Strong dynamics behind the formation of the $125$ GeV Higgs boson

0.21 MB·

by  M. A. Zubkov

Strong dynamics behind theformationofthe125 GeVHiggsboson
M.A. Zubkov
∗
University of Western Ontario, London, ON, Canada N6A 5B7
We consider the scenario, in which the new strong dynamics is responsible for the formation of the 125
GeV Higgs boson. The Higgs boson appears as composed of all known quarks and leptons of the Standard
Model. Thedescription of the mentioned strong dynamics isgiven usingthe ζ - regularization. Itallowsto
construct the effective theory without ultraviolet divergences, in which the 1/Nc expansion works naturally.
Itisshown, thatintheleadingorderofthe1/Nc expansion themassofthecompositeh-bosonisgivenby
Mh=mt/√2 125GeV,wheremtisthetop-quarkmass.
≈
4 PACSnumbers:
1
0
2
I. INTRODUCTION

r
a
M Inthepresentpaperwesuggestthescenario,inwhichtherecentlydiscovered125GeVh-boson[1,2]iscomposite.
AccordingtothesuggestedmodelitiscomposedofallfermionsoftheStandardModel(SM).Thescaleofthehidden

0 strong dynamics is supposed to be of the order of several TeV. We suppose, that the W and Z boson masses are
2 determinedbythecondensateofthe125GeVh-bosonaccordingtotheHiggsmechanism[3,4]. AllDiracfermion
massesaredeterminedbytheh-bosonaswell. InthepresentpaperwedonotconsidertheMajoranamassesatall

] andassume,thatneutrinoshaveextremelysmallDiracmasses[5]. Thelowenergyeffectivetheorycontainsthefour-
h
p fermioninteraction[8]betweenallSMfermions.Ourmodeldiffersfromtheconventionalmodelswithfour-fermion
- interactions[6], inwhichthe topquarkiscondensed. Themodeloftop- quarkcondensationwasfirstsuggestedin
p
[7] (and developed later in [9–13]). The idea, that the Higgs boson may be composed of the known SM fermions
e
wasdiscussed evenearlier, in [14], togetherwith the certainpreonmodels(however,in [14] therewas noemphasis
h
[ in thedominatingroleof thetop quarkandthe compositenessofthe Higgsbosonwas consideredtogetherwith the
compositenessofquarksandleptons). Therearetwomainaspects,inwhichthemodeldiscussedinthepresentpaper
4
differsfromtheconventionalmodelsoftop-quarkcondensation:
v
1
1. Thefour-fermioninteractionisnon-local. Itcontainstheform-factorsGthatcorrespondtotheinteraction
1
betweenthecompositeHiggsbosonandthepairfermion-antifermion.Theformfactorsdependonthreescalar
3
3 parametersofthedimensionofmasssquared:q2,p2,k2,wherepandkarethe4-momentaofthefermionand
. anti- fermionwhile q = p k is the 4 - momentumofthe compositescalar boson. G(p,k) = g(q2,p2,k2)
1 −
aredifferentfordifferentfermionsatlargedistances(bothtime-likeandspace-like),i.e. ifatleastoneofthe
0
4 quantities q2 , p2 , k2 ismuchsmaller,thantheElectroweakscaleMZ 90GeV.However,atsmallspace
| | | | | | ≈
1 - like and time - like distances (i.e. for q2 p2 k2 M2) those form- factorsbecome equalfor all
| | ∼ | | ∼ | | ∼ Z
: SM fermions. Unlikefor the otherfermionsthe mentionedform - factorfor the top - quarkis assumedto be
v
independentofmomenta.
i
X
2. Ourmodelwith the four- fermioninteractionis notrenormalizable. Therefore,it is theeffectivetheoryonly
r
a anditsoutputstronglydependsontheregularizationscheme. Contrarytotheapproachofthementionedabove
papersonthetop-quarkcondensationmodelswedonotusetheconventionalcutoffregularization.Weusezeta
regularization[15,16]. Itallowstoconstructtheeffectivetheorywithoutanyultravioletdivergencies.
The1/N expansionisagoodapproximationwithinoureffectivetheory. Thisisbecausethedangerousultraviolet
c
divergencesare absent. This is well - known, that these divergencesbreak the 1/N expansion in the NJL models
c
defined in ordinarycutoff regularization. For example, in [17] it is shown, that when the ultravioletcutoff is much
larger, than the generatedmasses, the nextto leading order1/N approximationto variousquantities doesnotgive
c
the correctionssmaller, than theleadingone. However,thisoccursbecauseof the termsthatare formallydivergent
in the limit of infinite cutoff. Therefore, in the regularization, that is free from divergencesthe next to leading ap-
proximationissuppressedbythefactor1/N comparedtotheleadingapproximation. Moreover,forthecalculation
c
∗onleaveofabsencefromITEP,B.Cheremushkinskaya25,Moscow,117259,Russia
2
of variousquantitiesrelated to the extra high energyprocesses (the processes with the characteristic energiesmuch
larger, than m ) the next- to leading orderapproximationis suppressedby the factor 1/N , where N = 24
t total total
is the total number of the Standard Model fermions (the color components are counted as well, so that we have
N =3generations (3colors+1lepton) 2isospincomponents=24).
total
× ×
Zetaregularization[15,16]providesthenaturalmechanismofcancellingtheunphysicaldivergentcontributionsto
variousphysicalquantitiesinthefieldtheories.Forexample,beingappliedtoquantumhydrodynamicsitprovidesthe
absenceof the unphysicaldivergentcontributionsto vacuumenergydue to the phononloops. The latter are known
to be cancelled exactly by the physics above the naturalcutoff of the hydrodynamics[18]. If zeta regularizationis
applied to quantum hydrodynamics,those divergencesdo not appear from the very beginning. The cancellation of
thementioneddivergencesinquantumhydrodynamicsoccursduetothe thermodynamicalstabilityofvacuum[18].
