Pis'ma v ZhETF, vol.88, iss. 11, pp.737-742
© 2008 December 10
Vacuum energy, cosmological constant and Standard Model physics
G. L. Alberghi+, A. Yu. Kamenshchik+*1\ A. Tronconi+, G.P. Vnccn'. G. Venturi+ + Dipartimento di Fisica and INFN, 40126 Bologna, Italy *L.D. Landau Institute for Theoretical Physics RAS, 117940 Moscow, Russia Submitted 28 October 2008
The conditions for the cancellation of one loop contributions to vacuum energy (both U.V. divergent and finite) coming from the Standard Model (SM) fields are examined. It is proven that this is not possible unless one introduces besides several bosons, at least one massive fermion having mass within specific ranges. On examining one of the simplest SM extensions satisfying the constraints one finds that the mass range of the lightest massive boson is compatible with the present Higgs mass bounds. We briefly address the possible treatment of the presence of interactions.
PACS: 04.62,+v, ll.55.Hx, 98.80.Cq, 95.36,+x
1. Introduction. Almost sixty years ago Pauli [1, 2] suggested that the vacuum (zero-point) energies of all existing fermions and bosons compensate each other. This possibility is based on the fact that the vacuum energy of fermions has a negative sign whereas that of bosons has a positive one. We note that such an idea is realized in a highly constrained way in supersymmet-ric models, although supersymmetry breaking must be present at probed energies in order to explain observed data. Subsequently in a series of papers Zeldovich [3] connected the vacuum energy to the cosmological constant, however rather than eliminating the divergences through a boson-fermion cancellation he suggested a Pauli-Villars regularization of all divergences introducing a spectrum of massive regulator fields. Covariant regularization of all contributions then leads to finite values for both the energy density and (negative) pressure corresponding to a cosmological constant, i.e. connected by the equation of state p = —e, where p is the pressure, while e is the energy density.
The approach, which we develop in this paper is based on the combination of these two ideas: as Pauli suggests the existing boson and fermion fields provide the exact cancellation of all the ultraviolet divergences in the vacuum energy, while as Zeldovich suggests, the remaining finite part of the vacuum energy leads to an effective cosmological constant. Thus, rather than use a regularization approach we shall assume that the actual particle content of a theory is such that U.V. divergences do not appear insofar bosons and fermions should compensate each other as Pauli suggested. Indeed we have previously examined [4] the problem of U.V. divergences of the vacuum energy for both Minkowski and de Sitter
e-mail: kamenshchik0bo.infn.it
space-times and formulated the conditions for the cancellation of all divergences. These conditions lead to strong restrictions on the spectra of possible elementary particle models. In this note we shall apply such considerations to the observed particles of the Standart Model (SM) and also study the finite part of the vacuum energy and the possibility of a cancellation for this contribution also, so as to obtain a result compatible with the observed value of the cosmological constant (almost zero with respect to SM particle masses). The cancellation of all one-loop contributions to the cosmological and Newton constants was also considered in the context of the induced gravity approach [5]. We shall instead consider Einstein gravity and obtain all constraints in Minkowski space which is implicit if all contributions (divergent and finite) compensate between fermions and bosons. Let us illustrate them in order.
The requirement that quartic divergences cancel is just that the numbers of bosonic and fermionic degrees of freedom be equal (Nb = AV). The conditions for the cancellation of quadratic and logarithmic divergences on a flat Minkowski background are
and
+ 3$>4f = 25>4f, (2)
respectively. Here the subscripts s, V and F denote scalar, massive vector and massive spinor Majorana fields respectively (for Dirac fields it is enough to put 4 instead of 2 on the right-hand sides of Eqs. (1) and (2)). For the case of a de Sitter spacetime, equations giving conditions for the cancellation of quadratic and logarithmic ultraviolet divergences are more involved.
Some examples of these conditions for simple particle physics models have been presented in Ref. [4].
As was shown in Ref. [3] the finite part of the vacuum energy density is equal to
£ =
J dnf(n)n4lnn,
(3)
where f(p) is the spectral function. The requirement that the finite part of the vacuum energy (and pressure) is very small compared with SM masses suggests that we also need a compensation between the finite parts of fermion and boson vacuum energies, obtaining
TO, In ms +3 m.y In mv — 2 ^ m% In nip = 0.
(4)
This leads to a zero cosmological constant (Minkowski space).
