The Standard Model in the Alpha gauge is not renormalizable

27 Pages·0.18 MB·English

by  Hung Cheng

The Standard Model in the Alpha Gauge is Not Renormalizable
Hung Cheng1
Department of Mathematics, Massachusetts Institute of Technology
Cambridge, MA 02139, U.S.A.
and
S.P. Li2
9
9 Institute of Physics, Academia Sinica
9
1 Nankang, Taipei, Taiwan, Republic of China

n
a
J

7
2

2
v
9
2
1
1
0
9
9
/
h
t
-
p
e
h
:
v
i
X
r
a
PACS: 11.10.Gh; 11.15.-q; 12.20.-m
Keywords: Standard Model; Ward-Takahashi identities; Renormalizability
1. E-mail: [email protected]
2. E-mail: [email protected]
Abstract
We study the Ward-Takahashi identities in the standard model with the gauge
fixing terms given by (1.1) and (1.2). We find that the isolated singularities of the
propagators for the unphysical particles are poles of even order, not the simple poles
peoplehave assumedthem to be. Furthermore, theposition ofthese poles areultravio-
letdivergent. Thusthestandardmodelinthealphagaugeingeneral, andtheFeynman
gauge inparticular, is notrenormalizable. We studyalso thecasewith thegaugefixing
terms (1.3), and find that the propagators remain non-renormalizable. The only gauge
withoutthesedifficulties istheLandaugauge. AsemphasizedbyBonneau[1], onemust
make a distinction between the renormalizability of the Green functions and that of
the physical scattering amplitudes.
1
1. Introduction
In this paper, we shall explore some of the exact and non-perturbative consequences of
the Ward-Takahashi identities in the standard model.
The organization of this paper will be as follows: in Sec. 2, we study the 2 2 mixing
×
matrix of propagators for the longitudinal W and the charged Higgs meson with the gauge
fixing term of W chosen to be
1
(∂µW+ +iα M φ+)(∂νW− iα M φ−), (1.1)
−α µ W 0 ν − W 0
W
where φ± are the charged Higgs fields, M is the bare mass1 of W, and α is the gauge
0 W
parameter. This study has been carried out by a number of authors[2,3]. Indeed, we could
have started where they left off, but we opt for making the presentation more self-contained.
In Sec. 3, we study the 3 3 mixing matrix of propagators for the longitudinal A, the
×
longitudinal Z and φ0, where φ0 is the unphysical neutral Higgs meson, (the imaginary part
of the neutral Higgs meson) with the gauge fixing term chosen to be
1 1
(∂ Zµ +α M φ0)2 (∂ Aµ)2, (1.2)
µ Z 0 µ
−2α − 2α
Z A
′
where M is the bare mass of Z, and α and α are gauge parameters.
0 Z A
In Sec. 4, we study these propagators when the gauge fixing terms are, instead of (1.1)
and (1.2),
1 1 1
(∂ Zµ)2 (∂µW+)(∂νW−) (∂ Aµ)2. (1.3)
−2α µ − α µ ν − 2α µ
Z W A
In Sec. 5, we discuss the meaning of our results.
In Appendix A, we list the BRST variations of the fields in the standard model, in
the gauge of (1.1) and (1.2). In Appendix B, we present an alternative way to derive the
Ward-Takahashi identities[4], and list the three Ward-Takahashi identities for propagators
involving the longitudinal W, in the gauge of (1.1). In Appendix C, we list the nine Ward-
Takahashi identities for propagators involving the longitudinal A and the longitudinal Z, in
the gauge of (1.2). In Appendix D, we present the derivations of the three relations satisfied
1The bare mass M0 is equal to 12g0v0, where v0 may be defined either as the classical vacuum value or
the quantumvacuumvalue ofthe Higgsfield. TheGreenfunctions differwithdifferentdefinitions ofv0,but
by gauge invariance, physical quantities remain the same with either definition.
2
by the 1PI amplitudes of the propagators in the 3 3 mixing matrix, in the gauge of (1.2).
×
In Appendix E, we list the twelve Ward-Takahashi identities in the gauge of (1.3), and derive
the relations satisfied by the 1PI self-energy amplitudes.
2. The Mixing of W and the Charged Higgs Meson
In the standard model, a longitudinal W meson may propagate into either a longitudi-
nal W or a Higgs meson with the same charge. Thus all of the following propagators
< 0 TW+(x)W−(0) 0 >, < 0 TW+(x)φ−(0) 0 >,
| µ ν | | µ |
< 0 Tφ+(x)W−(0) 0 >, < 0 Tφ+(x)φ−(0) 0 >,
| µ | | |
are none-zero and together they form a 2 2 mixing matrix. We shall denote the Fourier
×
transform of such a propagator with the symbol G and put
GW+W−(k) iDT (k2)T iα D (k2)L ,
≡ − WW µν − W WW µν
ikµ
GW+φ−(k) = Gφ+W−(k) α D (k2),
µ µ ≡ M W Wφ
0
Gφ+φ−(k) iDφ+φ−(k2),
≡
where
k k k k
µ ν µ ν
T g , L .
µν µν µν
≡ − k2 ≡ k2
The D functions are so defined that their unperturbed forms in the gauge of (1.1) are simply
1
D(0) (k2) = D(0) (k2) = (2.1)
WW φ+φ− k2 α M2
− w 0
They are finite and non-zero at α = 0. The unperturbed form for D is zero.
W Wφ
The charged Higgs meson mixes not with the transverse W but with the longitudinal W,
as is indicated by the factor kµ in the expression for GW+φ−. Thus, in the 2 2 mixing
×
matrix under discussion, this factor kµ can be replaced by K, where
K √k2
≡
For the same reason, kµkν/k2 can be replaced by unity. The 2 2 mixing matrix is therefore
×
equal to
iα D iKα D /M
W WW W Wφ 0
−
  . (2.2)
 iKαWDWφ/M0 iDφ+φ− 
 
