Table Of ContentA Condensate Mechanism of Conformal Symmetry Breaking and Higgs Particle
V.N. Pervushin,1 A.B. Arbuzov,1 R.G. Nazmitdinov,2,1 A.E. Pavlov,1,3 and A.F. Zakharov1,4
1Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
2Department de F´ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
3Moscow State Agri-Engineering University, 127550 Moscow, Russia
4 Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia
(Dated: November20, 2012)
A mechanism of spontaneous conformal symmetry breaking based on field condensates in the
Standard Model of strong and electroweak interactions is suggested. It is shown that an existence
of the top quark condensate can supersede the tachyon mass term in the Higgs potential in the
standardmechanism ofelectroweak symmetrybreaking. Considering theratioofafieldcondensate
2
tothecorrespondingmasspower(dependingonquantumstatistics)forvariousfieldsasaconformal
1 invariant,we obtain the Higgs boson mass to beabout 130±15 GeV.
0
2
PACSnumbers: 11.15.Ex,14.80.Bn
v
o
N Recently a few researchgroups reported upon the dis- constantderived fromthe muon life time measurements:
covery of scalar particles with almost similar masses v = (√2G )−1/2 246.22 GeV. The experimental
9 around 120 140 GeV [1–3]. The experimentalists use studies at FLeHrmCi [1, 2]≈and Tevatron [3] observe an ex-
1 −
anextremecautionintheidentificationoftheseparticles cessofeventsinthe datacomparedwiththebackground
with the long-waiting Higgs one. Indeed, the literature in the mass range around 126 GeV. Taking into ac-
] ∼
h contains a plethora of predictions on lower and upper countradiativecorrections,such a mass value makes the
p limits of the Higgs mass based on many different ideas, SM being stable up to the Planck mass energy scale [7].
- models and numerical techniques, which are close to the Nevertheless, the status of the SM and the problem of
p
e observed values. The question on a genuine mechanism the mechanism of elementary particle mass generation
h which provides an unambiguous answer about the Higgs are still unclear.
[ massisarealchallengetohighenergyphysicsandiscru- The idea on dynamical breaking of the electroweak
2 cial for the base of the Standard Model (SM) [4]. It is gauge symmetry with the aid of the top quark conden-
v especially noteworthy that in the SM the Higgs mass is satewascontinuouslydiscussedintheliteraturesincethe
0 introduced ad hoc. pioneering papers [8–10] (see also review [11], recent pa-
6 According to a general wisdom, all SM particles (may pers [12, 13], and references therein). Such approaches
4 be except neutrinos)ownmassesdue to the spontaneous suffer,however,fromformalquadraticdivergencesintad-
4
symmetrybreaking(SSB)ofthe electroweakgaugesym- pole loopdiagramsleading,inparticular,to the natural-
.
9 metry [5]. In particular, one deals with the potential (in ness problem (or fine tuning) in the renormalization of
0 notation of Ref. [6]): the Higgs boson mass.
2
All mentioned facts suggest that, it might be well to
1
λ2 examine that the SSB is also responsible for the Higgs
: V (φ)= (φ†φ)2+µ2φ†φ, (1)
v Higgs 2 field energy scale. To begin with, we suppose that there
i
X is a general mechanism of the SSB, which is responsible
whereonecomponentofthe complexscalardoubletfield
fortheappearanceofallSMfieldcondensates. Themain
r φ+
a φ = acquires a non-zero vacuum expectation feature of our approach is the assumption about the un-
(cid:18) φ0 (cid:19) derlying (softly broken) conformal symmetry which pro-
value φ0 = v/√2 if µ2 < 0 (the stability condition tectsthejumpoftheHiggsbosonmasstoacut-offscale.
λ2 > 0h isialways assumed). Note that the tachyon-like We will call this mechanism the spontaneous conformal
massterminthepotentialiscriticalforthisconstruction. symmetry breaking (SCSB). Evidently, in this case one
IncontrasttotheSSB,itbreakstheconformalsymmetry shouldrequiretheconservationofthe conformalsymme-
explicitly being the only fundamental dimension-full pa- try of the genuine theory fundamental Lagrangian. It
rameterin the SM. We recallthat the explicit conformal will be shown that the SCSB provides the breaking of
symmetry breaking in the Higgs sector gives rise to the the gauge, chiral, and conformal symmetries on equal
unsolvedproblemoffinetuningintherenormalizationof footing. Therefore, it allows also to introduce the uni-
the Higgs boson mass. That is certainly one of the most versalrelationbetween differentcondensates determined
unpleasant features of the SM. relative to the corresponding mass power depending on
In the classical approximation, from the condition of quantum statistics, see Eq. (14). Our basic conjecture is
the potential minimum one obtains the relationbetween that this relation holds approximately after the sponta-
the vacuum expectation value and the primary parame- neous breaking of the conformal symmetry.