That’s why zeta regularization is the natural way to incorporate the principle of the thermodynamical stability of
vacuumalreadyattheleveloflowenergyapproximation(i.e. atthelevelofthehydrodynamics).Itwassuggestedto
applythesimilarprinciplesofstabilitytosubtractdivergencesfromthevacuumenergyinthequantumtheoryinthe
presenceofgravity[19]andtosubtractquadraticdivergencesfromthecontributionstotheHiggsbosonmassesinthe
NJLmodelsofcompositeHiggsbosons[20–23].Nowweunderstand,thatthezetaregularizationallowstoimplement
suchprinciplesofstabilitytovarioustheoriesinaverynaturalandcommonway. That’swhythisregularizationmay
appeartobenotonlytheregularization,buttheessentialingredientofthequantumfieldtheory.
The approachpresentedin the given paperallows to derivethe relationM2 = m2/2 on the level of the leading
H t
orderofthe1/N expansion.Inthesimplestmodelsofthetop-quarkcondensationthemassofthecompositeHiggs
c
bosonis typicallyrelatedto themassof thetopquarkbythe relationM2 = 4m2. The resultingmass 350GeV
H t ≈
contradicts to the recent discovery of the 125 GeV h - boson. It is worth mentioning, that there were attempts to
constructthetheorythatincorporatestheexistenceofseveralcompositeHiggsbosonsM relatedtothetop-quark
H,i
massbythe NambuSum rule M2 = 4m2. Thosemodelsadmitthe appearanceofthe 125GeV h - bosonand
H,i t
predicttheexistenceofitsNamPbupartners[21–23]. Theapproachofthe presentpaperpredictstheonlycomposite
Higgsbosonwiththeobservedmassaroundm /√2 125GeV.
t
≈
Thepaperisorganizedasfollows. InSectionIIwedefinethemodelunderconsideration.InSectionIIIwepresent
theeffectiveactionandevaluatephysicalquantities(thepracticalcalculationsusingzetaregularizationareplacedin
Appendix).InSectionIVweendwiththeconclusions.
II. THEMODELUNDERCONSIDERATION
A. Notations
SMfermionscarrythefollowingindices:
1. Weilspinorindices. WedenotethembylargeLatinlettersA,B,C,...
2. SU(2) doublet indices (isospin) both for the right - handed and for the left - handed spinors are denoted by
small Latin letters a,b,c,... For the left - handed doublets index a may be equal to 0 and 1. For the right -
handedsingletsoftheStandardModelthisindextakesvaluesU,D,N,Ethatdenotetheupquark,downquark,
neutrinoandelectronofthefirstgenerationandthesimilarstatesfromthesecondandthethirdgenerations.
3. ColorSU(3)indicesaredenotedbysmallLatinlettersi,j,k,...
4. ThegenerationindicesaredenotedbyboldsmallLatinlettersa,b,c,...
Besides,space-timeindicesaredenotedbysmallGreeklettersµ,ν,ρ,...Letusintroducethefollowingnotations:
L1 = uL , L2 = cL , L3 = tL
(cid:18) dL (cid:19) (cid:18)sL (cid:19) (cid:18)bL (cid:19)
1 2 3
R =u , R =c , R =t
U R U R U R
1 2 3
R =d , R =s , R =b (II.1)
D R D R D R
and
3
1 = νL , 2 = νµL , 3 = ντL
L (cid:18) eL (cid:19) L (cid:18) µL (cid:19) L (cid:18) τL (cid:19)
1 2 3
=ν , =ν , =ν
RN R RN µR RN τR
1 2 3
=e , =µ , =τ (II.2)
RE R RE R RE R
B. Theaction
Inourconsiderationweneglectgaugefields. Thepartitionfunctionhastheform:
Z = Dψ¯DψeiS, (II.3)
Z
ByψwedenotethesetofallStandardModelfermions. TheactionS = d4xψ¯i∂γψ+S containstwoterms. The
I
firstoneisthekineticterm. ThesecondoneisrelatedtothehiddenstrongRinteractionbetweentheSMfermions,that
resultsintheappearanceoftheadditionalfour-fermionnon-localinteractionterm
1
S = d4x d4x d4y d4y d4z L¯a(x )Ra(x )Ga(x z,x z)+ ¯a(x ) a (x )Ga (x z,x z)
I MI2 Xa Z 1 2 1 2 (cid:16) b 1 U 2 U 1− 2− Lb 1 RN 2 N 1− 2−
+R¯a(x )La(x )ǫ Ga(x z,x z)+ ¯a(x ) a(x )ǫ Ga(x z,x z)
D 1 c 2 bc D 1− 2− RE 1 Lc 2 bc E 1− 2− (cid:17)
G¯a(y z,y z)L¯a(y )Ra(y )ǫ +G¯a(y z,y z)¯a(y ) a(y )ǫ
(cid:16) D 2− 1− d 1 D 2 bd E 2− 1− Ld 1 RE 2 bd
+G¯a(y z,y z)R¯a(y )La(y )+G¯a (y z,y z)¯a (y ) a(y ) , (II.4)
U 2− 1− U 1 b 2 N 2− 1− RN 1 Lb 2 (cid:17)
Here M2 is the dimensionalparameter(itwill be shownbelow thatit’s barevalue is negativeandis aroundM2
M2 I [90GeV]2). Functions Ga(y z,y z) for a = U,D,N,E are the form - factors mentioned inIth≈e
i−ntroZdu≈ctio−ndefined in coordinate spaace1(G−¯a me2a−ns complex conjugation, ǫ is defined in such a way, that ǫ =
a ab 12
ǫ =1). Inmomentumrepresentationwehave
21
−
V
S = L¯a(p)Ra(k)Ga(p,k)+ ¯a(p) a (k)Ga (p,k)
I MI2 Xa q=p Xk=p′ k′(cid:16) b U U Lb RN N
− −
+R¯a(p)La(k)ǫ Ga(p,k)+ ¯a(p) a(k)ǫ Ga(p,k)
D c bc D RE Lc bc E (cid:17)
G¯a(p,k )L¯a(k )Ra(p)ǫ +G¯a(p,k )¯a(k ) a(p)ǫ
(cid:16) D ′ ′ d ′ D ′ bd E ′ ′ Ld ′ RE ′ bd
+G¯aU(p′,k′)R¯Ua(k′)Lab(p′)+G¯aN(p′,k′)R¯aN(k′)Lab(p′)(cid:17), (II.5)
Here V is the four - volume. Fermionfields in momentumspace are denotedas ψ(q) = 1 d4xe iqxψ(x) while
V −
theform- factorsinmomentumspacearedefinedasGa(p,k) = d4xd4ye ipx+ikyGa(x,yR) fora = U,D,N,E.