 
3
We shall express the propagators by their 1PI amplitudes.
In a field theory without mixing, let the unperturbed propagator for a particle be i(k2
−
m2)−1, and let the 1PI self-energy amplitude be Π, then this propagator is the inverse of
i(k2 m2 Π). Now that the unperturbed propagators in the mixing matrix are given by
− − −
(2.1), the mixing matrix (2.2) is equal to the inverse of
i(k2 M2 α Π )/α iKΠ /M
− 0 − W WW W Wφ 0
  , (2.3)
 iKΠWφ/M0 −i(k2 −αWM02 −Πφ+φ−) 
 
where Π , for example, is the 1PI amplitudes for D . The 1PI amplitudes are functions
Wφ Wφ
of k2 and α, the dependence of which will be exhibited only when needed. Inverting the
matrix in (2.3), we find that the mixing matrix in (2.2) is equal to
−i(k2 −αWM02 −Πφ+φ−)αW −iKαWΠWφ/M0
 
iKα Π /M i(k2 α M2 α Π )
 − W Wφ 0 − W 0 − W WW 
  . (1)
(k2 −αWM02 −Πφ+φ−)(k2 −αWM02 −αWΠWW)+k2αWΠ2WφM0−2
Substituting the matrix elements in (2.4) into the Ward identity (B.4a), we get[2,3]
(M02 +ΠWW)(M02 −M02Πφ+φ−/k2) = (M02 +ΠWφ)2, (2.5)
As a side remark, (2.5) provides a subtraction condition. By setting k2 to zero in (2.5), we
find that
Πφ+φ−(0) = 0. (2.6)
This ensures the divergence of Πφ+φ− to be a logarithmic one. If we express the real part
of the neutral Higgs field as (v +H), and choose v to be the quantum value of the Higgs
0 0
field, then the Higgs potential has a term linear in H. This term acts as a counter term
which cancels all contributions of tadpole diagrams. With the absence of the contributions
of the tadpole diagrams, the self-energy amplitude Π is only logarithmically divergent,
WΦ
not quadratically divergent. By (2.5), so is Π . We also note that, with v the quantum
WW 0
±
vacuum value, the Higgs potential has a mass term for φ . When we calculate Πφ+Φ−, we
include the contributions of this mass term. But to incorporate the condition (2.6), we
4
make a subtraction for Πφ+Φ− at k2 = 0. The contributions of the mass term vanish after
subtraction.
The denominator in(2.4)issimplified if wetake advantageof(2.5). Wemay, forexample,
use (2.5) to eliminate Πφ+φ− from the denominator, which then becomes
J2
W , (2.7)
1+Π M−2
WW 0
where
k2
J k2 α M2 α Π + Π . (2.8)
W ≡ − W 0 − W WW M2 wφ
0
Thus we have
D = (k2 −αWM02 −Πφ+φ−)(1+ΠWWM0−2) , (2.9a)
WW
(J )2
W
(k2 α M2 α Π )(1+Π M−2)
Dφ+φ− = − W 0 − w WW WW 0 , (2.9b)
(J )2
W
and
Π (1+Π M−2)
D = Wφ WW 0 (2.9c)
Wφ
− (J )2
W
Note that the denominator in the expressions in (2.9) is the square of J . Since the poles
W
of these propagators come from the zeroes of J , and since J is a linear superposition of
W W
1PI self-energy amplitudes, which are analytic functions of k2, the order of the poles of the
propagators are always even.
Propagators in quantum field theories generally have simple poles. It is therefore easy to
be lulled into believing that poles of propagators are simple poles. We have found that this
is often not true when two fields mix under the auspices of the Ward-Takahashi identities.
In particular, the charged φ and the longitudinal W have the same unperturbed mass. Thus
the poles of the unperturbed propagators for these fields are both at k2 = α M2. As the
W 0
interactions are turned on and the propagators form a mixing matrix, the positions of the
poles change but the Ward-Takahashi identities force them to remain to be at the same
