of the validity of the Slavnov-Taylor identity
[9].
This strategy was first applied to the nonlinearly realized
SUð2Þ massive Yang-Mills theory [3]. The full set of oneloop counterterms and the self-energy have been obtained
in [4].
The extension to the electroweak model based on the
nonlinearly realized SUL ð2Þ UY ð1Þ gauge group introduces a number of additional nontrivial features [1,2]. The
direction of the spontaneous symmetry breaking fixes the
linear combination of the hypercharge and of the third
generator of the weak isospin, giving rise to the electric
charge. Despite the fact that both the hypercharge and the
SUL ð2Þ symmetry are nonlinearly realized, the Ward identity for the electric charge has a linear form on the vertex
functional.
The anomalous couplings are forbidden by the Uð1ÞY
invariance together with the weak power-counting.
However, two independent mass parameters for the vector
*daniele.bettinelli@physik.uni-freiburg.de
†
ruggferr@mit.edu
‡
andrea.quadri@mi.infn.it
1550-7998= 2009=79(12)=125028(13)
mesons are allowed. Thus the ratio of the vector meson
masses is not given by the Weinberg angle anymore.
As a first step toward a detailed analysis of the radiative
corrections of the nonlinearly realized electroweak theory,
we provide in this paper the vector meson self-energies in
the one-loop approximation.
The dependence of the self-energies on the second mass
parameter is important in order to establish a comparison
with the linear realization of the electroweak group based
on the Higgs mechanism.
We provide a rough estimate of both the extra mass
parameter and the scale of the radiative corrections. We
fix some of the parameters on measures taken at (almost)
zero momentum transfer, while the one-loop corrections
are confronted with measures at the resonant value of the
vector bosons energies. The resulting values are challenging: the departure from the Weinberg relation between the
vector meson mass is very small, and the scale of the
radiative corrections is of the order of a hundred GeV.
The aim of the present work is to provide the amplitudes
in D dimensions for future high order computations and to
provide a preliminary assessment of the predictivity of the
electroweak model based on the nonlinearly realized gauge
group, including the one-loop self-energy corrections.
Electroweak physics is described with very reasonable
parameters (the second mass parameter and the scale of
the radiative corrections).
The cancellations among unphysical states required by
physical unitarity can be easily traced out. The physical
amplitudes are shown to be free of infrared divergences. It
is also remarkable that they do not depend on the spontaneous symmetry breaking parameter v. This fact has been
discussed in Refs. [1,2,4].
The computation is done in the symmetric formalism on
the SUð2ÞL flavor basis. This choice greatly simplifies both
the Feynman rules and the actual computation; in fact,
symmetry arguments turn out to be very useful in the
125028-1
Ó 2009 The American Physical Society
D. BETTINELLI, R. FERRARI, AND A. QUADRI
PHYSICAL REVIEW D 79, 125028 (2009)
calculation of the invariant functions. The symmetric formalism puts emphasis on the fact that the entering parameters are not renormalized (e.g. as in the on-shell
renormalization procedure) and are fixed at the end, by
means of the comparison with the experimental data.
Moreover, the symmetric formalism makes the underlying
symmetric structure encoded by the local functional equation more transparent.
II. FEYNMAN RULES
mass eigenstates in the tree-level Lagrangian; Vjk is the
Cabibbo-Kobayashi-Maskawa (CKM) matrix. Similarly,
we use for the leptons the notation ðluj ; j ¼ 1; 2; 3Þ ¼
ðe ; ; Þ and ðldj ; j ¼ 1; 2; 3Þ ¼ ðe; ; Þ. The single
left doublets are denoted by Llj , j ¼ 1, 2, 3 for the leptons,
and Lqj , j ¼ 1, 2, 3 for the quarks. Color indices are not
displayed.
One also introduces the SUð2Þ matrix ,
¼
The classical action is written in order to establish the
Feynman rules. We omit all the external sources which are
needed in order to subtract the divergences at higher loops.
We refer to the previous publications [1,2] where the
procedure is described at length. The field content of the
electroweak model based on the nonlinearly realized
SUð2ÞL Uð1Þ gauge group includes the SUð2ÞL connection A ¼ Aa 2a (a , a ¼ 1, 2, 3 are the Pauli matrices),
the Uð1Þ connection B , the fermionic left doublets collectively denoted by L, and the right singlets, i.e.
!
u !
lLj
quLj
;
; j; k ¼ 1; 2; 3 ;
L2
ldLj
Vjk qdLk
(1)
!
u !
quRj
lRj
; d ; j ¼ 1; 2; 3 :
R2
ldRj
qRj
In the above equation the quark fields ðquj ; j ¼ 1; 2; 3Þ ¼
ðu; c; tÞ and ðqdj ; j ¼ 1; 2; 3Þ ¼ ðd; s; bÞ are taken to be the
1
ð þ ia a Þ;
v 0
y ¼ 1 ) 20 þ 2a ¼ v2 :
(2)
The mass scale v gives , the canonical dimension at D ¼
4. We fix the direction of spontaneous symmetry breaking
by imposing the tree-level constraint
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 ¼ v2 2a :
(3)
The SUð2Þ flat connection is defined by
F ¼ i@ y :
(4)
A. Classical action
Discarding the neutrino mass terms, the classical action
for the nonlinearly realized SUð2Þ Uð1Þ gauge group
with two independent mass parameters for the vector mesons can be written as follows, where the dependence on is explicitly shown:
2 1
1
g0
dD x 2 Tr G G F F þ M2 Tr gA 3 B y F
4
4
2
2 X 0
X g
g0
g0
þ M2 Tr gA 3 B y F 3
þ
6 Lþ
6 R
L i@
6 þ gA
6 þ YL B
R i@
6 þ ðYL þ 3 ÞB
2
2
2
2
L
R
X
1 3 y l
1 þ 3 y q
y q 1 3
Lj mquj R qj
Lj þ mqdk Vkj
y Lqj þ H:c: :
mlj R lj
(5)
Rk
þ
2
2
2
j
S ¼ ðD4Þ
Z
In D dimensions the doublets L and R obey
D L ¼ L;
D R ¼ R;
(6)
with D a gamma matrix that anticommutes with every
other .
The non-Abelian field strength G is defined by
G ¼ Ga a ¼ ð@ Aa @ Aa þ gabc Ab Ac Þ a ;
2
2
(7)
However, the same structure is required by the weak
power-counting requirement, as discussed in Ref. [2].