a − a
Accordingto our suppositionall functionsGa(p,k) = ga(q2,p2,Rk2) vary between fixed valuesmuch smaller than
a a
unityatsmallmomentaandunityatlargevaluesof q2 , p2 , k2 :
| | | | | |
Ga(p,k) = ga(q2,p2,k2) 1
a a →
(q2 , p2 , k2 M2 [90GeV]2);
| | | | | |∼ Z ∼
Ga(p,k) = ga(q2,p2,k2) κa
a a → a
(q2 , p2 , k2 M2)
| | | | | |≪ Z
Ga(p,k) = ga(q2,p2,k2) 1
| a | | a |≪
(q2 M2; p2 , k2 M2;
| |∼ Z | | | |≪ Z
exceptfortopquark) (II.6)
4
Here M 90 GeV is the Electroweak scale, κa are the dimensionless coupling constants that are all much
smaller Zthan≈unity except for the top quark. Becaause of the isotropy of space - time all Ga are the functions
a
of the invariants p2,k2, and q2 = (p k)2 only. Besides, in order to provide the left - right symmetric mass
matrix (see below) we require Ga(x,y−) = G¯a(y,x) that leads to G(p,k) = G¯(k,p) and to ga(q2,p2,k2) =
a a a
g¯a(q2,k2,p2). In principle, we may consider the real - valued form - factors G(x,y) = G(y,x) that would result
a
in Ga(p,k) = d4xd4ye ipx+ikyGa(x,y) = d4xd4ye ipy+ikxGa(x,y) = G¯a(k,p) = Ga( k, p). Since
G(k,ap) = G( kR, p)= g−((p k)2,ak2,p2)wewRouldarriv−eatthereaal-valuedforma -factorsinam−ome−ntumspace.
Weimplythat−Eq.−(II.6)isvali−dforthecomplex-valuedp2,q2,k2 intheupperhalfofthecomplexplane. Besides,
werequire,thatforthethetopquark
3
G (p,k) G (p,k) 1, (II.7)
t ≡ U ≈
foranyp,k. Wemaychoose,forexample,
q4+αaM4 p4+βaM4 k4+βaM4
ga(q2,p2,k2)= a I a I a I (II.8)
a q4+γaM4 × p4+λaM4 × k4+λaM4
a I a I a I
withsomeconstantsα,β,γ,λsuchthat αaaβaaβaa =κawhileαa,βa,γa,λa 1,andα3 =β3 =γ3 =λ3 =1. We
γaaλaaλaa a a a a a ≪ U U U U
mayalsointerpolatetheForm-factorsasfollows(forallfermionsexceptforthetop-quark):
gaa(q2,p2,k2)=(cid:26)1foκra|q<2|<,|p12o|,t|hke2r|w>iseM02 (cid:12) (II.9)
a (cid:12)
(cid:12)
withadimensionalparameterM suchthatM M (whereM istheZ -bos(cid:12)onmass). Forthetop-quarkweset
0 0 Z Z
G3(p,k) = g3(q2,p2,k2) = κ3 = 1. Letusi≪ntroducetheauxiliaryscalarSU(2)doubletH. Theresultingaction
U U U
receivestheform
S = d4x ψ¯i∂γψ H 2M2
Z h −| | I(cid:3)
d4x d4x d4z L¯a(x )Ra(x )Ga(x z,x z)Hb(z)+ ¯a(x ) a (x )Ga (x z,x z)Hb(z)
−Xa Z 1 2 h(cid:16)h b 1 U 2 U 1− 2− Lb 1 RN 2 N 1− 2− i
+ L¯a(x )Ra(x )G¯a(x z,x z)H¯b(z)ǫ + ¯a(x ) a(x )G¯a(x z,x z)H¯b(z)ǫ +(h.c.)
h c 1 D 2 D 2− 1− bc Lc 1 RE 2 E 2− 1− bci (cid:17)i
Inmomentumspacewehave:
S = V ψ¯(p)pγψ(p) H+(p)H(p)M2
Xp h − Ii
V L¯a(p)Ra(k)Ga(p,k)Hb(q)+ ¯a(p) a (k)Ga (p,k)Hb(q)+(h.c.)