ters µ and λ in the form v = 2µ2/λ. This quantity Following the ideas of Nambu [8, 9], we generate the
−
canbe defined aswellwiththepaidofthe Fermicoupling SCSB of the Higgs potential, using the top quark con-
2
densate. It is assumed that the general construction of interactionterminEq.(2). Invirtueoftheaboveresults,
the SM should remain unchanged. Let us start with the we have for the top quark Casimir condensate density
conformal invariant Lagrangian of Higgs boson interac-
m 1
tions tt¯ =4N t =4N γ m3, (8)
λ2 h iCas cV0 Xp 2 p2+m2t c· 0· t
L = h4 g h t¯t. (2) p
int t
− 8 − where N =3 is the number of colors.
c
Keepinginmindalltheseresults,wearereadytotreat
Here, for the beginning, we consider only the most in-
the contributionofthe topquarksto the effective poten-
tensive terms: the self-interaction and the Yukawa ones
tial, generated by the term Eq.(2):
of the top quark coupling constant g . Contributions of
t
other interaction terms will be considered below as well.
λ2
We assume that the O(4) symmetry of the Higgs sector V (h) = h4 g tt¯ h. (9)
cond t
8 − h i
has being spontaneously broken to the O(3) symmetry.
So that the whole construction should contain the ap- The extremum condition for the potential
pereance of the non-zeroHiggs field vacuum expectation dV /dh =0 yields the relation
cond h=v
value. |
In accord with the postulates of Quantum Field The- λ2
v3 =g tt¯ . (10)
ory(QFT),tocalculatephysicalquantities(includingthe 2 th i
mass spectrum) one has to apply the normalordering to
field operators. The normalordering in the Hamiltonian This relation follows from the fact that the Higgs field
ofinteractingscalarfieldsleadstothecondensatedensity has a zero harmonic v in the standard decomposition of
φφ thefieldhoverharmonicsh=v+H,whereH isthesum
h iCas of all nonzero harmonics with a condition d3xH = 0.
1 1 Here, the Yukawa coupling of the top quarRk g 1/√2
φφ = , (3) t ≈
h iCas V0 Xp 2 p2+m2 is known from the experimental value of top quark mass
m =vg 173.4 GeV.
p t t
≃
Thespontaneoussymmetrybreakingyieldsthe poten-
which we name the Casimir condensate density. Indeed,
tialminimumwhichresultsinthenonzerovacuumexpec-
by means of differentiation
tation value v and Higgs boson mass. The substitution
2 ∂ h=v+H into the potential (9) leads to the result
φφ = E , (4)
h iCas V ∂m2 Cas
0 m2 λ2v λ2
V (h)=V (v)+ HH2+ H3+ H4, (11)
this density is related to the Casimir energy [14] cond cond 2 2 8
1 which defines the scalar particle mass as
E = p2+m2. (5)
Cas
2
Xp p λ2
m2 3v2. (12)
H ≡ 2
In the continual limit of the QFT one has
We stress that this relation is different from the one
1 1 d3p 1 (m = λv) which emerges in the SM with the Higgs
= H
V0 Xp 2 p2+m2t ⇒Z (2π)32 p2+m2 potential (1).