B. Gauge fixing
(8)
In order to set up the framework for the perturbative
quantization of the model, the classical action in Eq. (5)
needs to be gauge fixed. The ghosts associated with the
SUð2ÞL symmetry are denoted by ca . Their antighosts are
denoted by c a , and the Nakanishi-Lautrup fields by ba . It is
also useful to adopt the matrix notation
b ¼ ba a ;
c ¼ c a a :
(9)
c ¼ ca a ;
2
2
2
In the above equation the phenomenologically successful
structure of the couplings has been imposed by hand.
The Abelian ghost is c0 , the Abelian antighost is c 0 , and the
Abelian Nakanishi-Lautrup field is b0 .
while the Abelian field strength F is
F ¼ @ B @ B :
125028-2
ONE-LOOP SELF-ENERGIES IN THE ELECTROWEAK . . .
For the sake of simplicity, we deal with the Landau
gauge here. All external sources are dropped since they
are not relevant for the present work. The complete set of
external sources is provided in Ref. [2]. Then the gaugefixing part of the classical action is
SGF ¼ ðD4Þ
Z
dD xðb0 @ B c 0 hc0
(10)
D½A cgÞ:
þ 2 Trfb@ A c@
C. Boson symmetric formalism
The bilinear part of the boson sector is
M2
2
2
2
gAa g0 B 3a @ a
v
2 2
þ GZ @ 3
þ b0 @ B þ ba @ A
a
v
2 þ 2
þ
þ
þb @ W
@
¼ M 2 g2 W
vg 2
M2 2
2
þ
@ ðG Þð1 þ Þ Z þ b @ W þ
vG Z
2
þ bZ @ Z þ bA @ A :
(11)
We use the notations
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G ¼ g2 þ g02 ;
MW ¼ gM;
g
g0
s¼
c¼ ;
G
G
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MZ ¼ ð1 þ ÞGM
(12)
and
1
1
Z ¼ ðgA3 g0 BÞ;
W þ ¼ pffiffiffi ðA1 iA2 Þ;
G
2
1
A ¼ ðg0 A3 þ gBÞ:
G
(13)
PHYSICAL REVIEW D 79, 125028 (2009)
hðA1 A1 Þþ i ¼ hðA2 A2 Þþ i ¼ hðW þ W Þþ i;
hðA1 A2 Þþ i ¼ hðA1 A3 Þþ i ¼ hðA2 A3 Þþ i ¼ 0;
1
hððgZ þ g0 AÞðgZ þ g0 AÞÞþ i
G2
1
¼ 2 ðg2 MZ þ g02 DÞ;
G
1
hðA3 BÞþ i ¼ 2 hððgZ þ g0 AÞðg0 Z þ gAÞÞþ i
G
gg0
¼ 2 ðMZ þ DÞ;
G
1
hðBBÞþ i ¼ 2 hððg0 Z þ gAÞðg0 Z þ gAÞÞþ i
G
1
¼ 2 ðg02 MZ þ g2 DÞ;
G
hðA3 A3 Þþ i ¼
where we used a shorthand notation, e.g.
p p
i
MZ ! 2
g
;
p MZ2
p2
p p
i
D ! 2 g 2 :
p
p
For the one-loop calculation of the vector boson selfenergies, one needs the usual Feynman rules and the trilinear couplings generated by the two mass invariants in
Eq. (5). The first mass invariant generates the trilinear
couplings
2 M2
g0
y
Tr 2 gA B T0 F trilinear
2
2
2
M
G
4 2 Z 3bc @ b c
¼
2
v
g X
4 2
Aa abc @ b c
v a¼1;2
gg0
ðg0 Z þ gAÞ A
a 3ab b
Gv
g0
0
8
ðg Z þ gAÞ 3bc @ b c :
Gv2
þ4
(14)
is given by
0
B
V
B
B
B
@ b
i
p2 m2
p
p2
b
p
p2
0
0
i 2pv 2
V
p p
ðg p2 Þ
In the symmetric notation we have
1
0 C
C
C:
i 2pv 2 C
A
v2
4m2
i
p2
(17)
D. Boson trilinear couplings
In the Landau gauge the propagator matrix for the bilinear
form
2
1
m2
2
F F þ
V @ 3 þ b@ V 4
v
2
(16)
(15)
(18)
The second mass invariant yields
2 g0
M2
y
y
Tr g A T0 B þ i @ T0 trilinear
2
2
M2
1
4 2 GZ 3bc ð@ b Þc
¼
2
v
1
8
4g GZ Aa ab3 b þ 2 gAa @ 3 3ab b :
v
v
(19)
We put everything together as follows:
125028-3
D. BETTINELLI, R. FERRARI, AND A. QUADRI
Sjboson
trilinear
PHYSICAL REVIEW D 79, 125028 (2009)
M2
g X
g
G g2 þ g02
4 2
¼
Aa abc @ b c þ 8 2 Aa @ 3 3ab b þ 4 2
þ
Z 3bc @ b c
2
v a¼1;2
v
v
G2
gG g02
gg0
g2 g0
A Aa 3ab b :
4
þ Z Aa 3ab b 8
A 3bc @ b c þ 4
(20)
v G2
Gv Gv2
E. Fermion contribution
The evaluation of the fermion loops requires a rule on
how to handle the 5 in dimensional regularization. Our
mechanism for the removal of divergences is based on a
regularization that respects the local gauge invariance;
therefore 5 must anticommute with any . At one loop
this is possible, as it is well known, since there are no chiral
anomalies. For higher loop calculations any trace involving
5 must be considered as an independent amplitude up to
the end of the subtraction procedure. Eventually we evaluate the limit at D ¼ 4 for physical amplitudes.
