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PHYSICAL MASSES AND THE VACUUM EXPECTATION VALUE
provided by CERN Document Server
OF THE HIGGS FIELD
Hung Cheng
Department of Mathematics, Massachusetts Institute of Technology
Cambridge, MA 02139, U.S.A.
and
S.P.Li
Institute of Physics, Academia Sinica
Nankang, Taip ei, Taiwan, Republic of China
Abstract
By using the Ward-Takahashi identities in the Landau gauge, we derive exact relations
between particle masses and the vacuum exp ectation value of the Higgs eld in the Ab elian
gauge eld theory with a Higgs meson.
PACS numb ers : 03.70.+k; 11.15-q
1
Despite the pioneering work of 't Ho oft and Veltman on the renormalizability of gauge
eld theories with a sp ontaneously broken vacuum symmetry,a numb er of questions remain
2
op en . In particular, the numb er of renormalized parameters in these theories exceeds that
of bare parameters. For such theories to b e renormalizable, there must exist relationships
among the renormalized parameters involving no in nities.
As an example, the bare mass M of the gauge eld in the Ab elian-Higgs theory is related
0
to the bare vacuum exp ectation value v of the Higgs eld by
0
M = g v ; (1)
0 0 0
where g is the bare gauge coupling constant in the theory.We shall show that
0
v
u
u
D (0)
T
t
g (0)v: (2) M =
2
D (M )
T
2
In (2), g (0) is equal to g (k )atk = 0, with the running renormalized gauge coupling constant
de ned in (27), and v is the renormalized vacuum exp ectation value of the Higgs eld de ned
in (29) b elow.
Consider the Ab elian-Higgs mo del with the Langrangian density given by
1
+ 2 + + 2
F F +(D ) (D )+ ( ) ; (3) L =
0
0
4
with
D (@ + ig V ) ;
0
and
F @ V @ V :
In the ab ove, V and are the gauge eld and the Higgs eld resp ectively. The subscript (0)
of the constants in (3) signi es that these constants are bare constants. As usual, we shall
put
v + H + i
0 2
p
= (4)
2
p
where v = :
0 0 0 1
To canonically quantize the theory given by this Lagrangian, we add a gauge- xing term
and the asso ciated ghost terms to the Lagrangian. The e ective Lagrangian obtained is
1
2 2
L L ` i(@ )(@ )+i M + i g M H; (5)
ef f 0 0
0
2
In (5)
` @ V M ; (6)
0 2
is a constant and and are ghost elds. The Lagrangian in (5) is the e ective Lagrangian
in the -gauge. It is invariant under the following BRST variations:
V = @ ; H = g ; = g (v + H );
0 2 2 0 0
1
`; =0: (7) i =
3
There exists a sp eci c formula relating the vacuum state j0 > in this e ective theory to
that in the original gauge theory. This vacuum state satis es
Qj0 >=0: (8)
In (8), Q is the BRST charge, the commutator (anticommutator) of which with a b osonic
(Grassmann) eld is the BRS variation for this eld, e.g.,
[iQ; V ]=V : (9)
The Ward-Takahashi identities are easily derived from (8) and (9). For example, wehave,
4
as a consequence of (8) ,
< 0j[iQ; T i (x) (y )] j0 >=0; (10)
2 +
where T signi es time-ordering. However, the anticommutator ab ove is indeed the BRST
variation of Ti(x) (y). Thus we obtain, by (9), the Ward-Takahashi identity
2
1
@ V (x) M (x)) (y )+Ti(x)(y)(M + g H (y ))j0 >=0: (11) < 0jT (
0 2 2 0 0
All other Ward-Takahashi identities can b e derived in a similar way. It is particularly
convenient to study these Ward-Takahashi identities in the limit ! 0, i.e., in the Landau
gauge. 2
The propagators for H; , and the ghost eld will b e denoted by
2
2 2 2
iZ (k ; ) iZ (k ; ) iZ (k ; )
H
2
; ; and
2 2 2 2
k 2 k k
p
resp ectively, where k is the momentum of the particle and 2 is the physical mass of the
Higgs meson. The propagator for the vector meson will b e denoted as:
2 2
k k Z (k ; ) Z (k ; ) k k
L T
i(g ) i ; (12)
2 2 2 2 2
k k M k k
2
where the subscripts T and L signify transverse and longitudinal resp ectively, and where k
2
is really k + i, with p ositive and in nitesimal. In the lowest order of p erturbation, all the
Z functions are equal to unity. Finally, there is the propagator
Z
2
k M Z (k )
0 L2
ik :x
dxe < 0jTV (x) (0)j0 > ; (13)
2
2 2
k k
which is not zero since mixes with the longitudinal comp onentofV as so on as the
2
interaction is turned on. In the unp erturb ed order, Z = 0 as this mixing is due to the
L2
interaction of with other elds.
In the Landau gauge, the longitudinal part of (12) vanishes. The propagator in (13) also
vanishes at = 0. In addition, the ghost eld is decoupled from the other elds at =0.
2
Thus Z (k ; 0)=1.We shall take advantage of these simpli cations o ccuring in the Landau
gauge.