− Xa q=Xp kh b U U Lb RN N i
−
V L¯a(k)Ra(p)G¯a(p,k)H¯b(q)ǫ + ¯a(k) a(p)G¯a(p,k)H¯b(q)ǫ +(h.c.) , (II.10)
− Xa q=Xp kh c D D bc Lc RE E bc ii
−
whereH(q)= 1 d4xH(x)e iqx.
V −
TheimportantpRropertyofthefunctionsga(q2,p2,k2)isthattheWickrotationfromthepositivevaluesofp2,q2,k2
a
tonegativevaluesofp2,q2,k2(i.e. positivevaluesofEuclideanmomentasquared)keepstherelation
Ga(p,k)=ga(q2,p2,k2) 1
a a →
(q2 , p2 , k2 M2 [90GeV]2) (II.11)
| | | | | |∼ Z ∼
ThiswillallowustoevaluateloopintegralsusingtheEuclideanexpressons(seeAppendix). Itisworthmentioning,
thattheinvarianceofEq.(II.11)undertheWickrotationisseendirectlyintheformoftheForm-factorsofEq.(II.8).
5
C. Fermionmasses
InthefollowingwefixUnitarygaugeH = (v+h,0)T,wherevisvacuumaverageofthescalarfield. Wedenote
the Dirac 4 - componentspinorsin this gaugecorrespondingto the SM fermionsby ψa. We omitangle degreesof
a
freedomtobeeatenbythegaugebosons. InthisgaugethepropagatorQa ,a = U,D,N,E oftheSMfermionhas
a
theform:
[Qaa]−1 =pˆγ−vGaa(p,p) (II.12)
Foreachfermionexceptforthetop-quarkthereisthepoleofthispropagatorat p2 M2 thatgivesthefermion
| | ≪ Z
mass
a a a
m vG (0,0) vκ (II.13)
a ≈ a ≈ a
Forthetopquarkwe havem3 = v. Thevalueofv isto becalculatedusingthegapequationthatisthe extremum
U
conditionfortheeffectiveactionasafunctionofh. TheeffectiveactionofEq. (II.10)canberewrittenas
S = d4x ψ¯(x)(i∂γ M)ψ(x) ψ¯(x) Gˆ ψ (x)
h
Z (cid:16) − − h i
M2(v+h(x))2 , (II.14)
− I (cid:17)
whereM isthemassmatrix.Itisdiagonal,withthediagonalcomponentsma(a=U,D,N,E;a=1,2,3)givenby
a
κav. Weuseherethefollowingnotationfortheh-dependentoperatorGˆ :
a h
[Gˆ ξ](x )= d4zd4x ξ(x )G(x z,x z)h(z) (II.15)
h 1 2 2 1 2
Z − −
ThisoperatorishermitianasfollowsfromtheconditionG(x,y)=G¯(y,x). Weimply,thattheh-bosonofEq.(II.10)
isresponsiblefortheformationofmassesofW andZ bosons.Hereandbelowweomitindicesa,aforGandκ. The
sumovertheseindicesisimpliedinthefollowingexpressions. Gandκareconsideredasthediagonalmatriceswith
thediagonalelementsGa andκa correspondingly.
a a
Theinteractionatthemomentaoffermions p2 , k2 [90GeV]2 andthemomentumoftheHiggsboson q2 =
(p k)2 [90GeV]2givesrisetotheU(N | )|(w|ith|N∼ =24)symmetricinteractionbetweenthereal-v|alu|ed
total total
| − |∼
HiggsfieldexcitationshandtheSMfermions. Theinteractionatmomenta (p k)2 0correspondstotheHiggs
| − | ≈
fieldcondensate. TheexistingpolesoffermionpropagatorsgivethemassesofallSMquarksandleptons. Thevalue
of v is to be determined through the requirement, that δ S (h) = 0 at h = 0, where S is the effective action
δh eff eff
(obtainedaftertheintegrationoverthefermions).
III. EFFECTIVELOWENERGYACTIONANDTHEEVALUATIONOFPHYSICALQUANTITIES.
A. Effectiveactionfortheh-boson.GapequationandHiggsbosonmass.
The effective action for the theory with action Eq. (II.14) as a function of the field h is obtained as a result of
the integration over fermions. In zeta regularization this effective action is calculated in Appendix up to the terms
quadraticinh. Thiscorrespondstotheleading1/N approximation.
c
Thetotalone-loopeffectiveactionfortheh-bosonreceivestheform:
S[h]= d4x M2(v+h)2 Cˆv2(v+h)2+h(x)Z2(w(pˆ2)pˆ2 M2)h(x) , (III.1)
Z h− I − h − H i
6
where
w(p2) 1, (p2 M2)
≈ | |∼ H
w(p2) 1/8, (p2 M2)
≈ | |≪ H
N µ2
Z2 totallog
h ≈ 16π2 m2
t
M2 =4N m2/N ;
H c t total
N µ2
Cˆ = c log (III.2)
8π2 m2
t
According to our supposition κ = 1, and the mass of the top - quark is given by v = m . Here we neglect the
t t
imaginarypartoftheeffectiveactionthatismuchsmaller,thantherealpartforthemomentasquaredoftheh-boson
smaller,than4m2. Forp2 4m2thedecayoftheh-bosonintothepairtt¯shouldbetakenintoaccountthroughthe
t ≥ t
imaginarypartofeffectiveaction.