The fermion contribution can be easily cast into a global
formula,
and the longitudinal part of the contribution of the fermions,
4
1
L ½ABST ¼ 2 Tr ðAS þ BTÞðiðm2 þ M2 Þm
2
p
þ iðm2 M2 ÞM Þ þ Hðm; MÞ
1
mMðAS BTÞp2 þ ðAS þ BTÞ
2
ðM4 þ m4 2m2 M2 p2 ðm2 þ M2 ÞÞ :
(24)
½ABST h0jð c ðxÞ ðA þ B5 Þ c ðxÞ c ð0Þ
ðS þ T5 Þ c ð0ÞÞþ j0i
Z
dp Z dq
¼ Tr
ðA þ B5 Þ
ð2ÞD ð2ÞD p
6 þ 6q þ m ipx
e ðS þ T5 Þ
ðq þ pÞ2 m2
6q þ M
;
(21)
2
q M2
where A, B, S, T are matrix elements corresponding to the
flavor and the color of the fermions with mass m and M,
and they can be obtained from the classical action (5). In
particular, the neutral sector is
3
GZ c
s2 Q 3 5 c þ eA c Q c ;
4
4
0
gg
e
:
(22)
G
One then gets the transverse part of the contribution of
the fermions (for the notations see the Appendix),
T ½ABST
1
ð2 DÞ
Tr iðAS þ BTÞ
ðm þ M Þ þ Hðm;MÞ
D1
2
ð2 DÞ
ðAS þ BTÞðp2 þ M2
mMðAS BTÞD þ
2
1 1
þ m2 Þ 2 ðAS þ BTÞðm2 M2 ÞiðM m Þ
p 2
1
1
2 Hðm;MÞ mMðAS BTÞp2 þ ðAS þ BTÞ
2
p
ððm2 M2 Þ2 p2 ðm2 þ M2 ÞÞ ;
(23)
III. SELF-ENERGY AMPLITUDES IN D
DIMENSIONS
The presentation of the results (Landau gauge in D
dimensions) is as follows. First we report the result of the
calculation for the transverse and for the longitudinal parts,
p p
p p
¼ g 2 T þ 2 L :
p
p
(25)
For each of them we report the contributions of the single
graphs. Subsequently, we evaluate the diagonal amplitudes
T on shell, where we discuss the validity of physical
unitarity and the absence of any infrared singularity.
Finally, the on-shell amplitudes are taken at D ¼ 4.
We omit, for the sake of conciseness, the self-energies at
D ¼ 4 for generic momentum (the procedure is straightforward). We do not use on-shell renormalization: MW and
MZ are dummy parameters, as are c and s. The massless
photon is a source of some infrared problems in the Landau
gauge. In the Appendix the analytical tools needed to
handle these difficulties are provided [see Eqs. (A2)–
(A5)].
¼4
A. The D ¼ 4 amplitudes
The D ¼ 4 amplitudes are recovered as the finite parts in
the Laurent expansion of the generic dimensional regularized amplitudes, normalized by the factor
ðD4Þ :
(26)
This point has been discussed at length in Refs. [6,8]. In
this procedure we essentially encounter the following
cases.
125028-4
ONE-LOOP SELF-ENERGIES IN THE ELECTROWEAK . . .
ðD4Þ m jD4 ¼
m2
ð4Þ2
2 2
m
1 þ þ ln
;
D4
42
(27)
ðD4Þ Hðm; MÞðp2 ÞjD4
i
2
þ lnð4Þ
¼
2
ð4Þ D 4
2
Z1
m
þ
dx ln 2 ð1 xÞ
0
2
M
p2
þ 2 x 2 xð1 xÞ :
(28)
PHYSICAL REVIEW D 79, 125028 (2009)
All other limit expressions can be reduced to Eqs. (27)–
(29).
B. The counterterms
The counterterms are given by the pole parts of the same
Laurent expansion taken with a minus sign and finally
multiplied by the common factor ðD4Þ . In Eq. (29) the
pole in D 4 dangerously multiplies a nonlocal term.
However, we shall find that GðMÞ always enters with a
factor p2 M2 .
IV. WW SELF-ENERGY
In the Appendix we give the value of the last integral in
Eq. (28).
ðD4Þ GðMÞðp2 ÞjD4
i
1
2
þ ln4
¼
ð4Þ2 p2 M2 D 4
2
1
M
½M2 p2 ½M2 p2 þ 2
ln
þ
ln
: (29)
p M 2 p2
M2
2
At D 4 we use
4
ðD2 2ÞðD2 2Þ
þ OððD 4Þ2 Þ:
’
D4
ðD 4Þ
(30)
We first list the contributions to the transverse part.
A. Transverse WW self-energy
The Goldstone bosons contribution to the transverse part
of WW is
02
2 2 1
MZ
1
2 g þ G
iGoldstone
M
¼
i
þ
TWW
4ðD 1Þ W
G
MZ2 p2
Gð0Þ 2 gg0 2 p2
Hð0; 0Þ
MW
þ
þ
D1
G
4
4ðD 1ÞG2 MZ2
2
fp2 MW
ðg02 þ G2 Þ2 þ g2 MZ2 ½2ð3
Hð0; MZ Þ
4G2
2
2
2
ðMZ p Þ
2
½g02 þ G2 2 MW
4þ
:
ðD 1ÞMZ2 p2
(32)
Then
4D
@
GðMÞjM¼0
@M2
i
D
2
2
½p 1 þ ð 1Þ 2
2
ð4Þ2
D
p2
2 log 1þ
2
ð4Þ2
4
þ OððD 4Þ2 Þ
D4
2i
2
2 2
1þ
¼
½p
D4
ð4Þ2
p2
þ log :
ð4Þ2
iW
TWW ¼ Gð0Þ
2 þ G2 p2 ð1 þ Þg þ
þ 2DÞg02 MW
The Faddeev-Popov contribution is
iFP
TWW ¼ g2
p2 Hð0; 0Þ:
2ðD 1Þ
(33)
