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Pole masses of gauge bosons
F.Jegerlehner, M.Yu.Kalmykov, O.Veretin.
DESY Zeuthen
JINR Dubna
Karlsruhe U.
ACAT 2002, Moscow
. Pole Mass
. Expansion procedure and calculation
. MS renormalization and two­loop results

Pole mass. General consideration.
The position of the pole s P of the massive (gauge) boson
propagator in a QFT is defined by equation:
s P -m 2
- #(sP , m 2 , · · ·) = 0
For gauge bosons # is transversal part of the one­particle
irreducible self­energy (depends on all SM parameters).
The pole s P is
. gauge invariant
. infrared stable
. complex in general: Ims p #= 0
We can define pole (on­shell) mass M and width # via
s P = M 2
- iM#
By iterative solution of pole formula up to 2­loops we have:
s P = m 2
+ # (1) (m 2 , m 2 )
+ # (2) (m 2 , m 2 ) +# (1) (m 2 , m 2 )# (1)# (m 2 , m 2 ).
# (L) is the bare (m = m 0 ) or MS­ renormalized (m the
MS­mass) L­loop contribution to #, and the prime denotes
the derivative with respect to p 2 . In this way we need to
evaluate propagator type diagrams and their derivatives at
p 2 = m 2 .
Kalmykov, ACAT'2002, Moscow 2

Pole mass and #­Z mixing.
In neutral sector because of mixing two neutral bosons Z
and # form a 2 â 2 matrix propagator
D -1 (p 2 ) = # #
p 2
-# ## (p 2 ) # #Z (p 2 )
#Z# (p 2 ) p 2
-m 2
Z - # ## (p 2 )
# #
The equation for the pole is now modified
s P -m 2
Z -#ZZ (s P ) -
# 2
#Z
(s P )
s P -# ## (s P )
= 0.
# Mixing term # 2
#Z starts contributing at two­loop
# Photon term # ## only contributes beyond the two­loop
Solution (up to 2­loops):
s Z = m 2
Z
+ # (1)
ZZ
(m 2
Z
)
+ # (2)
ZZ
(m 2
Z
) +# (1)
ZZ
(m 2
Z
)# (1)
ZZ
# (m 2
Z
) +
# # (1)
#Z
(m 2
Z
) # 2
m 2
Z
.
Kalmykov, ACAT'2002, Moscow 3

Diagrams and topologies
To be computed on­shell (p 2 = m 2
Z ):
#(p 2 ) = # 1 (p 2 ) +# 2 (p 2 ) + . . .
2
P =
+
+ H
P =
+
H H H
+ + +
H
H
H
H +
H
H H
+ +
H
H
+
H
H
H
+ + + +
+
+
+ + +
1
Bosonic contribution
Number of diagrams linearR # gauge nonlinearR # gauge
one­loop : # 50
two­loop : 1P I T otal 1P I T otal
Z 616 2348 410 1837
W 792 4084 537 2942
With one fermion family
two­loop : 1P I T otal 1P I T otal
Z 802 4410 550 3631
W 990 7780 669 5604
Kalmykov, ACAT'2002, Moscow 4

Gauge
The gauge fixing Lagrangian is
L g.f. = -
1
# W
F + F - -
1
2# (# µ A µ ) 2
-
1
2# Z
(# µ Z µ
- # Z MZh) 2
where for the linear R # gauge is
F ± = # µ W ±
µ # i# WMW # ±
and the nonlinear R # gauge is defined as
F ± = # µ W ±
µ # i# WMW # ± # ieA µ W ±
µ ± ig
sin 2 # W
cos # W
Z µ W ±
µ
K. Fujikawa '73
D.A. Dicus & C. Kao '94
The old vertices:
{A µ , Z µ } W ±
µ # # absent
{A # , Z # , A µ A # , A µ Z # , Z µ Z # } W ±
µ W #
# modified
{A µ , Z µ } # ± # ± modified
W ± # ± {c # , #Z } modified
The new vertices:
{A µ A µ , A µ Z µ , Z µ Z µ } # ± # ± new
{A µ , Z µ } W ± # ± {c # , #Z } new
Kalmykov, ACAT'2002, Moscow 5

