ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 7, с. 949-955
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
EFFECTIVE MASS OF W BOSON IN A MAGNETIC FIELD
©2014 V. V. Skalozub*
Dnipropetrovsk National University, Ukraine Received December 2, 2013
Simple representation for the average value of the W-boson one-loop polarization tensor in a magnetic field B = const, calculated in the ground state of the tree-level spectrum, is derived. It corresponds to Demeur's formula for electron in QED. The energy of this state, describing effective particle mass, is computed by solving the Schwinger—Dyson equation. As application, we investigate the effective mass squared at the threshold of the tree-level instability, B — Bc = m2/e, and show that it is positive. In this way the stability of the W-boson spectrum is established. Some peculiarities of the results obtained and other applications are discussed.
DOI: 10.7868/S0044002714070174
1. INTRODUCTION
Nowadays, physics of charged vector particles in strong magnetic fields has obtained new stimulus for investigation. This concerns, first of all, p-meson physics where the effective Lagrangian describing electromagnetic interactions has been derived in different approaches [1—4]. In [5, 6] on the base of this Lagrangian the properties of the p-meson vacuum in strong magnetic fields of the order eB > m2p, where mp is particle mass, were investigated and the superconducting state having a structure similar to Abrikosov's lattice was observed. Such type structure was derived already in electroweak theory for the W-boson vacuum [7, 8, 9]. Other important reasons are the existence of extremely strong magnetic fields in the Universe as well as in collisions of beams of protons and heavy ions at modern colliders. In the latter case, they influence characteristics and properties of particles, in particular, W bosons that is important for various decay processes which are investigated.
Recently the ground state projection of the SU(2) polarization tensor for charged gluons in Abelian chromomagnetic field has been calculated and studied at high temperature in [10]. Simple expression for this function was derived which just corresponds to the Demeur formula for electron in magnetic field in QED. The obtained results (Eqs. (17), (22) in [10]) can be modified to find the ground state energy for charged massive vector particle. Of course, a number of other contributions has to be added in different models.
In what follows, we apply the results of [10] to calculate the ground state energy, (t|n(p||,B)|t), for the W boson, accounting for the one-loop diagrams. Since on the ground state It) the full W-boson polarization tensor is diagonal [11], it is possible to write down and solve the Schwinger—Dyson (SD) equation for this state and in this way obtain a nonperturbative effective mass M(B) (or effective ground state energy) of the particle. This mass can be calculated for arbitrary values of the field strength B (at least numerically) that can be useful for various applications.
In the present paper, the calculated expression is investigated in the limit of B — Bc, where Bc = = m2/e is the critical magnetic field strength for the W-boson tree-level spectrum
pjj = pj + m2 + (2n + l)eB - 2eaB (1) (n = 0, l,..., a = 0, ±1)
in a homogeneous magnetic background, B = const, described by the potential
Aexi = Bx\5^2,
(2)
E-mail: skalozubv@daad-alumni.de
where pn is a momentum component along the field direction, e is electric charge, n is Landau level number, a is spin projection. In (1) a tachyon mode is present in the ground state (|t) = In = 0, a = +1)) for the field strength B > Bc = m2/e. Considering this limit, we show that the effective energy calculated from the SD (or gap) equation remains real for realistic mass of Higgs particle. Thus, radiation corrections act to prevent the instability of the vacuum. Other obvious applications are in W-boson physics for different processes in strong magnetic fields.
The noted problem has been investigated already in one-loop order for the W bosons in the Georgy— Glashow model of electroweak interactions (see review [9]). As it was found, the result depends on the value of the Higgs boson mass mH. For heavy Higgs particle, K = mH/mw > 1-2, the spectrum stabilization takes place. For light Higgs particle, K < 1.2, the instability was found. However, this problem was not investigated in detail for the standard model. Other note, the absence of an adequate representation for the ground state projection of the W-boson polarization tensor, similar to Demeur's formula [12], made investigations of W bosons bulky and complicated.
In the next section, we calculate the W-boson polarization tensor and its mean value in the ground state of the spectrum (1). In Section 3, we derive the SD equation for the ground state projection (t|n(B)\t) and investigate the limit B ^ Bc. General conclusions and discussion are given in the last section.
2. W-BOSON POLARIZATION TENSOR
In what follows, we use Euclidean space—time and the representation of the polarization tensor for gluons given in [13]. It is reasonable to rewrite color gluon field, V/a (a = 1,2,3), in terms of charged,
= T2{Vl ± iVP> 3nd neUtra1' ^ = C°m"
ponents. In momentum representation, the initial expression reads
nAA' (p) = (3)
dk
{r\vpGv
(p - k)rX'v'p' Gpp
(2n)4
+ (p - k)\G(p - k)ky G(k) +
(k) +
tadpol
+ kxG(p - k)(p - k)x>G(k)} + nAuP
where the second and third lines result from the ghost contribution and the tadpole contribution is given by
n
tadpol AA'
(4)
dp
{2Gaa'(p) - Saa'Gpp(p) - Gaa'(p)} .