The vector boson tadpole contribution is
itadpole
TWW ¼ i
(31)
ðD 1Þ2 2
g ðMW þ c2 MZ Þ:
D
(34)
The W loop is
g2 g02 p6
ðD 2Þg2 g02 p4
g2 g02
2
Hð0; 0Þ
þ GðMW Þ
ðMW
p2 Þ2
2 2
2 2
2 2
4ðD 1ÞG MW
ðD 1ÞG MW
4ðD 1ÞG2 MW
p
4
2
½MW
þ 2ð2D 3Þp2 MW
þ p4 þ Hð0; MW Þ
þ ðD 2Þp4 iMW
g2 g02
2
4
2
ðMW
þ p2 Þ½ðD 2ÞMW
þ 2ð3 2DÞp2 MW
2 p2
ðD 1ÞG2 MW
g2 g02
4 þ 2½Dð2D 1Þ 2p2 M2 þ Dð4D 7Þp4 Þ: (35)
ðDð4D 7ÞMW
W
2 2
4ðD 1ÞDG2 MW
p
The ZW loop is
125028-5
D. BETTINELLI, R. FERRARI, AND A. QUADRI
iZW
TWW ¼
PHYSICAL REVIEW D 79, 125028 (2009)
g4 Hð0; 0Þp6
g4 Hð0; MW Þ
4 þ 2ð2D 3Þp2 M2 þ p4 ðM2 p2 Þ2 ½MW
W
2 2
2
2
4ðD 1ÞG MW MZ 4ðD 1ÞG2 MW
MZ2 p2 W
iMZ g4
g4 Hð0; MZ Þ
2
2 2
4
2 2
4
ðM
p
Þ
½M
þ
2ð2D
3Þp
M
þ
p
þ
fDð4D 7Þp4
Z
Z
Z
2 M2 p2
2 M2 p2
4ðD 1ÞG2 MW
4ðD
1ÞDG
Z
Z
2 2½Dð2D 1Þ 2M2 Þp2 þ D½M4 þ ð4D 7ÞM2 M2 þ ð7 4DÞM4 g
þ ðDð4D 7ÞMW
Z
W
Z W
Z
iMW g4
2 þ ð7 4DÞDM2 Þp2 þ D½ð4D 7ÞM4
fDð4D 7Þp4 þ ðð4D2 2D 4ÞMW
Z
W
2 2
4ðD 1ÞDG2 MW
p
2
ð4D 7ÞMZ2 MW
MZ4 g þ
g4 HðMZ ; MW Þ
ððMW MZ Þ2 p2 ÞððMW þ MZ Þ2 p2 Þ
2 M 2 p2
4ðD 1ÞG2 MW
Z
4 þ 2ð2D 3ÞðM2 þ p2 ÞM2 þ M4 þ p4 þ 2ð2D 3ÞM 2 p2 g:
fMW
Z
W
Z
Z
The charged lepton contribution to transverse TWW is
obtained from Eq. (23) by using
AS BT ¼ 0;
AS þ BT ¼
g2
:
4
2
iGoldstone
¼ iMZ MW
LWW
X g2 1
M2
ð2 DÞiMl þ 2l iMl
2 D 1 l¼e;;
p
Ml2
2
2
þ Hð0; Ml Þðp þ Ml Þ ð2 DÞ 2 :
p
(38)
The quark contribution to the transverse part of WW is
given by Eq. (23), where
AS BT ¼ 0;
ðAS þ BTÞab
g2
¼ 3Vab Vab
;
4
ðG2 þ g02 Þ2 1
1
þ
p2 MZ2
4G2
g2 g02 p2
Hð0; MZ Þ
4G2
2
ðG2 þ g02 Þ2 MW
1
ðMZ2 p2 Þ2 þ Hð0; 0Þ
2
2
2
4
4G MZ p
2 M
2 W 2 2g2 g02 MZ2 þ ðG2 þ g02 Þ2 p2
G MZ
g2 2 p2
:
(41)
þ1
2
Gð0ÞMW
(37)
Therefore
ileptons
TWW ¼
(36)
The Faddeev-Popov contribution to the longitudinal part of
WW is
iFP
LWW ¼ g2 2
p Hð0; 0Þ:
2
(42)
(39)
The vector boson tadpole contribution to the longitudinal
part is
with Vab the CKM matrix. Thus
g2 X
1
Vab Vab
ima ð2 DÞ þ 2
2ðD 1Þ ab
p
1
ðm2a Mb2 Þ þ iMb ð2 DÞ 2 ðm2a Mb2 Þ
p
þ Hðma ;Mb Þ ð2 DÞðp2 þ Mb2 þ m2a Þ
1
(40)
2 ððm2a Mb2 Þ2 p2 ðm2a þ Mb2 ÞÞ :
p
itadpole
LWW ¼ i
iquarks
TWW ¼ 3
B. Longitudinal WW self-energy
In a similar way we list the contributions for the longitudinal parts of LWW . The Goldstone boson contribution
to the longitudinal part of WW is
ðD 1Þ2 2
g ðMW þ c2 MZ Þ:
D
(43)
The W loop contribution to the longitudinal part is
2
g2 g02 MW
g2 g02
2
ðMW
p2 Þ2 iMW
iW
LWW ¼ GðWÞ
2 2
4G p
2G2 p2
7
2 þ 2D 3 þ 2 p2
2D MW
2
D
2
02
gg
4
þ Hð0; MW Þ 2 2 ½2ðD 2ÞMW
2G p
2
p2 MW
þ p4 :
The ZW loop contribution to the longitudinal part is
125028-6
(44)
ONE-LOOP SELF-ENERGIES IN THE ELECTROWEAK . . .
iZW
LWW ¼
PHYSICAL REVIEW D 79, 125028 (2009)
2
iMZ g4 MW
Hð0; MW Þg4 3
Hð0; MZ Þg4 3
2 2
2 2
2 2
2
2
p
ðM
M
p
Þ
þ
ðM
M
p
Þ
ðM
p
Þ
þ
2D
3
þ
W
Z
W
Z
2 p2
D
4G2 MZ2 p2
4G2 MW
2G2 p2 2MZ2 W
iMW g4 MZ2
7
2 2
7
2
2
2
2
2
p
ðM
þ 2D ðMW
MZ2 Þ ðM
p
Þ
þ
2D
3
þ
þ
2D
M
Þ
Z
Z
W
2
2
D
2
2G2 p2 2MW
HðMZ ; MW Þg4 2
4 þ ðð4D 6ÞM2 2p2 ÞM2 þ ðM2 p2 Þ2 g:
ðMW MZ2 Þ2 fMW
Z
W
Z
2
4G2 MW
MZ2 p2
The charged lepton contribution to the longitudinal LWW
is obtained from Eqs. (24) and (37),
ileptons
LWW
ðp2 þ Ml2 ÞMl2 g:
self-energy is
iWW
TZZ ¼ Hð0; 0Þ
g2 X
¼ 2
fiMl2 Ml þ Hð0; Ml Þ
2p l¼e;;
(46)
The quark contribution to the longitudinal LWW is obtained from Eqs. (24) and (39),
iquarks
LWW ¼ 3
g4 p6
HðMW ; MW Þ
4
4ðD 1ÞG2 MW
2 p2 Þ
g4 ð4MW
4
½4ðD 1ÞMW
þ 4ð2D 3Þp2
4
4ðD 1ÞG2 MW
2 þ p4 Hð0; M Þ
MW
W
2
g4 ðMW
p2 Þ2
4 p2
2ðD 1ÞG2 MW
4 þ 2ð2D 3Þp2 M2 þ p4 ½MW
W
g2 X
Vab Vab
fima ðm2a Mb2 Þ
2p2 ab
iMW
iMb ðm2a Mb2 Þ Hðma ; Mb Þ
½ðm2a Mb2 Þ2 p2 ðm2a þ Mb2 Þg:
(45)
g4
2 2
2ðD 1ÞDG2 MW
p
4
2
½DMW
þ ð5D 4Þp2 MW
þ Dð4D 7Þp4 :
(47)
(51)
V. ZZ SELF-ENERGY
The neutrino contributions to the transverse TZZ are
obtained from Eq. (23) by using
We first list the contributions to the transverse part.