Let us take the Fourier transform of (11) and then take the limit ! 0. We get, after
some algebra
2 2
Z (k )+Z (k )=Z; (14)
L2
2
where
1
< 0j(v + H )j0 >: (15) Z
0
v
0
2 2 2
In the ab ove, Z (k )isZ (k ;0), and similarly for Z (k ). Note that Z is the ratio of
L2 L2
2
the quantum exp ectation value of with the classical value of the vacuum state of and is
2
indep endentof k .
Next we turn to the Ward-Takahashi identity obtained by setting
1
< 0j[iQ; T i ( @ V M )] j0 > (16)
0 2 +
3
to zero. Taking the limit ! 0 for (16) requires a little care, as there are terms in (16)
1
which are of the order of . These terms will cancel and we are left with the terms which
are nite as we take the limit ! 0. We get
2 2 2
1 Z (k ; ) 2M M
L
0 0
2 2
lim + Z (k )+ Z (k )=0: (17)
L2
2
2 2
!0
k k
Let us denote the 1PI (one-particle-irreducible) amplitude for the propagation of V to
V by
2 2
g A(k ; )+k k B(k ; )
which can b e written as
k k k k
2 2 2 2
)A(k ; )+ [A(k ; )+k B(k ; )]: (18) (g
2 2
k k
2
We also denote the 1PI amplitude for the propagation of to by C (k ; ); and the 1PI
2 2
k
2
into by i amplitude for the propagation of V D (k ; ): By writing out the unrenor-
2
M
0
malized propagators in terms of A; B ; C and D ,we nd that the mixing of with the
2
longitudinal comp onentof V is given by
2 2 2
D M +A+k B
0
2 2
]+ 0( ); (19a) Z (k ; )=1+ [
L
2
2 2
k M (k +C)
0
2
k
2
Z (k ; )= +0( ); (19b)
2
2
k +C
and
2
D k
2
Z (k ; )= +0( ): (19c)
2L
2
2
k +CM
0
From the last two equations of (19), we get, setting ! 0,
2 2 2 2 2
Z (k ) D (k ) 1 M
L2 0
= : (20)
2
2 2 2 2
k Z (k ) k + C (k ) M
0
2
Thus, the equation (19a) gives
2 2
2 2 2 2 2 2
Z (k )
M + A(k )+k B(k ) M Z (k ; ) 1
L
L
0 0
2
= : (21) lim
2 2 2
!0
k k Z (k )
2
Substituting the ab oveinto the Ward-Takahashi identity (17), we get
2
Z
2 2 2 2 2
M + A(k )+k B(k )=M ; (22)
0 0
2
Z (k )
2 4
where (14) has b een utilized.
The unrenormalized propagator for the transverse comp onents of V at =0 is
2
i(g k k =k )
:
2
2 2
k M A(k )
0
Thus
2 2
k M
2
; (23) Z (k )=
T
2
2 2
k M A(k )
0
or
2 2
M k
2 2 2
M + A(k )=k + : (24)
0
2
Z (k )
T
>From (22) and (24), we get
2 2 2
M k Z
2 2 2 2
k + + k B (k )=M : (25)
0
2 2
Z (k ) Z (k )
T
2
2
Finally,we set k = 0, obtaining
2 2 2
Z v Z Z (0)
T
0 2 2 2
; = g Z (0): (26) M = M
T
0 0
Z (0) Z (0)
2 2
2
We de ne the running renormalized coupling constant g (k )as
q
2
2
Z (k ) (27) g(k )g
T 0
Also, we de ne the vacuum exp ectation value of the renormalized scalar eld as
v + H
0
q
v < 0j j0 >: (28)
Z (0)
2
By (15), wehave
v Z
0
q
v= : (29)
Z (0)
2
From (22), (23) and (24), we get (2).
Finally,we mention that it is also p ossible to relate the mass of a fermion to the vacuum
exp ectation value of the Higgs eld. Similar to (1), wehave, in the classical theory,
m = f v ;
0 0 0
where m is the bare mass of the fermion, and f is the bare Yukawa coupling constant. In
0 0
5
the quantum theory,wehave , similar to (2)
2
m = fv=[1 + a (m )]: (30)
r
2
where a (p )isaninvariant amplitude in the renormalized fermion propagator.
r 5
References
1. G. t'Ho oft and M. Veltman, Nuclear Physics,B44, 189, (1972).
2. See, for example, the review article by G. Bonneau, Int'l Journal of Mo dern Physics A,
Vol. 5, No. 20, 3831 (1990).
3. H. Cheng and E.C. Tsai, Physics Review, D40, 1246, (1989). See also H. Cheng in
Physical and Nonstandard Gauges , edited by Gaigg et. al, Springer-Verlag (1989).
4. The metho d we use to obtain the Ward-Takahashi identities here was shown to one of
us(H. Cheng) by E.C. Tsai in a private communication.
5. H. Cheng and S.P. Li, \How to Renormalize a Quantum Gauge Field Theory with Chiral
Fermions", (submitted for publication, 1996). 6