p p With the aid of Eqs.(10),(12), the squared scalar par-
d3x 1
=m2 γ m2. (6) ticle mass can be expressed in terms of the t quark con-
Z (2π)32√x2+1 ≡ 0· densate:
Thus,theCasimircondensatedensityofamassivescalar 3g tt¯
m2 = th i. (13)
fieldintheabsenceofanyadditionalscaleisproportional H v
to its squared mass
The conjecture on universality of the conformal invari-
φφ ant ratios of the field condensate densities and the cor-
φφ =γ m2 h iCas γ , (7)
h iCas 0· ⇒ m2 ≡ 0 responding mass powers (see Eqs. (6)–(8)), allows us to
determinethe tquarkcondensatedensitywiththeaidof
where γ0 is a dimensionless conformal quantity with the light quark one. Using the relation
a zero conformal weight (see discussion on conformal
weights in [15]). tt¯ qq¯
h i = h i (14)
The normal ordering of a fermion pair (we intention- m3 m3
t q
ally interchange the order of fermion fields to deal with
positive condensates) ff¯=: ff¯: + ff¯ yields the con- we consider the left and right hand sides as scale invari-
densate density of the fermion fieldhff¯i in the Yukawa ants. However their numerators and denominators are
h i
3
scaledependent. Thereforewehavetochoosetheproper of the vector field condensate contributions for the mass
scales. For the left hand side, the scale is naturally de- formula (17) at the tree level for the SM
fined by the known t quark mass. We define the scale of
the right hand side by the light quark condensate den- 3λ2 3 ZZ
lsiimtyithqoq¯fit.hIetQisCrDathloewr-aecnceurrgayteplyhedneotmeremnionleodgyin[6t]h:e chiral ∆m2H= 4 hHHi+ 8g2(cid:18)2hWWi+ cohs2θWi (cid:19), (20)
qq¯ (250 MeV)3. (15) whereg andθW arethe Weinberg couplingconstantand
h i≃ the mixing angle. In Eq.(20) the first term is a con-
At this scale the light quark possesses the constituent tribution to the square mass due to the very scalar field
mass m 330 MeV estimated in the QCD-inspired condensate HH . Takingintoaccountthevaluesofcou-
model [1q6]≈. With the aid of Eq.(14) one determines the pling constahnts, imixing angle, masses, and condensates,
top quark condensate value we arrive to the following result
htt¯i≈(126 GeV)3. (16) m =m0 1+4∆m2H 1/2 m0 (1+0.02), (21)
Such a large value of the top quark condensate does not H H(cid:20) v2 (cid:21) ≈ H ·
where m0 is given by Eq.(17). If there exist additional
affectthelowenergyQCDphenomenology,sinceitscon- H
heavy fields interacting with the SM Higgs boson, their
tribution is very much suppressed by the ratio of the
condensates would contribute to the Higgs boson mass.
corresponding energy scales (squared).
By means of Eqs.(14),(15), in the tree approximation In this way, we suggest the condensate mechanism of
we obtain for the scalar particle mass spontaneous conformal symmetry breaking in the Stan-
dard Model. We suppose that this mechanism in its ori-
(m0 )2 =(130 15 GeV)2. (17) ginisrelatedtothevacuumCasimirenergy. Thisenables
H ±
us to avoid the problem of the regularization of the di-
Here, we have assigned 10% uncertainty into the ratio vergenttadpole loopintegral. The topquarkcondensate
light quark condensate and its constituent mass. supersedes the phenomenological negative square mass
The tentative estimate of the Higgs boson mass given term in the Higgs potential. The number of free param-
above is rather crude. In order to improve this value we eters in the SM lagrangian is reduced, since the tachyon
consider below the contributions of other condensates at masstermisdropped. Thecondensatemechanismallows
thetreelevel. Themasscanbe alsoaffectedbyradiative toestablishrelationsbetweencondensatesandmassesin-
corrections which will be analyzed elsewhere. Under the cluding the Higgs boson one.
assumptionofγ universality,the normalorderingofthe
0 Togetnumericalresultsweusetheadditionalassump-
field operators HH =:HH :+ HH yields tion that the condensates of all SM fields normalized to
h i
theirmasses(totheproperpower)anddegreesoffreedom
HH
h i =γ . (18) representa universalconformalinvariant. This allowsto
m2H 0 to estimate the value of the top quark condensate and
consequently the Higgs boson mass in agreement with
The normalordering of the vector fields V V defines the
i j recent experimental data.
vector field condensates normalized on each degree of
freedom
VV =M2 γ , V =W±,Z, (19)
h i V · 0 Acknowledgments
calculated in the gauge V = 0. Here, M is a corre-
0 V
sponding mass of the vector field. Transverse and lon- TheauthorswouldliketothankA.EfremovandYu.A.
gitudinal components are considered on equal footing in Budagov for useful discussions. VNP was supported in
thereducedphasespacequantizationofthe massivevec- part by the Bogoliubov–Infeld program. AEP and AFZ
tor theory [17]. As a result, one obtains the upper limit are grateful to the JINR Directorate for hospitality.
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