Strong Interactions in Spacelike and Timelike Domains. Dispersive Approach

211 Pages·2016·4.37 MB·English

by  Alexander V. Nesterenko (Auth.)

Strong Interactions in Spacelike and
Timelike Domains
Strong Interactions in
Spacelike and Timelike
Domains
Dispersive Approach
Alexander V. Nesterenko
BogoliubovLaboratoryofTheoreticalPhysics
JointInstituteforNuclearResearch
Dubna,Moscowregion,RussianFederation
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About the Author
Dr. Alexander V. Nesterenko is a senior researcher at the Bogoliubov
LaboratoryofTheoreticalPhysics,JointInstituteforNuclearResearch,Dubna,
Russian Federation. He graduated with honors from Moscow State University
where he obtained a PhD in Theoretical Physics. He was a postdoctoral
researcher at the École Polytechnique, France, and University of Valencia,
Spain.Heactivelyworksintheareaofthetheoreticalparticlephysicsandisa
refereeforseveralAPSandIOPjournals.AnexperiencedlecturerinQuantum
Field Theory and Quantum Chromodynamics, he has published two textbooks
basedonhislecturecourse.
vii
Preface
The landmark discovery of the asymptotic freedom in Quantum Chromody-
namics (QCD) has eventually resulted in an overwhelming success of the
perturbative description of the high-energy strong interaction processes in
the spacelike (Euclidean) domain. At the same time the QCD perturbation
theory appears to be directly inapplicable to the studies of hadron dynamics
in the timelike (Minkowskian) domain. Furthermore, the low-energy strong
interactionprocessesentirelyfalloutofthescopeofperturbativeapproachand
canbehandledonlywithinnonperturbativemethods.
In general, there is a diversity of the nonperturbative approaches to QCD,
which originate in various aspects of the tangled dynamics of the colored
fields. A certain clue on the low-energy hadron dynamics is supplied by the
corresponding dispersion relations, which convert the underlying kinematic
restrictionsonaphysicalprocessonhandintotheintrinsicallynonperturbative
constraintsontherelatedfunctionsthatshoulddefinitelybeaccountedforwhen
onegoesbeyondthescopeofperturbationtheory.Furthermore,thedispersion
relations provide the only way of the self-consistent description of the strong
interactionprocessesinthetimelikedomain.
The book primarily focuses on the mutually interrelated hadronic vacuum
polarization function, R-ratio of electron-positron annihilation into hadrons,
and the Adler function, which govern various strong interaction processes
in the spacelike and timelike domains at the energy scales spanning from
low to high energies. Specifically, the book presents the essentials of the
dispersion relations for these functions, recaps their perturbative calculation,
anddelineatesthedispersivelyimprovedperturbationtheory.Thelattermerges
the nonperturbative constraints imposed by the pertinent dispersion relations
on the functions on hand with corresponding perturbative input in a self-
consistentway,getsridofinnateobstaclesoftheQCDperturbationtheory,and
considerablyextendsitsapplicabilityrangetowardthelowenergies.Thebook
alsoelucidatesthebasicsofthecontinuationofthespacelikeperturbativeresults
into the timelike domain, which is essential for the study of electron-positron
annihilationintohadronsandtherelevantstronginteractionprocesses.
ThebookisbasedonaportionofacourseoflecturesonQCD,whichwas
delivered by the author at the Moscow State University and at the Moscow
InstituteofPhysicsandTechnology.Thetopicscoveredinthebook,beingofa
ix
x Preface
directrelevancetothenumerousongoingresearchprogramsinparticlephysics
and future collider projects, can be of benefit to graduate and postgraduate
students, academic lecturers, and scientific researchers working in the related
fieldsoftheoreticalandmathematicalphysics.
A.V.Nesterenko
Dubna,Russia
June2016
Acknowledgments
The author expresses his heartfelt gratitude to A.C. Aguilar, A.B. Arbuzov,
B.A. Arbuzov, A.P. Bakulev, G.S. Bali, V.V. Belokurov, A.V. Borisov,
N.Brambilla,G.Cvetic,F.DeFazio,A.E.Dorokhov,H.M.Fried,R.Kaminski,
A.L. Kataev, D.I. Kazakov, S.A. Larin, M. Loewe, K. Milton, S. Narison,
J.Papavassiliou, M. Passera, J. Portoles, G.M. Prosperi, F. Schrempp,
A.V. Sidorov, C. Simolo, I.L. Solovtsov, O.P. Solovtsova, O.V. Teryaev,
A.Vairo,andH.Wittigfortheinterest,stimulatingcomments,valuableadvices,
fruitfuldiscussions,andcontinuoussupport.