Recall,thattheresultingexpressionsforvariousquantitiesinzeta-regularizationcontainthescaleparameterµ.We
identifythisparameterwiththetypicalscaleoftheinteractionthatisresponsiblefortheformationofthecomposite
Higgsboson. Below itwillbeshown, thatthe valueofµ consistentwith theobservedmasses ofHiggs, W, andZ-
bosonsisµ 5TeV.
≈
ThevacuumvaluevofH satisfiesgapequation
δ
S[h]=0 (III.3)
δh
Thenwegetthenegativevalueforthebaremassparameterofthefour-fermioninteraction:
N µ2
M2 = c m2log (III.4)
I −8π2 t m2
t
ThisnegativevalueM2 M2 means,thatthebarefour-fermioninteractionatthehighenergyscaleµ m of
I ≈ − Z ≫ t
Eq.(II.4)isrepulsive.However,theappearanceofthevacuumaveragemeans,thatthisrepulsiveinteractionissubject
tofiniterenormalization: atlowenergies µ, wheretheHiggsbosonmassisformed,therenormalizedinteraction
≪
betweenthefermionsisattractive.Onecansee,thatthereisonlyonepoleofthepropagatorforthefieldh.Asaresult
theHiggsmassisindeedgivenby
M2 4N m2/N =m2/2 125GeV
H ≈ c t total t ≈
B. WandZbosonmasses.Thenewinteractionscaleµ.
We may considerEq. (III.1) as the low energyeffectiveactionat most quadraticin the scalar field. There exists
theeffectiveactionwrittenintermsofthefieldH. ItshouldcontaintheλH 4 terminordertoprovidethenonzero
vacuumaverage H = (v,0)T. Comparingthecoefficientsatthedifferen|tte|rmsofthiseffectiveactionwiththatof
h i
Eq. (III.1)wearriveat
Z2
S[H]= d4x H+(x)Z2w( D2) D2 H(x) h(H 2 v2)2 (III.5)
Z (cid:16) h − (cid:16)− (cid:17) − 8 | | − (cid:17)
HerethegaugefieldsoftheSMaretakenintoaccount. Asaresulttheusualderivative∂ ofthefieldH issubstituted
bythecovariantoneD =∂+iBwithB =Ba σa+YB (whereY = 1isthehyperchargeofH).Functions
w,Z aredefinedinAppendix.TheydependsStUro(n2)glyontheUp(a1r)ticularchoice−oftheform-factorsGaandsatisfythe
h a
conditionsofEq.(III.2).
In orderto calculate gaugebosonmasses we needto substitute into Eq. (III.5) the SU(2) U(1) gaugefield B
⊗
insteadofiDand(v,0)T insteadofH. AsaresultweobtaintheeffectivepotentialforthefieldB:
V m2Z2w( B 2) B 2 (III.6)
B ≈ t h k k k k
7
Herewedenote B 2 =(1,0)B2(1,0)T = (B3 B )2+[B1 ]2+[B2 ]2 . Uptothetermsquadratic
k k (cid:16) SU(2)− U(1) SU(2) SU(2) (cid:17)
inB weget:
η2
V(2) B 2 (III.7)
B ≈ 2 k k
The renormalized vacuum average of the scalar field η should be equal to 246 GeV in order to provide the
≈
appropriatevaluesof the gauge boson masses. In order to evaluate the gauge boson masses in the given modelwe
cannotneglectnontrivialdependenceofw( B 2)onB inEq. (III.6). Todemonstratethisletusevaluatethetypical
k k
valueofB thatentersEq. (III.6). Ourcalculationfollowstheclassicalmethodforthecalculationoffluctuationsin
quantumfieldtheoryandstatisticaltheory(see,forexample,volume5,paragraph146of[30]). Thetypicalvalueof
Bisgivenbytheaveragefluctuation B¯ 2 withinthefour-volumeΩofthelinearsize 1 ,whereM istheZ-
hk k i ∼ MZ Z
bos1onmasMs,2w.h(iWleekBc¯aklc2u=late|Ω1t|hReΩinktBegkr2adl4inx.spTahceed-irteimcteeostfimEuatceligdievaensshikgBnakt2uire≈an−d∂t|hΩe∂|nMrZ2otlaotgeRbadcBketh−e|Ωfi|nMaZ2lkrBesku2lt=.)
A|Ωs|MaZ2res∼ultinZEq. (III.6)wemaysubstitutew(M2) 1insteadofw( B 2). Thisgives
Z ≈ k k
η2 N µ2
S B 2 totalm2log B 2
B ≈ 2 k k ≈ 16π2 t m2 k k
t
B 2 M2 [90GeV]2 (III.8)
k k ∼ Z ≈
That’swhywearriveatthefollowingexpressionforη
2N µ2
η2 2Z2v2 = totalm2log (III.9)
≈ h 16π2 t m2
t
Fromhereweobtainµ 5TeV.