point. Thus the two simple poles merge to form a double pole.
The function J contains divergent integrals. Since no subtraction conditions are avail-
W
able, we expect that the location of the zero of J be ultraviolet divergent. This is confirmed
w
5
by a perturbative calculation. We find that J has a zero at
w
I g2 3 s2α α s4α
k2 = α M2 1+ + 0 A W Z , (2.10)
W 0(cid:26) µ20 8π2ǫ "2 − 4 − 2 − 4c2 #(cid:27)
where µ2 is the bare mass squared of the physical Higgs and is equal to λ v2/2, λ and v are
0 0 0 0 0
the bare four point coupling and the vacuum expectation value of the φ-field respectively. s
and c are the sine and cosine of the Weinberg angle. I is the contribution from the tadpole
and ǫ = 4 D , with D the space-time dimension. I is given by
−
3λ g2 1
I = + 0 1+ I
2 2 2c2 2
(cid:20) (cid:18) (cid:19)(cid:21)
v2 α λg2 1
+ 0 3λ2 +α λg2 + Z 0 +3g4 1+
64π2ǫ W 0 2c2 0 2c4
(cid:20) (cid:18) (cid:19)(cid:21)
.
I is the quadratic divergent term and is given by
2
dDp 1
I = .
2 (2π)D p2
Z
For comparison, the pole of the transverse part of the W is located at
2I g2 17 3 α α 1
k2 = M2 1+ + 0 W Z c2 + .
0 µ2 8π2ǫ 3 − 4c2 − 2 − 2 2c2
(cid:26) 0 (cid:20) (cid:18) (cid:19)(cid:21)(cid:27)
In (2.10), we keep only the terms which are ultraviolet divergent and we have not included
the contributions of quark or lepton loops. Since the ratio of (2.10) and the transverse mass
pole in the above expression is infinite, the double pole in (2.10) cannot be made finite.
Let the Fourier transform of < 0 Tiη+(x)ξ−(0) 0 > be denoted as
| |
Gη+ξ− iDη+ξ− ,
≡
where η+ andξ− arehermitian ghost fields associated with W. The Ward-Takahashiidentity
(B.4b) gives
1+Π M−2
Dη+ξ− = Wφ 0 , (2.11)
J (1+g E)
W 0
where E is defined in (B.5). The ghost propagator has only one factor of J in the denom-
W
inator.
6
Since the pole of a function cannot be removed by multiplying the function by a con-
stant, the propagators cannot be made finite by wavefunction renormalizations. The stan-
dard model in the alpha gauge in general, and the Feynman gauge in particular, is not
renormalizable.
Let us set α = 0. In this limit, (2.8) becomes
W
J = k2(1+Π M−2). (2.12)
W Wφ 0
Thus we have, in the Landau gauge,
1
Dφ+φ− = , (2.13a)
k2 Πφ+φ−
−
and
1
Dη+ξ− = , (2.13b)
k2(1+g E)
0
with the propagators GW+φ− and GW+W− vanishing at α = 0. Because of (2.6), both
L W
propagators in (2.13) have a simple pole at k = 0. These propagators are renormalizable by
wavefunction renormalizations.
3. The Mixing of A, Z and φ0
In this section we discuss the A, Z, and φ0 mixing with the gauge fixing terms given by
(1.2).
The longitudinal Z meson mixes with φ0 as well as with the longitudinal A, where φ0 is
the imaginary part of the neutral Higgs meson and is unphysical. Thus the propagators for
these fields form a 3 3 matrix. Let the propagators be denoted as
×
GAA(k) iDT (k2)T iα D (k2)L ,
µν ≡ − AA µν − A AA µν
GZZ(k) iDT (k2)T iα D (k2)L ,
µν ≡ − ZZ µν − Z ZZ µν
GAZ(k) = GZA(k) iDT (k2)T iα α D (k2)L ,
µν µν ≡ − AZ µν − A Z AZ µν
kµ
GAφ(k) = GφA(k) α D (k2),
µ − µ ≡ AM′ Aφ
0
kµ
GZφ(k) = GφZ(k) α D (k2),
µ − µ ≡ ZM′ Zφ
0
Gφ0φ0(k) iD (k2),
φ0φ0
≡
7
where M′ is the bare mass of Z. In the above, the function GAZ, for example, is the Fourier
0 µν
transform of
< 0 TA (x)Z (0) 0 > .
µ ν
| |
These propagators form the mixing matrix
′
iα D iα α D Kα D /M
− A AA − A Z AZ A Aφ 0
 