The contributions of the charged tadpole diagrams are taken into consideration. Only these tadpoles are relevant to the problem of interest. The vertex factor,
rXvp = (k - 2p)pô\v + bpV(p - 2k)x + (5)
+ $px(p + k)v,
completes the description of the vector part of the polarization tensor. These formulas hold also in a background field, provided the corresponding expressions for the propagators are used. We take a homogeneous
magnetic background field in the representation given in Eq. (2). In this case the operator components fulfill the commutation relation
[p^,pv] = ieF^v (F12 = B). (6)
In what follows, where it will be not misleading, we write B instead of eB or even put B = 1, for short. In the above formulas (3) and (4) we omitted the coupling factors e2. These factors as well as other factors proper to different models of interest can be accounted for in the final expressions. Below, we will also use the notation l2 = l2 + l2 and h2 = p2 + + p2, where we write l3 and l4 for the momenta in parallel to the background field p\\ = l3 and imaginary time, respectively. Other information relevant to the massless case is given in [10, 13].
To obtain the results for the electroweak sector, one has to take into account the masses of the W, Z, and Higgs bosons, and add the contributions of the latter two particles.
First, we incorporate the masses in the representation of the polarization tensor as given in Eq. (51) of [13]. It results from the proper time representation of the propagators,
G(p - k) = j ds e-sm2 e-s(p-k)2
(7)
G(k) = J dt e-tM2 e-tk2 0
GAA' (p - k) = J ds e-sm2 e-s(p-k)2EAA',
Eaa' = e
-2ieFs AA',
for charged and neutral particles, and integration over k in Eq. (3). The mass M is M = 0,mZ,mH for photon, Z, and Higgs boson, correspondingly.
The representation for the SU(2) sector of the standard model (W bosons, massive ghosts and photons) is obtained in terms of the integral over the parameters s and t,
n
AA'
= ds dte-sm Q(s, t) x
(8)
x z Mj+M*f\ +nAar
i,3
with
@(s,t) =
exp(-ff)
(47r)2(s + i)VÄ'
0
0
0
0
Here, the following notations are used:
H
st s + t
l2 + m(s,t)(2n + 1)B, (10)
, 1
m(s, t) = s + - in —, 2
A = ß± = t + sinh(s)e:
±s
©(s,%2 = 1
exp
(4n)2 (s + t)ß-
(11
For the projection of the functions Mj we use representation (55) in [13]. Calculation of these terms is given in the appendix of [10]. At this place we mention that under the tachyonic projection we get directly a representation suitable for further calculations. The presence of particle masses is reflected in a simple factor in the integrand of Eq. (8) and does not influence any computation procedures applied in the massless case.
Note that expression (8) is calculated in the Feynman—Lorentz—'t Hooft gauge
P W- - mr = iC-, (12)
in which the mass of charged ghost, C±, and Goldstone, 4>±, fields equals the W-boson mass m.
Detailed calculations of (t|n|t) are given in [10] and not modified for m =0. Only the contribution from M33 + Mgh requires an additional consideration. As it is shown in [13], Eqs. (87)—(89), this part can be written in the form,
MH+gh =
(13)
= — / ds dt e
0
^ d© d©
OAA'^--1"
ds dt
where ©(s, t) is the function in Eq. (9) and the matrix Ew = e-j!"sF. These combining into derivatives allow for carrying out one of the parameter integrations. Using (t^xy t = 1, (tExy t = e2s,
e(< = 0,t) = ± = = (14)
and integrating by part we get in the projection
f dsdte-sm2 (t|M33+gh ©(s,t)t = (15)
(4n)2
dq
q
1 e~qm e2q q sinh(g)
m
(4n)2
ds dt e-
(s + t)ß-
which are equivalent to Eqs. (23)—(26) in [13]. The sum over i, j in (8) follows the subdivision introduced in [13] and the functions M^j, are given by Eq. (53) in [13].
Now we take the tachyonic projection of nxx,, Eq. (8). In doing so we note especially n = 0 (for B = 1) and the function © simplifies,
where in the last line function (11) is substituted. To complete this part, we write down the remaining (except M33+gh) terms coming from the main diagram (3)
s +1
(16)
and hence
n(£ Mij )
t=
(4n)2 J s + t
ds dt ont 2 e x
(17)
4 . ts + te2s 2
— +4 ß- s +1
l2
exP ( -Tbl2 ~ s
Here, Mtj reminds about the omitted terms. The contributions from the tadpoles, (4), take the form
1
<i|ntp|i> = —TTZV? X
dq _ qe
qm
(4n)2
2 + cosh(2<?) + 3 sinh(2g) \ sinh(q) J
(18)
Then we have to add the contri
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