∼
C. BranchingratiosandtheHiggsproductioncross-sections
The Higgs boson production cross - sections and the decays of the Higgs bosons are described by the effective
lagrangian[27]:
2m2 m2 α
L = WhW+W + ZhZ Z +c s hGa Ga
eff η µ µ− η µ µ g12πη µν µν
α m m m
+c hA A c bh¯bb c chc¯c c τhτ¯τ. (III.10)
γπη µν µν − b η − c η − τ η
HereG andA arethefieldstrengthsofgluonandphotonfields,ηtheconventionallynormalizedvacuumaverage
µν µν
ofthescalarfieldη 246GeV. Thiseffectivelagrangianshouldbeconsideredatthe treelevelonlyanddescribes
the channels h g≈g,γγ,ZZ,WW,¯bb,c¯c,τ¯τ. The fermions and W bosons have been integrated out in the terms
→
correspondingtothedecaysh γγ,gg,andtheireffectsareincludedintheeffectivecouplingsc andc (thesimilar
g γ
→
constantsintheSM aregivenbyc 1.03,c 0.81,see[27]). TheeffectivelagrangianEq. (III.10)issimilar
g γ
to that of the Standard Model. This≃results from≈ou−r initial requirement, that the Form - factors Ga(p,k) 1 for
(p k)2 M2; p2 , k2 M2 forallfermionsexceptforthetopquark. Thisreducesconsideraablythe≪ratesof
| − | ∼ Z | | | | ≪ Z
thedirectdecaysoftheHiggsbosontolightfermions.However,unliketheSMthevaluesofc ,c ,c maydifferfrom
b c τ
unityandaregivenby
c g(M2,m2,m2)/κ3,
b ≈ H b b D
c g(M2,m2,m2)/κ2
c ≈ H c c U
c g(M2,m2,m2)/κ3 (III.11)
b ≈ H τ τ E
ThesimilardecayconstantsmaybedefinedforallSMfermions: cfaa ≈ g(MH2,m2f,m2f)/κaa. Weassume,thatthese
constantsarenotmuchlargerthanunity,sothatthedecaysintotheheavyfermionsdominatelikeintheSM.
8
The consideration of constants c and c is more involved. Those constants contain the fermion loops. Let us
g γ
consider for the definiteness the constant c (the consideration of c is similar). The contribution of each fermion
g γ
(otherthanthetop-quark)isgivenbytheloopintegral. Thetypicalvalueofmomentumcirculatingwithintheloop
isof theorderof thelargestofthe two externalparameters: M andm . Asa resultwe cannotneglectthe form-
H f
factorG(p,k) g(M2,M2,M2)forthelightfermionscomparedtothatofthetopquark.Withinthisintegralthere
arethreefermi∼onpropHagatoHrsandHtwoverticesproportionaltoγµ. Wearriveattheexpressionthatisproportionalto
m that results in δc(f) = 2mfδr(f). We may estimate quantity δr(f) as follows. It is given by the dimensionless
f g MH g g
loop integral. For the definiteness let us consider the expression for the Form - factors given by Eq. (II.9). We
insert G(p,k) g(M2,M2,M2) 1 into expressionfor the loop integral. However, while evaluatingthis loop
integralweare∼toconsHiderthHeregHion≈ofmomentasquaredboundedfrombelowbyM2. Besides,weshouldomitthe
0
contributionofthepartofthisexpressionrelatedtothedecayoftheHiggsbosonintotheintermediatestatecomposed
oftwolightfermions. Suchacontributionisrelatedtotheimaginarypartofthefermionpropagatorproportionalto
δ(p2 m2). Therefore,duringtheevaluationofsuchacontributionrelatedtotheintermediatefermionstatesonmass
− f
shell one should substitute the value of the form - factor g(M2,m2,m2) 1 instead of g(M2,M2,M2) 1.
H f f ≪ H H H ≈
Thus,thiscontributionissuppressedforthelightfermions. Thevalueg(M2,M2,M2) 1istobesubstitutedinto
H H H ≈
the remaining (the main) part of the amplitude, and we should omit the imaginary - valued expressions during the
calculation.Fortheroughevaluationofr(f) weusetheexpressionforA (τ)giveninEq.(2.5)of[27].Wesubstitute
g f
intotheexpressionthevalueτ = MH2 insteadofτ = MH2 andmultiplytheresultingexpressionbythetwofactors:
0 4M2 4m2
0 f
mt MH√2 = 2√2√τ (takesintoaccountthevertexg(M2,M2,M2) 1insteadofM /m )and√τ (takes
M0 ≈ M0 0 H H H ≈ 0 t 0
intoaccountthatwedealwithδr(f) insteadofδc(f)):
g g
3√2 τ 1 1 1/τ +1 2
δr(f) τ 0− log − 0 (III.12)
g ≈ τ0 (cid:16) 0− 4 h (cid:12)p1 1/τ0 1(cid:12)i (cid:17)
(cid:12) − − (cid:12)
(cid:12)p (cid:12)
Asitwasexplainedabove,weomittedtheimaginarypartofthelogarithm. Onecansee,thatthevalueδr = δr(f)
g g
calculatedinthiswaydoesnotdependonthefermionflavor. Thisvaluerangesbetween 20atM =10GeVand
0
∼−
0 atM = 35GeV.We shouldnoticethatthisisveryroughevaluationthatshouldbeconsideredastheorderof
0
∼
magnitudeestimateonly. Theoutputfromthisestimateisthattheexpressionforthequantityr dependsstronglyon
g
theparticularformoftheForm-factors,isnegativeandisoftheorderof 10.Nevertheless,wefeelthisinstructive
∼−
togivetheestimateforc usingtheparticularformofr ofEq.(III.12). Namely,thecontributionsofthetop-quark,
g g
bottom quarkand charm quarkdominate. The ratio c /c(SM) (where c(SM) is the SM value) rangesfrom 1.0
g g g
∼ −
forM = 10GeVto +1.0forM = 35GeV.Weconclude,therefore,thatthecontributionofthelightfermions
0 0
dependsstronglyonth∼eparticularformoftheFrom-factorsga. However,thereexiststhepossibility,thatthevalue
a
ofthe gluonfusioncross- sectionmatchesthe presentexperimentalconstraints(see Fig. 12of [28] and Fig. 47 of
[29]).Inparticular,thevaluesc /c(SM) 1.0givethesamecross-sectionfortheprocessgg hastheStandard
g g
∼± →
Model.