A. The transverse ZZ self-energy
The Goldstone contribution to the transverse part of the
ZZ self-energy is
iGoldstone
¼ iMW
TZZ
2
ðMW
2
2
i
TZZ ¼ 3p
ðG2 þ g02 Þ2
þ Hð0; MW Þ
½M4 þ 2ð2D 3Þ
2ðD 1ÞG2 p2 W
2 þ p4 þ Hð0; 0Þ
p2 MW
p2
4ðD 1ÞG2
½g4 2ðG2 þ g02 Þg2 ðG2 þ g02 Þ2 :
(48)
The Faddeev-Popov contribution to the transverse part of
the ZZ self-energy is
iFP
TZZ ¼ g4
1
p2 Hð0; 0Þ:
2
G 2ðD 1Þ
g4 ðD 1Þ2
MW :
¼ i 22
D
G
G2 ð2 DÞ
Hð0; 0Þ:
4 D1
(53)
For the charged leptons, as well for the up and down
quarks, the contribution to the self-energy TZZ has the
same form (23),
X
1
charged fermions
iTZZ
iðAS þ BTÞð2 DÞmj
¼ 4
D1
j
þ Hðmj ; mj Þ 2m2j ðð2 DÞBT þ ASÞ
ð2 DÞ
ðAS þ BTÞ ;
(54)
p2
2
where the sum is over the flavors. For the leptons
(49)
The vector boson tadpole contribution to the transverse
part is
itadpole
TZZ
(52)
AS þ BT ¼
Therefore the neutrinos yield
02 2
þ p ÞðG þ g Þ
2ðD 1ÞG2 p2
2
G2
:
8
AS BT ¼ 0;
(50)
The WW loop contribution to the transverse part of the ZZ
AS BT ¼ G2 s2 ½12 þ s2 ;
(55)
AS þ BT ¼ G2 ½18 12s2 þ s4 :
For up quarks
AS BT ¼ 3G2 s2 ½12Qu þ s2 Q2u ;
AS þ BT ¼ 3G2 ½18 12s2 Qu þ s4 Q2u ;
125028-7
Qu ¼ 23:
(56)
D. BETTINELLI, R. FERRARI, AND A. QUADRI
PHYSICAL REVIEW D 79, 125028 (2009)
The fermion contribution to the longitudinal part of ZZ is
given by
X
ifermions
¼ 8BT
m2j Hðmj ; mj Þ;
(62)
LZZ
For the down quarks
AS BT ¼ 3G2 s2 ½12Qd þ s2 Q2d ;
AS þ BT ¼ 3G2 ½18 þ 12s2 Qd þ s4 Q2d ;
Qd ¼ 13:
j¼leptons;quarks
(57)
where B, T are taken from Eqs. (55)–(57).
B. The longitudinal ZZ self-energy
VI. SELF-ENERGY
Now we discuss the longitudinal part of ZZ self-energy.
The Goldstone contribution to the longitudinal part of
the ZZ self-energy is
iGoldstone
¼ iMW
LZZ
We first list the contributions to the transverse part.
A. The transverse self-energy
ðG2 þ g02 Þ2 2
ðMW þ p2 Þ
2G2 p2
ðG2 þ g02 Þ2 2
p
2G2
ðG2 þ g02 Þ2 2
Hð0; MW Þ
ðMW p2 Þ2 : (58)
2G2 p2
þ Hð0; 0Þ
The Goldstone contribution to the transverse part of the
self-energy is
iGoldstone
¼ iMW
T
g2 p2 g02
2ðD 1ÞG2
g2 g02
þ Hð0; MW Þ
2ðD 1ÞG2 p2
þ Hð0; 0Þ
The Faddeev-Popov contribution to the longitudinal part of
the ZZ self-energy is
iFP
LZZ ¼ g2 c 2 2
p Hð0; 0Þ:
2
(59)
The vector boson tadpole contribution to the longitudinal
part of the ZZ self-energy is
¼i
itadpole
LZZ
g4 ðD 1Þ2
MW :
2
D
G2
g4
g4
2 p2 Þ2 i
iWW
¼
Hð0;
M
Þ
ðM
W
MW
LZZ
2G2 p2 W
2G2 p2
1
2
MW
þ ½Dð4D 7Þ þ 4p2 :
(61)
D
iWW
T ¼ Hð0; 0Þ
4
2
½MW
þ 2ð2D 3Þp2 MW
þ p4 :
iFP
T ¼ e2
p2 Hð0; 0Þ:
2ðD 1Þ
(64)
The vector boson tadpole contribution to the transverse
part of the self-energy is
itadpole
¼i
T
g2 g02 ðD 1Þ2
MW :
2
D
G2
(65)
The WW loop contribution to the transverse part of the self-energy is
g2 g02 p6
g2 g02
2 p2 Þ½4ðD 1ÞM4 þ 4ð2D 3Þp2 M2 þ p4 HðMW ; MW Þ
ð4MW
W
W
2 4
4
4ðD 1ÞG MW
4ðD 1ÞG2 MW
Hð0; MW Þ
iMW
g2 g02
2
4
2
ðMW
p2 Þ2 ½MW
þ 2ð2D 3Þp2 MW
þ p4 4 2
2ðD 1ÞG2 MW
p
g2 g02
4 þ ð5D 4Þp2 M2 þ Dð4D 7Þp4 :
½DMW
W
2 2
2ðD 1ÞDG2 MW
p
The electromagnetic interaction gives
AS BT ¼ eQ;
and then
(63)
The Faddeev-Popov contribution to the transverse part of
the self-energy is
(60)
The WW loop contribution to the longitudinal part of the
ZZ self-energy is
g2 g02
ðM2 þ p2 Þ
2ðD 1ÞG2 p2 W
AS þ BT ¼ eQ;
¼4
ifermion
T
(67)
X
e2
Q2 ið2 DÞmj
D 1 j¼l;q;color
þ
p2 ð2 DÞ þ 4m2j
Hðmj ; mj Þ :
2
For small p2 one gets
125028-8
(66)
(68)
ONE-LOOP SELF-ENERGIES IN THE ELECTROWEAK . . .
ifermion
¼ Oðp2 Þ:
T
(69)
B. The longitudinal self-energy
The Goldstone contribution to the longitudinal part of
the self-energy is
0 2 2
2
1
p
2 gg
2
iGoldstone
¼
M
1
Hð0; MW Þ
M
W
L
2
2p2 W G
MW
2
p4
p
2 Hð0; 0Þ iMW
þ1 :
(70)
2
MW
MW
The Faddeev-Popov contribution to the longitudinal part of
the self-energy is
iFP
L ¼ e2 2
p Hð0; 0Þ:
2
PHYSICAL REVIEW D 79, 125028 (2009)
It is remarkable that the sum of all the contributions (70)–
(73) amounts to zero photon longitudinal self-energy. This
is in agreement with the Ward identity for QED derived in
Ref. [1].