As is known the observed number of fermionic degrees of freedom in the SM is much higher than the number of bosonic degrees of freedom [6]. Indeed Np is equal to 96 (if we consider the neutrinos as massive particles) while the number of bosonic degrees of freedom, carried by the photon, the gluons and the W± and Z° bosons is equal to 27. Thus we need an additional 69 boson degrees of freedom, one of which is the Higgs boson. Ideally we would like to obtain some minimal extension of the SM, which would not modify the fermionic degrees of freedom while just adding hypothetical bosons.
Our main result is a proof that, within the given framework, such an extension does not exist. In other words, we show that on introducing new bosonic fields, which provide the cancellation of the ultraviolet divergences in the vacuum energy density, the finite part of the effective cosmological constant is always positive and of order of the mass of the top quark to the fourth power, which is much higher than the value of the effective cosmological constant compatible with the cosmological observations. This leads to the necessity of introducing new heavy fermions. Indeed we shall find explicit realizations with zero finite energy once one introduces at least one fermion with a suitable mass.
Following a general analysis, for the sake of simplicity we shall consider an explicit minimal extension of the SM with a few massive bosons and weakly coupled, practically massless, others so as to satisfy the requirement Nb = Np. Such a possibility is viable in effective action approaches and, for example, has been considered recently in such scenarios as unparticle physics [7]. In this minimal framework we shall analyze the boson masses allowed by the cancellation constraints.
It is obvious that one may study vacuum energy in the more modern and general setting [8] of effective actions, renormalization group flows, and even attempt to include the effects of condensates, however we feel that it is worthwhile to first examine the full consequences for SM physics of Pauli's original suggestion.
In next section we study the consequences of our equations for the SM particle spectrum and in the last section our results are summarized and discussed.
2. The Standard Model and vacuum energy balance. Let us begin by observing that the mass of the top quark toj « 170 GeV is much higher than the masses of all other fermions (the bottom quark has the mass toj « 4.5 GeV while the mass of the heaviest r-lep-ton is mT « 2 GeV). Thus, on considering Eqs. (1), (2) and (4) we can limit ourselves to only taking into account the contributions of the top quark, whose mass is conveniently used as the reference unit mass, and of the massive vector bosons. Then the mass of W71*1 bosons is mw ~ 0.47rrat while that of the Z° boson is niz ~ 0.53TOt, with mt = 1.
Quantities describing the contributions of the heavy fermion and boson degrees of freedom in the conditions (1), (2) and (4) are then:
R
12m t — 6 to'
w
3 m%
11.5,
(5)
h = 12TOt2 - 6to^ - 3m| « 9.83, (6)
L = 12m\ In ml — 6m^ In to^ — 3to^ In m% « 0.743.
(7)
If we denote the masses squared of some hypothetic massive boson fields by xi,x2, • • • ,xn (x, > 0,Vz) then their values should satisfy the conditions
i=1
y~]xj = h,
i= 1
y^a;- In Xi
= L.
(8)
(9)
(10)
i= 1
We shall now proceed as follows:
- Firstly we shall find a lower bound to the number of massive boson degrees of freedom due to the constraints (8), (9) which define a surface S in the space of the
- Secondly we shall study on S, for positive x,, the extrema of the function <j> by using the method of Lagrange multipliers. We shall derive analytic formulae for the extrema of the function <j> on the constraint surface S and show that its minimum is much higher when the value of L coming from Eq. (7).
- Finally we shall obtain for <j>, its minimum (and maximum) value as a function of the SM particle content plus possible additional fermions in order to investigate the conditions for the satisfaction of the condition in (10).
The constraints (8), (9) have a simple geometrical sense [4]: they describe a sphere and a plane in the n-dimensional space and their intersection S is an (n — 2)-dimensional sphere, eventually to be sliced on the pos-itivity boundary of the The distance of the plane from the origin of the coordinates is h/y/n. In order to have an intersection between the sphere of radius R and the plane it is then necessary to have
n > h2/R2 и 8.4.
(И)
Thus, the number of massive bosonic degrees of freedom should at least be equal to 9. In general it is convenient to introduce the integer value no for such a threshold
n0 = [h2/R2
(12)
so that n > no is the requirement to have a non empty S. Now, in order to see when the condition (10) is also satisfied, it is convenient to calculate the minimum value of the function <j> = xi 1nXi on constraint surface S. Let us consider an auxiliary function
i= 1
( n \ ( n ^
ч«=1
(13)
whe
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.