Thecontributionofthelightfermionstothedecayconstantc mayberoughlyestimatedinthesimilarway. Now,
γ
theb,c-quarksandτ -leptondominate,andtheircontributionisgivenbyδc(b,c,τ) = 3(1/3)2m +3(2/3)2m +
γ b c
(cid:16)
m 2 δr . In the Standard Model value c(SM) the contribution of the W - boson dominates. The modeling
τ(cid:17)6MH g γ
expression for r of Eq. (III.12) gives the value of c /c(SM) that ranges from 1.4 for M = 10 GeV to 1.0
g γ γ 0
∼ ∼
for M = 35 GeV. One can see, that the constant c also depends on the particular form of the Form - factors.
0 γ
Therecertainlyexiststhechoiceoftheform-factorssuchthattheresultingexpressionsforc ,c matchthepresent
g γ
experimentalconstraints given by Fig. 12 of [28] and Fig. 47 of [29]. There is a hint in the present experimental
results. Thebestfittotheratio c /c(SM) reportedbyATLASis 1.2whilethebestfitreportedbyCMSis 1.4.
γ γ
| | ∼ ∼
Atthesametimeourroughestimatedescribedabovegivesthisratiowithintheinterval(1.0,1.4). Thebestfittothe
ratio c /c(SM) reportedbyATLASis 1.0,thebestfitreportedbyCMSis 0.7whileourestimategivesthisratio
g g
| | ∼ ∼
withintheinterval(0.0,1.0).
We conclude, that in the model suggested here the branching ratios of Higgs decay into the light fermions, the
branchingratioforthedecayintotwophotonsandtheproductioncross-sectionfortheprocessthatgoesthroughthe
gluonfusiongg hdependstronglyontheparticularformoftheForm-factorsofEq. (II.5). Inthetwolattercases
→
thedependenceontheparticularformoftheForm-factorsandthepotentialdeviationfromtheSMvalueisthemost
9
dramatic. Nevertheless, in this subsection we demonstrated, that the values of the branching ratios and production
cross-sectionscanbemadematchingthepresentexperimentalconstraints(given,forexample,in[28]and[29])bya
certainchoiceoftheForm-factors.
IV. CONCLUSIONSANDDISCUSSION
Inthispaperwesuggest,thatthereexiststhehiddenstrongdynamicsbehindtheformationofthe125GeVHiggs
boson. Accordingto ourscenarioitis composedofallknownSM fermions. Thecorrespondinginteractionhasthe
formofthenon-localfour- fermiontermofEqs. (II.4), (II.5). Thescale ofthe giveninteractionbetweentheSM
fermionsis of the order of µ 5 TeV. The form - factors entering this term of the action dependon the values of
the fermion momenta p,k. At≈large values (p k)2 , p2 , k2 [90GeV]2 all form - factors are equal to unity,
| − | | | | | ∼
and the interaction with real - valued excitation of the Higgs boson becomes identical for all SM fermions. In the
oppositelimitofsmallmomentathesymmetryislost,andtheform-factorsaredifferentfordifferentfermions. This
allowsto obtaintheobservedhierarchyofmassesforquarksandleptons. TheHiggsproductioncross-sectionand
thebranchingratiosoftheHiggsbosondecaydependstronglyontheparticularformoftheForm-factors. Themost
dramaticdependenceisencodedintheconstantsc ,c ofeffectivelagrangianEq. (III.10)whichareverysensitiveto
g γ
thenewphysics. TheseconstantsarerelatedtothecrosssectionoftheHiggsbosonproductionthatgoesthroughthe
gluonfusionandthebranchingratioforthedecayoftheHiggsbosonintotwophotons.AcertainchoiceoftheForm
- factorsgivesthe observablebranchingratiosandproductioncross- sectionsthatdonotdeviatesignificantlyfrom
theSMvalues,andthepresentmodelmatchesexistingexperimentalconstraints.ThefurtherinvestigationoftheLHC
datamaygivemoreinformationandmorestrongconstraintsontheexperimentallyalloweddependenceoftheForm-
factorsgaofEq. (II.5)onthevaluesofmomenta.
a
Inourmodelthecondensateofthecompositeh-bosonisresponsibleforthefermionmasses. Thisdistinguishesit
fromtheones,inwhichthe125GeVh-bosonisresponsibleforthemassesofWandZwhiletherearethesourcesof
thefermionmassesdifferentfromtheh-boson(see,forexample,[24–26]). Itisworthmentioning,thatintheusual
modeloftop-quarkcondensation[7],theinteractionscalewasextremelyhighΛ 1015 GeVwhileinourcasethe
∼
interactionscaleisµ 5TeV.