VII. Z SELF-ENERGY
We first list the contributions to the transverse part.
A. The transverse Z self-energy
The Goldstone contribution to the transverse part of the
Z self-energy is
iGoldstone
¼ Hð0; 0Þ
TZ
(71)
2 þ p2 Þ Hð0; M Þ
ðMW
W
The vector boson tadpole contribution to the longitudinal
part of the self-energy is
itadpole
L ¼ i
2 02
g g ðD 1Þ
2
MW :
D
G2
2
(72)
The WW loop contribution to the longitudinal part of the
self-energy is
2
g2 g02
p2 MW
2p2
þ
þ
þ 2Dp2
iWW
¼
i
MW
L
2
2
D
G2 p2
1
2 2
3p2 þ Hð0; MW Þðp2 MW
Þ :
(73)
2
g0 g3 p2
gg0 ðkG2 þ g02 Þ
þ iMW
2
2ðD 1ÞG2 p2
2ðD 1ÞG
4 þ 2ð2D 3Þp2 M2 þ p4 :
½MW
W
iWW
TZ ¼ Hð0; 0Þ
(74)
(75)
The Faddeev-Popov contribution to the transverse part of
the Z self-energy is
iFP
TZ ¼ g3 g0
1
p2 Hð0; 0Þ:
2 2ðD 1Þ
G
(76)
The vector boson tadpole contribution to the transverse
part of the Z self-energy is
itadpole
¼i
TZ
For the longitudinal part we get
ifermion
¼ 0:
L
gg0 ðkG2 þ g02 Þ
2ðD 1ÞG2 p2
g3 g0 ðD 1Þ2
MW :
2
D
G2
(77)
The WW loop contribution to the transverse part of the Z
self-energy is
g3 g0 p6
g3 g0
2
4
2
HðM
;
M
Þ
ð4MW
p2 Þ½4ðD 1ÞMW
þ 4ð2D 3Þp2 MW
þ p4 W
W
4
4
4ðD 1ÞG2 MW
4ðD 1ÞG2 MW
Hð0; MW Þ
iMW
g3 g0
2
4
2
ðMW
p2 Þ2 ½MW
þ 2ð2D 3Þp2 MW
þ p4 4 2
2ðD 1ÞG2 MW
p
g3 g0
4
2
½DMW
þ ð5D 4Þp2 MW
þ Dð4D 7Þp4 :
2 p2
2ðD 1ÞDG2 MW
The fermion contribution to the transverse part of Z is
X
ðASÞj
ið2 DÞmj
¼
4
ifermion
TZ
D1
j¼leptons;quarks;color
ðD 2Þ
þ 2m2j ;
þ Hðmj ; mj Þ p2
(79)
2
where the constants A, B, S, T are as follows.
Neutrinos:
AS BT ¼ 0;
AS þ BT ¼ 0:
(78)
Charged leptons:
AS BT ¼ AS þ BT ¼ eG½14 s2 :
(81)
Up quarks:
AS BT ¼ AS þ BT ¼ eQu G½14 s2 Qu ;
Qu ¼ 23:
(82)
Down quarks:
AS BT ¼ AS þ BT ¼ eQd G½14 þ s2 Qd ;
(80)
125028-9
Qd ¼ 13:
(83)
D. BETTINELLI, R. FERRARI, AND A. QUADRI
PHYSICAL REVIEW D 79, 125028 (2009)
B. The longitudinal Z self-energy
The Goldstone contribution to the longitudinal part of
the Z self-energy is
0
02
1 1
gg
2 g
iGoldstone
þ
G
¼
M
W
LZ
2 p2
G
G
2
2
p
2
MW
1 Hð0; MW Þ
2
MW
2
p4
p
2 Hð0; 0Þ iMW
þ 1 : (84)
2
MW
MW
Hð0; 0Þ
2
2
½ð3 þ 2DÞMW
þ 2ð2 þ DÞp2 2g2 g02 ð1 þ kÞMW
2 ½ð3 þ 2DÞM2 þ 2ð2 þ DÞp2 g
2g04 ð1 þ kÞMW
W
(89)
and
Hð0; MZ Þ
g3 g0 1 2
p Hð0; 0Þ:
G2 2
g3 g0 ðD 1Þ2
MW :
2
D
G2
2 p2 g2 þ 2g02 ðk þ 1ÞM2 g:
f½ð2k þ 1ÞMW
W
(90)
(85)
The vector boson tadpole contribution to the longitudinal
part of the Z self-energy is
itadpole
¼i
LZ
(86)
The WW loop contribution to the longitudinal part of the
Z self-energy is
g3 g0
p2 M2
2p2
WW
iLZ ¼ 2 2 iMW þ W þ
þ 2Dp2
2
2
D
Gp
1
2 2
3p2 þ Hð0; MW Þðp2 MW
Þ :
(87)
2
Thus they are zero on shell. Similarly, one can prove that
2
the coefficients of Gð0Þ and of GðMW Þ
on-shell p2 ¼ MW
are zero in generic D dimensions.
B. TZZ terms proportional to Hð0; 0Þ and Hð0; MW Þ
We collect the terms in the self-energy in Eqs. (48)–(51)
proportional to Hð0; 0Þ and Hð0; MW Þ:
Hð0; 0Þ p2 g4 4
fp MZ4 g;
4 G2
D 1 4MW
¼ 0:
g4 Hð0; MW Þ
2
ðp2 MZ2 Þð2MW
MZ2 p2 Þ
4 p2
2ðD 1ÞG2 MW
4 þ 2ð2D 3Þp2 M2 þ p4 :
½MW
W
(88)
VIII. PHYSICAL UNITARITY FOR DIAGONAL
ELEMENTS
It is simple to trace, at the one-loop level, the contributions due to unphysical modes (Faddeev-Popov ghosts,
Goldstone bosons, and scalar parts of the vector mesons).
These contributions have to cancel when we evaluate the
transverse part of the self-energies on shell. Here we show
that this cancellation works in generic D dimensions when
the transverse part is taken on shell.