xi
Introduction
The theoretical description of the strong interaction processes is utterly based
onthequantumnon-Abeliangaugefieldtheory,namely,QuantumChromody-
namics (QCD). This theoryoriginates inthenotion of quarks and gluons. The
formeraretheconstituentsofallhadronsthatwereproposedinthemid-1960s
inthemilestoneworksbyGell-Mann[1],Zweig[2,3],andPetermann[4](see
alsopaper[5]),whereasthelatterarethequantaofamasslessgaugevectorfield
providing interaction between quarks. Both quarks and gluons carry a specific
quantumnumber,whichplaysthecrucialroleinhadronphysics.Thisquantum
number was also suggested in the mid-1960s in the papers by Greenberg [6],
Han and Nambu [7, 8], Bogoliubov, Struminsky, and Tavkhelidze [9, 10], and
named“color”afterward[11](seealsopapers[12–14]).
Thorough theoretical and experimental investigations revealed that the
strong interactions possess two distinctive features. First, the strength of the
interactionbetweencoloredobjectsdecreaseswhenthecharacteristicenergyof
the process on hand increases. Inother words, the QCD invariant charge α =
s
g2/(4π), which is also called the strong coupling, vanishes at large momenta
transferred, which constitutes the asymptotic freedom of QCD. Second, free
quarks and gluons, as well as any other colored final state, have never been
observedexperimentally,whichconstitutestheso-calledconfinementofcolor.
Thesetwophenomenaaregovernedbythestronginteractionsattheopposite
energyscales.Specifically,theasymptoticfreedomtakesplaceathighenergies,
or in the so-called ultraviolet domain, that corresponds to small spatial quark
separations. As for the confinement of color it is related to low energies, or
the so-called infrared domain, that, in turn, corresponds to large spatial quark
separations.
The asymptotic freedom in the non-Abelian gauge field theory was dis-
covered in the early 1970s by ’t Hooft [15], Gross and Wilczek [16], and
Politzer [17] (see also papers [18–21]). In turn this finding gave rise to an ex-
tensiveemploymentofperturbationtheoryinthestudyofthestronginteraction
processesathighenergies.However,inpracticetheresultsofperturbativecalcu-
lationsinQCDappeartobeofalimitedapplicability.Thisisprimarilycaused
by the fact that the perturbative approach in Quantum Field Theory entirely
reliesontheassumptionthatthevalueofthecorrespondinginvariantchargeis
smallenough.Inparticular,itisthisassumptionthatallowsonetoapproximate
anexperimentallymeasurablephysicalobservablebyperturbativepowerseries
xiii
xiv Introduction
in the respective coupling, which drastically simplifies the theoretical analysis
oftheprocessonhand.InQuantumElectrodynamics(QED)theaforementioned
assumption is valid for all experimentally accessible energies1 that makes
the QED perturbative calculations reliable for all practical purposes. On the
contrary, in QCD the foregoing assumption of smallness of the perturbative
strongrunningcouplingisvalidforhighandintermediateenergiesonly.Inthe
experimentally accessible infrared domain the perturbative QCD comes out of
the “small coupling” regime, which makes perturbation theory inapplicable to
the study of the strong interaction processes at low energies. In other words,
the theoretical description of hadron dynamics in the infrared domain wholly
remainsbeyondthescopeofperturbationtheoryandcanonlybeperformedby
makinguseofnonperturbativemethods.Additionally,itisnecessarytooutline
thattheQCDperturbationtheoryisdirectlyapplicabletothestudyofthestrong
interaction processes only in the spacelike (Euclidean) domain, whereas the
self-consistent description of hadron dynamics in the timelike (Minkowskian)
domainisbasedonpertinentdispersionrelations.
Infact,thetheoreticaldescriptionofthestronginteractionprocessesatlow
energiesremainsoneofthemostchallengingissuesoftheelementaryparticle
physicssincethediscoveryoftheQCDasymptoticfreedom.Ingeneral,thereis
avarietyofmethodsthatenableonetoexplorethetangleddynamicsofcolored
fields in the infrared domain and to effectively describe the nonperturbative
aspects of the strong interactions. In particular, over the past decades the
confinementofcolorhasbeenaddressedintheframeworkofsuchapproaches
as, for example, the string models of hadrons [22–26], the phenomenological
potential models of quark-antiquark interaction [27–32], the quark bag mod-
els [33–35], the analytic gauge-invariant QCD [36–41], the holographic QCD
[42–45],thelatticesimulations[46–54],andmanyothers.