Evaluation of 2­loop self­energies
One can choose one of the approaches:
# Analytical # Numerical # Semianalytical
Each of them meets with the typical problems:
. Reduction of tensor integrals to set of scalar integrals
. Extraction of UV­ and IR­poles
. Analytical/numerical evaluation of scalar integrals
for di#erent region of variables (numerical stability)
Let us consider some of the approaches within our task:
. J. Fleischer & O.V. Tarasov '94
Small momentum expansion and construction of Pad’e
approximants.
Demand a lot of coe#cients of expansion, specially for
on­shell diagrams.
. A. Ghinculov & Y.­P. Yao '98; '01
Each two­loop diagram is reduced to ten scalar integrals
and their derivatives which can be evaluated numerically.
Applicable to diagrams (their combination) which are
free of mass (infrared) singularities.
. G. Passarino '01; '02
Present talk
. V.A. Smirnov '90; '95; '99; '01;
The asymptotic expansion with respect to small
parameter(s) existing in the initial diagram.
It is applied for special values of kinematical variables.
Kalmykov, ACAT'2002, Moscow 6

. Di#erential Equation Method
For each diagram the linear system of partial di#erential
equations in masses and momenta can be constructed,
which can be used for analytical A. Kotikov '91 or
numerical S. Laporta '00 calculation of diagram.
. Di#erence Equation Method
A linear system of di#erence equations is constructed and
solved analytically O.V. Tarasov '96; '00 or numerically
S. Laporta '00 .
Main application is the calculations of master integrals.
Not yet applied to real multi­loop calculations.
. G. Weiglein, R. Scharf & M. B˜ohm '94
S. Bauberger,F.A. Berends, M. B˜ohm, M. Buza, '95
S. Bauberger, M. B˜ohm, '95
Reduction of two­loop self­energy diagrams to scalar
integrals without numerator which can be calculated
numerically or analytically.
. O.V. Tarasov '97
Exact reduction of tensor two­loop propagator type
diagrams with arbitrary masses and momentum to set of
scalar master integrals.
Several masses led to the cumbersome expressions, which
are di#cult to simplify.
The result is expressible in terms of master­integrals
which demand a special consideration
. FORM based package ON­SHELL2
Single scale propagator­type diagrams with arbitrary
power of propagators are reduced to a set of master­
integrals which calculated analytically.
Kalmykov, ACAT'2002, Moscow 7

Evaluation by expansion
In order to check the gauge invariance we perform all
calculations in R # gauge with three independent gauge
parameters # W , # Z , # # . # Vector boson propagator reads
D µ# (p) = i
p 2
-m 2
Z
# -g µ# + (1 - #)
p µ p #
p 2
-m 2
Z
#
Then there are several scales: m V , # # V m V , mH , m t
We perform expansion in 3 steps:
1. Taylor (naive) expansion in (# V - 1): i.e., propagator
of the vector bosons look like
D V
µ#
(p) # -g µ# + (1 - # V ) p µ p #
p 2
-m 2
V
- (1 - # V ) 2 m 2
V p µ p #
(p 2
-m 2
V
) 2
2. Expansion in small parameter
sin 2 # W = 1 -
m 2
W
m 2
Z
< 0.25.
After expansion in sin 2 # W : the diagrams without
Higgs are the two­loop on­shell diagrams can be
evaluated analytically by FORM package ONSHELL2.
J. Fleischer & M.Yu. Kalmykov '00
Diagrams with massless fermion loop demand special
consideration.
3. Large mass expansion in m 2
Z /m 2
H < 0.64
Package TLAMM L.V. Avdeev et al., '97
Kalmykov, ACAT'2002, Moscow 8