A. TWW : The unphysical Hð0; 0Þ, Hð0; MZ Þ and
GðMW Þ, Gð0Þ
We collect the terms in the self-energy in Eqs. (32)–(36)
proportional to Hð0; 0Þ and Hð0; MZ Þ, in order to check
2 since
physical unitarity. They must vanish at p2 ¼ MW
they are due to the presence of unphysical modes
(Goldstone and longitudinal parts of the vector bosons).
We get
(91)
and similarly
For the longitudinal part, one has
ifermion
LZ
2 p2 Þ
ðMW
4
½G4 ðk þ 1Þ2 MW
4 2
4ðD 1ÞG4 ðk þ 1ÞMW
p
2
þ 2ð2D 3Þg2 G2 ðk þ 1Þp2 MW
þ g4 p4 The Faddeev-Popov contribution to the longitudinal part of
the Z self-energy is
iFP
TZ ¼ 2 Þ
g2 ðp2 MW
2
fg4 p2 ðp2 þ MW
Þ
2
4
4ðD 1ÞG ðk þ 1ÞMW
(92)
Thus, physical unitarity is again working for the self-mass
of Z.
C. T : The limit p2 ¼ 0
We collect the terms in the self-energy in Eqs. (63)–(66)
proportional to Hð0; 0Þ and then we put p2 ¼ 0 in order to
check that the photon remains with zero mass. One verifies
that
2 02 1
p4
1 Hð0; 0Þp2 g g
þ
1
¼ 0: (93)
2
4
p2 ¼0
2
2
G
4M
D1
W
For the terms involving Hð0; MÞ, one needs the identity
iM
p2
Hð0; MÞ ¼ 2 1 2 ðD 4Þ þ Oðp4 Þ:
(94)
M
M
It is then straightforward to verify that
lim iT ¼ 0:
p2 ¼0
Thus the mass of the photon remains null.
125028-10
(95)
ONE-LOOP SELF-ENERGIES IN THE ELECTROWEAK . . .
D. Unitarity for the TZ
PHYSICAL REVIEW D 79, 125028 (2009)
massless except the top with Mtop ¼ 174:2 GeV) are
The unitarity properties of TZ are strictly connected to
the process where this graph contributes (e.g. Z ! l þ l).
Thus, more graphs are necessary in order to verify physical
unitarity. This subject is outside the scope of the present
work.
IX. W AND Z SELF-MASSES
By using the procedure of extracting the finite parts from
the D-dimensional amplitudes described in Sec. III, we
evaluate the self-masses for W and Z bosons. Since we
have already thoroughly examined the properties of the
amplitudes in D dimensions at the on-shell momenta, the
self-masses can be evaluated by any computer algorithm.
We do not reproduce the results in the present paper.
X. PARAMETER FIT
In this section we provide an estimate of the parameters
introduced in the model. The parameters g, g0 , M can be
fixed by experiments that are essentially at low momentum
transfer: for instance, , G , and the e scattering that
provides a precise value of sinW . Our calculation of the
self-energies can be checked on the physics of the vector
bosons W, Z. The physical masses are the input for the
determination of the extra parameters of the model: and
.
For the processes at nearly zero momentum transfer, we
can use the Particle Data Group [10] values,
¼ 1=137:059 991 1ð46Þ;
(96)
G ¼ 1:166 37ð1Þ 105 GeV2 ;
and from e scattering [11],
sin 2 W ¼ 0:2324 0:011:
(97)
Z ¼ 2:203 GeV
ðexp : ð2:4952 0:0023Þ GeVÞ;
W ¼ 1:818 GeV
ðexp : ð2:141 0:041Þ GeVÞ:
(101)
These values are quite encouraging for the calculation of
further radiative corrections. However, one should consider
only the order of magnitude of these numbers. In fact, they
depend strongly on the value of sin2 W . Only a fit including other sensitive quantities will be able to reduce their
variability.
XI. COMPARISON WITH THE STANDARD MODEL
The comparison with the linear theory is performed in
the limit ¼ 0; i.e. the tree-level Weinberg relation between the masses of the intermediate vector mesons W and
Z holds. The standard model self-energy corrections are
given by the same diagrams of the nonlinear model evaluated at ¼ 0, plus the amplitudes involving one internal
Higgs line. The latter have been collected in [12].
We list below the amplitudes contributing to the transverse part of the W self-energy with an internal Higgs line.
The results are valid in the Landau gauge and in the limit
D ¼ 4. The Higgs tadpole is
iHiggs
TWW
With these inputs we can evaluate the values for the other
two parameters by imposing the conditions on the mass
corrections,
2 ¼ ð80:428 0:039Þ2 GeV2 ;
ðgMÞ2 þ MW
M2 G2 ð1
þ Þ þ
MZ2
¼ ð91:1876 0:0021Þ2
GeV2 :
(99)
One gets
¼ 0:0085;
¼ 283 GeV:
(100)
¼
ig2
:
4 mH
(102)
The Higgs-gauge bubble is
2
2
g2
MW
m2H
MW
gauge
i
1
þ
¼
i
mH
iHiggs
MW
TWW
12
p2
p2
p2
2i
2 þ ðm2 M2 Þ2 1 þ p2
M
H
W
ð4Þ2 W
p2
2
þ 10MW
2m2H HðMW ; mH Þ
We get
pffiffiffiffiffiffiffiffiffiffi
e
e
4
¼ 0:6281;
g0 ¼ ¼ 0:3456;
g¼ ¼
s
c
s
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(98)
1
M ¼ pffiffiffi
¼ 123:11 GeV:
4 2GF
tad
m2H 2 2
1 2 p Hð0; mH Þ :
p
The Higgs-Goldstone bubble is
g2
m2H
Higgs Goldstone
iTWW
i 1 þ 2 mH
¼
12
p
2
2i
2 p
m
3
ð4Þ2 H
2
2
mH
2
þ 1 2 p Hð0; mH Þ :
p
(103)
(104)
We list here the various contributions to the transverse part
of the Z self-energy with an internal Higgs line. The Higgs
tadpole is
The widths of the vector mesons obtained from the imaginary parts of the self-energies (all fermions are taken to be
125028-11
tad
iHiggs
¼
TZZ
iG2
:
4 mH
(105)
D. BETTINELLI, R. FERRARI, AND A. QUADRI
PHYSICAL REVIEW D 79, 125028 (2009)
The Higgs-gauge bubble is
G2
MZ2 m2H
M2
Higgs gauge
i 1 þ 2 2 MZ i 2Z mH
iTZZ
¼
12
p
p
p
2i
1
M2 þ ðm2H MZ2 Þ2 2 þ p2
ð4Þ2 Z
p
þ 10MZ2 2m2H HðMZ ; mH Þ
m2 2 2
p
Hð0;
m
Þ
1 H
H :
p2
(106)
The Higgs-Goldstone bubble is
G2
m2H
Higgs Goldstone
i 1 þ 2 mH
¼
iTZZ
12
p
2i
p2
2
m
3
ð4Þ2 H
2 2
mH
2
þ 1 2 p Hð0; mH Þ :
p
ACKNOWLEDGMENTS
(107)
These results can be used in order to estimate the numerical
impact of the Higgs corrections to the self-masses. We
choose as a reference value mH ¼ 165 GeV and evaluate
the corrections with the same input parameters in Eq. (98)
and ¼ 283 GeV. The shifts in the self-masses are
Higgs
MW
¼ 0:629 GeV;
parameter v is not a physical constant. The scheme is very
rigid and it should be checked by comparison with the
experimental measures.