iα α D iα D Kα D /M′ (3.1)
 − A Z AZ − Z ZZ Z Zφ 0 
 
 
 Kα D /M′ Kα D /M′ iD 
 − A Aφ 0 − Z Zφ 0 φ0φ0 
 
The unperturbed forms for these propagators are
1
D(0)(k2) = ,
AA k2
1
D(0)(k2) = D(0) (k2) = ,
ZZ φ0φ0 k2 α M′2
− Z 0
with all other unperturbed forms vanishing. Thus the matrix in (3.1) is the inverse of
i(k2 α Π )/α iΠ KΠ /M′
− A AA A − AZ Aφ 0
 
iΠ i(k2 α M′2 α Π )/α KΠ /M′ (3.2)
 − AZ − Z 0 − Z ZZ Z Zφ 0 
 
 
 KΠ /M′ KΠ /M′ i(k2 α M′2 Π ) 
 − Aφ 0 − Zφ 0 − − Z 0 − φ0φ0 
 
where Π , for example, is the 1PI amplitude for D . There are nine Ward-Takahashi
AA AA
identities for these propagators, the derivation of which is given in Appendix C. Three of
these identities give the following three relations among the 1PI amplitudes,[2,3]:
(M′2 +Π )(M′2 M′2Π /k2) = (M′2 +Π )2, (3.3a)
0 ZZ 0 − 0 φ0φ0 0 Zφ
(M′2 +Π )Π = Π2 , (3.3b)
0 ZZ AA AZ
and
(M′2 +Π )Π = (M′2 +Π )Π . (3.3c)
0 Zφ AZ 0 ZZ Aφ
The derivation of these relations is presented in Appendix D. Note the resemblance of (2.5)
with (3.3a), indeed with (3.3b) and (3.3c) as well, if one takes into account that the photon
is massless.
8
Next we calculate the inverse of (3.2) and equate it to the matrix in (3.1). We defer the
details to Appendix D and give only the results here:
1
D (k2) = , (3.4a)
AA
k2
[k2(k2 α M′2)+α α M′2Π k2Π ] 1+ ΠZZ
D (k2) = − Z 0 A Z 0 AA − φ0φ0 M0′2 , (3.4b)
ZZ k2J2 h i
Z
Π
D (k2) = D (k2) = AZ , (3.4c)
AZ Aφ0
− k2J
Z
(α M′2Π +Π k2) 1+ ΠZZ
D (k2) = A 0 AA Zφ M0′2 , (3.4d)
Zφ − k2J2 (cid:16) (cid:17)
Z
and
(k2 α M′2 α Π α Π ) 1+ ΠZZ
D (k2) = − Z 0 − Z ZZ − A AA M0′2 , (3.4e)
φ0φ0 J2 (cid:16) (cid:17)
Z
where
Π
J = k2 α M′2 α Π +k2 Zφ . (3.5)
Z − Z 0 − Z ZZ M′2
0
Note that all the propagators above except D have a double pole at the simple zero
AA
of J . The existence of this double pole is again easy to understand. The unperturbed
Z
longitudinal Z and the unperturbed φ0 have the same mass, while the unperturbed A has
zero mass. Thus D and D have simple poles at the same position, while D has a
ZZ φ0φ0 AA
simple pole at k2 = 0. As the interactions are turned on, the Ward-Takahashi identities
require that the position of the former two poles remain to be the same, while that of the
last pole remains to be zero. Thus the propagators in (3.4) have a double pole as well as a
simple pole at k2 = 0.
A perturbative calculation shows that the zero of J is located at
Z
I g2 3c2 α s2
k2 = α M′2 1+ + 0 Z α . (3.6)
Z 0 (cid:26) µ20 8π2ǫ " 2 − 4c2 − 2 W#(cid:27)
These propagators cannot be made finite by wavefunction renormalizations. One can easily
check that this double pole is different from the pole of the transverse part of the photon-Z
mixing sector.
From the Ward-Takahashi identities (C.4)–(C.7), we get
(1+g cF )J +α e F Π
D (k2) = 0 Z Z Z 0 Z AZ , (3.7a)
ηAξA k2J (1+e F +g cF )
Z 0 A 0 Z
9