∼
InordertodefinetheeffectivetheorywiththeinteractiontermofEqs. (II.4),(II.5)weusezetaregularizationthat
givesthefiniteexpressionsfortheconsideredobservables. Bareinteractiontermatthescaleofµisrepulsive. How-
ever,theloopcorrectionsprovidetheappearanceofthecondensateforthecompositescalarfield. Thisdistinguishes
oureffectivetheorydefinedusingzetaregularizationfromthemodelswiththefour-fermioninteractiondefinedusing
theusualcutoffregularization.Gapequationappearsasanextremumconditionfortheeffectiveactionconsideredasa
functionofthefieldh(representstheexcitationsofthecompositescalarfieldaboveitscondensate). Thevalueofthe
HiggsbosonmassM2 m2/2followsfromthesymmetryoftheinteractiontermthattakesplaceatsmalldistances
H ≈ t
(i.e. large momenta (p k)2 , p2 , q2 M2 [90GeV]2). At these values of momentawe have the effective
| − | | | | | ∼ Z ≈
interactiontermbetweenthecompositeHiggsbosonandtheSMfermionsoftheform:
S = V L¯a(p)Ra(k)Hb(q)+ ¯a(p) a (k)Hb(q)+(h.c.) (IV.1)
I − Xa q=Xp kh b U Lb RN i
−
V L¯a(k)Ra(p)H¯b(q)ǫ + ¯a(k) a(p)H¯b(q)ǫ +(h.c.) , ((p k)2 , p2 , k2 M2)
− Xa q=Xp kh c D bc Lc RE bc ii | − | | | | |∼ Z
−
All SM fermionsenter this interaction term in an equal way. This form of the interaction term points out, that the
ultravioletcompletionoftheStandardModelpreservesthe symmetryofEq. (IV.1)alreadyatthevaluesoffermion
and scalar boson momenta of the order of Electroweak scale M 90 GeV. (Recall, that this implies the light
Z
∼
fermionstobefaroutofthemassshell.) We suppose,thattheappearanceoftheeffectiveinteractionoftheformof
Eq. (IV.1) maybecheckedalreadyusingtheexistingdata. Atleast, wemayobservetheregimeofapproachingthe
effectiveinteractionbetweenthefermionsandtheHiggsbosontothatofEq.(IV.1).
Finally, we would like to remark, that the interaction term of the specific form Eq. (II.4) suggested here may be
too idealized. The actual interactionbetween the fermionsthat leadsto the formationof their masses may be more
complicated.However,weexpectthatifNaturechoosesthepatternofthecompositeHiggsbosonsuggestedhere,the
interactionbetweenthecompositeHiggsbosonandthe SM fermionsshouldanywayhavetheformof Eq. (IV.1)at
smallenoughdistances. Thisis, probably,the mainoutputof thepresentpaper: we suggesttolookcarefullyat the
existingexperimentaldatainordertoconfirmorrejecttheexistenceoftheinteractionoftheformofEq. (IV.1).
10
Acknowledgements
The author is grateful to V.A.Miransky for useful discussions and careful reading of the manuscript, and to
G.E.Volovik,whonoticedtherelationM2 m2/2soonafterthediscoveryofthe125GeVHiggsboson.Theauthor
H ≈ t
isbenefitedfromdiscussionofthemodelwithYu.Shylnov,andespeciallyfromthediscussionswithZ.Sullivanofthe
experimentalconstraintsontheHiggsbosoncrosssectionsandthedecaybranchingratios. Theworkissupportedby
theNaturalSciencesandEngineeringResearchCouncilofCanada.
Appendix.Applicationofzeta-regularizationtechniquetothecalculationofvariouscorrelators.
A. Evaluationofthefermiondeterminant
HereweusethemethodforthecalculationofvariousGreenfunctionsusingzeta-regularizationdevelopedin[16].
InordertocalculatethefermiondeterminantweperformtherotationtoEuclideanspace-time. Itcorrespondstothe
change:t ix4,ψ¯ iψ¯,γ0 Γ4,γk iΓk(k =1,2,3).(Thenewgamma-matricesareEuclideanones.) The
→− → → →
resultingEuclideanfunctionaldeterminanthastheform:
Z[h] = det PˆΓ+iM +iGˆ (IV.2)
h
h i
= Dψ¯Dψexp d4xψ¯ PˆΓ+i(M +Gˆ ) ψ
h
Z (cid:16)Z h i (cid:17)
HereψinvolvesallSMfermions,M isthefermionmassmatrixthatoriginatesfromthecondensateofh.Theoperator
Gˆ dependsonthefunctionsG(x,y)andh(x)andisgivenbyEq.(II.15). Thetransformationψ Γ5ψ,ψ¯ ψ¯Γ5
h
→ →−
resultsinZ[h]=det PˆΓ+i(M +Gˆ ) =det PˆΓ i(M +Gˆ ) . Therefore,
h h
h i h − i
1/2
Z[h]= det PˆΓ+i(M +Gˆ ) PˆΓ i(M +Gˆ ) (IV.3)
h h
(cid:16) h ih − i(cid:17)
WegetthefermionicpartoftheEuclideaneffectiveaction:
1
S [h]= Trlog Pˆ2+M2+(2MGˆ +Gˆ2 Γ[∂,Gˆ ]) (IV.4)
f −2 h h h− h i
WedenoteA=Pˆ2+M2,andV =2MGˆ +Gˆ2 Γ[∂,Gˆ ]. Itisworthmentioning,thatinmomentumspacewe
h h− h
have
[[∂,Gˆ ]ξ](P)=i Qξ(K)g( Q2, P2, K2)h(Q) (IV.5)
h
− − −
Q=XP K
−
Inzeta-regularizationwehave:
1 1
Sf[h]= ∂s µ2s dtts−1Trexp (A+V)t (IV.6)
2 Γ(s) Z h− i
At the end of the calculation s is to be set to zero. Here the dimensional parameter µ appears. We identify this
dimensionalparameterwiththeworkingscaleoftheinteractionthatisresponsiblefortheformationofcompositeh-
boson.Furtherweexpand:
( t)2 1
Trexp (A+V)t =Tr e−At+( t)e−AtV + − due−(1−u)AtVe−uAtV +... (IV.7)
h− i (cid:16) − 2 Z0 (cid:17)