The calculation has been performed in the Landau gauge
and by using the symmetric formalism whenever it was
possible. We checked the physical unitarity and the absence of v in the measurable quantities.
A very simple evaluation has been performed for the
parameters of the classical action, by using leptonic processes. The parameter that describes the departure from the
Weinberg relation between MW and MZ is very small, and
the scale of the radiative corrections is of the order of a
hundred GeV. This means that the model is on solid
ground, and it is reasonable to make further efforts for
the evaluation of the radiative corrections in other
processes.
This work is supported in part by funds provided by the
U.S. Department of Energy under Cooperative Research
Agreement No. DE FG02-05ER41360. R. F. acknowledges
the warm hospitality of the Center for Theoretical Physics
at MIT, Massachusetts, where part of this work was
completed.
APPENDIX: LIMIT D ¼ 4 FOR THE
LOGARITHMIC INTEGRAL
MZHiggs ¼ 0:531 GeV:
(108)
Higgs
MW
and
These estimates are rather intriguing.
Higgs
strongly depend on the value of (they vary by
MZ
more than 20% in the range from ¼ 200 GeV to ¼
350 GeV). Compensations of the Higgs contributions to
electroweak observables may be triggered by a change in
the scale of the radiative corrections. A more refined fit
to the electroweak precision observables is required in
order to discriminate between the linear and the nonlinear
theory.
We collect, in this appendix, some relevant formulas.
1 Z D
i
d q 2
;
D
ð2Þ
q m2
1 Z D
1
1
d q 2
Hðm; MÞ :
ð2ÞD
q m2 ðp þ qÞ2 M2
m (A1)
The following identities allow one to prove the cancellation of infrared divergences due to the massless photon.
XII. CONCLUSIONS
The one-loop evaluation of self-energies for the vector
mesons in the electroweak model based on a nonlinearly
realized gauge group has been explicitly performed in D
dimensions. The finite amplitudes in D ¼ 4 have been
achieved according to the procedure suggested by the local
functional equation associated with the local invariance of
the path integral measure. In practice, this implies the
minimal subtraction of poles in D 4 on properly normalized amplitudes. Thus, in this model the Higgs sector is
absent and the parameters are fixed by the classical
Lagrangian (no free parameters for the counterterms and
therefore no on-shell renormalization). Two new parameters appear: a second mass term parameter and a scale of
radiative corrections. The spontaneous symmetry breaking
125028-12
@
Hðm; MÞjm2 ¼0
@m2
ð3 D2 Þ Z 1
i
¼
dxð1 xÞ
ð4ÞðD=2Þ ð2Þ
0
GðMÞ ½M2 x p2 xð1 xÞðD=2Þ3 :
(A2)
Gð0Þ lim GðMÞ
M¼0
¼
ð3 D2 Þ
i
2 ðD=2Þ3
½p
ð2Þ
ð4ÞD=2
ðD2 2ÞðD2 1Þ
:
ðD 3Þ
(A3)
ONE-LOOP SELF-ENERGIES IN THE ELECTROWEAK . . .
PHYSICAL REVIEW D 79, 125028 (2009)
ð1 D2 Þ 2 ðD=2Þ2
1
1
ðm Þ
m ¼ lim
¼ 0:
2
m¼0 m
m¼0 ð4ÞD=2
ð1Þ
(A4)
By following the Feynman prescription, one obtains the
following:
For 0 < p2 < ðM mÞ2 , > 0 and then the integral is
pffiffiffiffi
b ða þ b þ cÞ
ln
þ
2 þ lnða þ b þ cÞ þ
2a
c
2a
pffiffiffiffi
2c þ b pffiffiffiffi ;
(A10)
ln
2c þ b þ lim
ð4 D2 Þ
@
i
½p2 ðD=2Þ4
GðMÞj
¼
M¼0
@M2
ð4ÞD=2 ð2Þ
ðD2 2ÞðD2 2Þ
:
ðD 4Þ
(A5)
For the integral in Eq. (28), use the notation
a ¼ p2 ;
b ¼ p2 þ M2 m2 ;
c ¼ m2 (A6)
i.e.
Z1
dx lnðp2 x2 þ ½M2 m2 p2 x þ m2 Þ
0
¼
Z1
dx lnðax2 þ bx þ cÞ:
(A7)
0
If one or two masses are zero, one gets
Z1
dx lnðx½p2 x þ M2 p2 Þ
0
M2 M2 ¼ 2 þ lnjp2 M2 j þ 2 ln
2
2
p M p
i
for p2 > ðM þ mÞ2 , > 0 and then the integral is
pffiffiffiffi
b ða þ b þ cÞ
ln
þ
2 þ lnða þ b þ cÞ þ
2a
c
2a
pffiffiffiffi
pffiffiffiffi
2c þ b pffiffiffiffi i
;
(A11)
ln
a
2c þ b þ for p2 ¼ ðM mÞ2 or p2 ¼ ðM þ mÞ2 , ¼ 0 and then
the integral is
2 þ lnða þ b þ cÞ þ
p2 M 2
ðp2 M2 Þ:
p2
(A8)
¼ ½m2 þ M2 p2 2 4m2 M2 :
(A9)
Let
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b ða þ b þ cÞ
ln
;
2a
c
(A12)
for ðM mÞ2 < p2 < ðM þ mÞ2 , < 0 and then the
integral is
b
b
lnc þ 1 þ
ln½a þ b þ c
2 2a
2a
pffiffiffiffiffiffiffiffi þb
b
1 2a
1 pffiffiffiffiffiffiffiffi
p
ffiffiffiffiffiffiffiffi
þ
tan
tan
; (A13)
a
where 2 < tan1 ðxÞ < 2 , x 2 R.
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