ONSHELL2
www­zeuthen.desy.de/#kalmykov/onshell2/onshell2.html
J. Fleischer, M.Yu. Kalmykov & A. Kotikov '99
A.I. Davydychev & M.Yu. Kalmykov '01
J. Fleischer & M.Yu. Kalmykov '00
All two­loop on­shell self­energy diagrams with one mass,
with arbitrary power of propagators are reduced to 13
scalars integrals plus bubble­type integrals and product of
the one­loop integrals.
J111
V1001 J011
V1111
F00001
F10100
F01100 F00101
F 11111 F10110
F00111 F10101
Bold and thin lines correspond to the mass and massless
propagators, respectively.
Kalmykov, ACAT'2002, Moscow 9

Massless fermion contribution
In contrast to the bosonic corrections where all diagrams
can be expanded from very beginning in the small parameter
sin 2 # W the individual diagrams with massless fermion
loop develop the thresholds singularity which look like
ln j sin 2 # W . To control these terms it is desirable to have the
exact analytical result without expansion. Using Tarasov's
recurrence relations we reduce all diagrams with massless
fermion loop to following set of master­integrals
F20200
M
M
M
M
M m
M
M
M
m
M
m
J002
F00002
a
s 1
s 2
a
(a,s ,s ,b)
1 2
V1002
b
F20100
b
s
V2001
V2002
(a,b,s)
The analytical results for these diagrams are presented by
Scharf, Tausk, '94 We performed an independent analytical
calculations of these master integrals.
After summing of all contribution the singular terms,
ln j sin 2 # W are canceled.
Kalmykov, ACAT'2002, Moscow 10

Renormalization.
We have s P parametrized in terms of bare parameters
s P = f(e 0 , mZ,0 , mW,0 , mH,0 ).
Renormalization of charges and masses:
e 0 = µ # e Z e
m 2
0
= m 2
Zm
We need: + Z e , ZH , Z t up to 1­loop
+ ZW , ZZ up to 2­loops
We don't need: - field renormalization
- renormalization of ghost sector
- gauge parameters renormalization
The charge MS renormalization constant can be calculated
in the unbroken theory
Z e = 1 + e 2
16# 2 #
# -
7
2
+ 1
3 nG # 10
9 N c + 2 ## ,
where e, mZ , mW , mH are MS parameters. N c is
color factor (N c = 3) and nG is a number of fermion
families (n G = 3).
Kalmykov, ACAT'2002, Moscow 11

Mass renormalization.
m 2
0
= m 2 # 1 + g 2
16# 2
Z (1,1)
#
+ g 4
(16# 2 ) 2
Z (2,1)
#
+ g 4
(16# 2 ) 2
Z (2,2)
# 2
#
The mass renormalization constants cannot be calculated
from the unbroken theory. The calculation of the Feynman
diagrams in the Standard Model is required.
One­loop results:
Z (1,1)
H
= -
3
2 -
3
4
m 2
Z
m 2
W
+
3
4
m 2
H
m 2
W
+ #
fermion
1
2
m 2
f
m 2
W
Z (1,1)
W
= -
3
4
m 2
H
m 2
W
- 3 m 2
W
m 2
H
-
3
2
m 4
Z
m 2
H m 2
W
+
3
4
m 2
Z
m 2
W
-
17
3
+ #
fermion
# 2
m 4
f
m 2
W m 2
H
-
1
2
m 2
f
m 2
W
# + 1
3 nG (N c + 1)
Z (1,1)
Z
= -
3
4
m 2
H
m 2
W
- 3 m 2
W
m 2
H
-
3
2
m 4
Z
m 2
W m 2
H
+ 11
12
m 2
Z
m 2
W
-7
m 2
W
m 2
Z
+
7
6
+ #
fermion
# 2
m 4
f
m 2
W m 2
H
-
1
2
m 2
f
m 2
W
#
+
1
3 nG # m 2
Z
m 2
W
# 11
9 N c + 3 # + m 2
W
m 2
Z
# 20
9 N c + 4 # #
-
1
3 nG # 22
9 N c + 6 #
Kalmykov, ACAT'2002, Moscow 12