Theoretical particle physics widely employs various methods originated in
thecorrespondingdispersionrelations(see,e.g.,books[55–59]andpapers[60–
63]).Inparticular,theserelationshaveprovedtheirefficiencyinsuchissuesas
the extension of the applicability range of chiral perturbation theory [64–74],
theprecisedeterminationofparametersofresonances[75,76],theassessment
ofthehadroniclight-by-lightscattering[77–80],aswellasmanyothers.Basi-
cally,thedispersionrelationsprovideanadditionalsourceoftheinformationon
thepertinentphysicalprocesses,whichdoesnotrelyontheperturbationtheory.
Specifically, such relations convert the physical kinematic restrictions on the
processonhandintotheintrinsicallynonperturbativeconstraintsontherelated
functions that should definitely be accounted for when one comes out of the
applicabilityrangeoftheperturbativeapproach.Furthermore,asnotedearlier,
the relevant dispersion relations provide the only way to properly analyze the
stronginteractionprocessesinthetimelikedomain.
1.InQEDtheperturbativeapproachfailsintheultravioletasymptoticattheenergyscaleofthe
orderofthePlanckmass(seealsoSection2.2).
Introduction xv
A key role in the study of various strong interaction processes in the
spacelike and timelike domains is played by the so-called hadronic vacuum
polarization function (cid:4)(q2), the related function R(s), and the Adler func-
tionD(Q2).Inparticular,thesefunctionsgovern,forexample,suchprocessesas
theelectron-positronannihilationintohadrons,inclusiveτ leptonandZ boson
hadronicdecays,aswellasthehadroniccontributionstosuchobservablesofthe
precision particle physics as the muon anomalous magnetic moment (g−2)
μ
and the running of the electromagnetic fine structure constant. The theoretical
analysis of these processes constitutes a decisive self-consistency test of QCD
and the entire Standard Model, that, in turn, puts strong limits on a possible
new fundamental physics beyond the latter. Additionally, the aforementioned
strong interaction processes are characterized by the energy scales spanning
from low to high energies, so that their theoretical exploration constitutes
a natural framework for a thorough investigation of both perturbative and
intrinsicallynonperturbativeaspectsofhadrondynamics.Itisalsoworthwhile
to note that most of the processes mentioned earlier are of direct relevance
to the physics at the future collider projects, such as the Future Circular
Collider (FCC) at CERN (specifically, itsFCC-ee part),the Circular Electron-
PositronCollider(CEPC)inChina(specifically,thefirstphaseofthisproject),
theInternationalLinearCollider(ILC),theCompactLinearCollider(CLIC),as
well as the E989 experiment at Fermilab, the E34 experiment at Japan Proton
AcceleratorResearchComplex(J-PARC),andothers.
Thebookmainlyfocusesonthedescriptionoftheaforementionedhadronic
vacuum polarization function (cid:4)(q2), the function R(s), which is also called
R-ratio of electron-positron annihilation into hadrons, and the Adler func-
tion D(Q2). In particular, the book highlights the basics of the dispersion
relations for these functions, delineates their perturbative calculation, and
presentsthedispersivelyimprovedperturbationtheory(DPT).Thelattermerges
the intrinsically nonperturbative constraints for the functions on hand, which
originate in the pertinent dispersion relations, with corresponding perturbative
input in a self-consistent way. In addition, it overcomes inherent obstacles of
the QCD perturbation theory and substantially extends its applicability range
toward the infrared domain. The book also elucidates the essentials of the
continuation of the spacelike perturbative results into the timelike domain,
which is indispensable for the study of the electron-positron annihilation into
hadronsandtherelatedstronginteractionprocesses.
The layout of the book is as follows. Chapter 1 presents the basics of the
dispersionrelationsforthefunctionsonhand.Specifically,Section1.1describes
the function R(s), derives in the leading order of perturbation theory the total
crosssectionsoftheelectron-positronannihilationintoamuon-antimuonpairas
wellasintohadrons,discussesthekinematicrestrictionsfortheprocessonhand,
anddepictstheexperimentaldataonR-ratio.Section1.2dealswiththehadronic
vacuumpolarizationfunction(cid:4)(q2),discussesitspropertiesinthecomplexq2-
plane,derivesthedispersionrelationfor(cid:4)(q2),andpresentstheexperimental