2­loop constants in MS. Exact results:
Z (2,1)
W
= 63
64
m 4
H
m 4
W
-
3
4
m 2
H
m 2
W
-
3
8
m 2
H m 2
Z
m 4
W
- # 301
192
+ # 55
432 N c +
5
16 # nG # m 4
Z
m 4
W
+ # 17
12
+ # 29
108 N c +
3
4 # nG # m 2
Z
m 2
W
+ # 31
12 - # 44
27 N c + 4 # nG # m 4
Z
m 2
H m 2
W
- # 17
2 - # 40
27 N c +
8
3 # nG # m 2
Z
m 2
H
- # 176
3 -
4
3
(N c + 1)n G
# m 2
W
m 2
H
+ # 59
24
+ # 22
27 N c + 2 # nG # m 6
Z
m 2
H m 4
W
-
53
3
+ # 343
216 N c +
31
24 # nG
. Z (2,2)
W , Z (2,1)
Z , Z (2,2)
Z have similar structure.
. Massive top corrections m t generate terms proportional
to powers of m t .
. The terms proportional to m 4
H , m 2
H are generated by
pure bosonic contributions.
Kalmykov, ACAT'2002, Moscow 13

Renormalization group analysis
We can check two­loop coe#cients Z (2,2) and Z (2,1) in
mass renormalization formula
m 2
0
= m 2 # 1 + g 2
16# 2
Z (1,1)
#
+ g 4
(16# 2 ) 2
Z (2,1)
#
+ g 4
(16# 2 ) 2
Z (2,2)
# 2
#
= m 2 # 1 + z (1)
#
+ z (2)
# 2
+ . . . #
using renormalization group.
Introduce anomalous mass dimension
# = µ 2 d
dµ 2
log m 2 .
From
0 = µ 2 d
dµ 2
log m 2
0
we have
# =
1
2 g
#
#g
z (1) ,
# # + # i
# g i
#
#g i
+ # i
# i m 2
i
#
#m 2
i
# z (n) = 1
2 g
#
#g
z (n+1) ,
where
z (n) = # k=1
# g 2
16# 2
# k
Z (n,k) .
Kalmykov, ACAT'2002, Moscow 14

Pole structure
Check of 1/# 2 pole:
From previous formula we can derive
2Z (2,2) = # Z (1,1)
# 2
+2 16# 2
g 3
# (1)
g Z (1,1) + # i
Z (1,1)
m i
m 2
i
#
#m 2
i
Z (1,1)
Check 1/# pole:
Use relation between charges
e 2 = g 2 # 1 -
m 2
W
m 2
Z
# and
1
e 2
=
1
g 2
+
1
g #2
.
Di#erentiating the former formulae w.r.t. log µ 2 we derive
# g 2
16# 2
# 2
# Z (2,1)
Z - Z (2,1)
W # c 2 = g 2
# (2)
g #
g #3
s 4
-
# (2)
g
g
s 2 c 2 .
#­functions for g and g # are calculated from symmetric
phase theory D.R.T. Jones '82
M.E. Machacek & M.T. Vaughn '83
# g # | Nc=3 = # 1
12
+
10
9
nG # g #3
16# 2
+ # 1
4
+
95
54
nG # g #5
(16# 2 ) 2
+ # 3
4
+
1
2
nG # g #3 g 2
(16# 2 ) 2
# g | Nc=3 = # -
43
12
+
2
3
nG # g 3
16# 2
+ # -
259
12
+
49
6
nG # g 5
(16# 2 ) 2
+ # 1
4
+
nG
6
# g 3 g #2
(16# 2 ) 2
Kalmykov, ACAT'2002, Moscow 15

MSRenormalization
MSrenormalized relation can be written as
sZ = m 2
Z +
# # #
# (1)
0,V
# # # MS
+
# # #
# (2)
0,V
+ # (1)
0,V
# (1)
0,V
# # # # MS
where
# # #
# (1)
0,V
# # # MS
= m 2
V (µ)
e 2
16# 2 sin 2
#W
lim
##0
# 1
#
Z (1,1)
V
+X (1)
0,V
#
where we have introduced new function
# (1)
0,V = m 2
0,V
g 2
0
16# 2
X (1)
0,V
# # #
# (2)
0,V
+ # (1)
0,V
# (1)
0,V
# # # # MS
= lim
##0
# ## (2)
0,V
+ # (1)
0,V
# (1)
0,V
#
+m 2
V (µ)
1
#
# e 2
16# 2 sin 2
#W
# 2
# #Z (1,1)
V
+
# # #g 2
g 2
# # + # j
Z (1,1)
m 2
j
#
#m 2
j
# #X (1)
0,V
+m 2
V (µ) # e 2
16# 2 sin 2
#W
# 2
# # 1
#
Z (2,1)
V +
1
# 2
Z (2,2)
V
# # # #
where the sum runs over all species of particles j = Z, W, H, t and
# # #g 2
g 2
# # =
cos 2
#W
sin 2
#W # Z (1,1)
W - Z (1,1)
Z #
+ sin 2
#W
#
#-7 +
2
3
nG
# # 10
9
N c + 2
# # # #
Kalmykov, ACAT'2002, Moscow 16

Results
M 2
V
m 2
V
= 1 +
e 2
16#s 2
X 1,V + # e 2
16#s 2
# 2
X 2,V ,
X 2,V =
m 4
H
m 4
V
5
# i=0
A i s 2i
,
A i =
5
# j=0
A V
j
# m 2
V
m 2
H
# j
, V = Z, W
All parameters in MSscheme.
Six coe#cients calculated analytically.
Expansion in powers and log's (i.e. is asymptotic expansion not naive
Taylor expansion).
Expansion coe#cients A V
i,j given by a small set of transcendental
constants, like
S 0 =
#
# 3
, S 1 =
#
# 3
ln 3,
S 2 =
4
9 # 3
Cl 2 (#/3), S 3 = #Cl 2 (#/3),
After summing all diagrams with massless fermion loop all singularites
of type ln j sin 2
#W canseled # infrared finiteness of the massless
fermion contribution to the pole mass.
After UV renormalization the pole mass is a finite expression. #
infrared finiteness of the bosonic contribution to the pole mass.
Pole mass of gauge boson is infrared finite quantity
To restore gauge invariance of pole s p the tadpole shall be taking
into account.
RG invariance also requires the tadpole ­ ## s contribution
Kalmykov, ACAT'2002, Moscow 17

MS mass in terms of on­shell mass
Inverse of our ``master formula'' express all MS parameters in terms of
on­shell ones:
m 2
V = M 2
V - “
# (1)
V -
# # #
# (2)
V + # (1)
V # (1)
V
# # # # MS
- # j
(#m 2
j ) (1) #
#m 2
j
“
# (1)
V - (#e) (1) #
#e
“
# (1)
V
# # # # # # m 2
j =M 2
j , e=e OS
,
where the sum runs over all species of particles j = Z, W, H and
(#m 2
j ) (1)
= -Re “
# (1)
V
# # # # # # m 2
j =M 2
j , e=e OS
stands for the self­energy of the jth particle at p 2 = m 2
j in the
MSscheme and parameters replaced by the on­shell ones. Note that in
the above relation we had to perform a change from the MS to the
on­shell scheme also for the electric charge.
Unphysical terms, propartional to m 4
H drop out in sin 2
#W
sin 2
#W = 1 -
m 2
W
m 2
Z
# 1 -
M 2
W
M 2
Z
# â
# 1 -
cos 2
#W
sin 2
#W
# (# (1)
W - # (1)
Z )(1 - # (1)
Z ) + # (2)
W - # (2)
Z # #
where m 2
V /M 2
V = 1 + # (1)
V + # (2)
V
Kalmykov, ACAT'2002, Moscow 18