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Strong Interactions in Spacelike and
Timelike Domains
Strong Interactions in
Spacelike and Timelike
Domains
Dispersive Approach

Alexander V. Nesterenko
Bogoliubov Laboratory of Theoretical Physics
Joint Institute for Nuclear Research
Dubna, Moscow region, Russian Federation

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD

PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
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About the Author

Dr. Alexander V. Nesterenko is a senior researcher at the Bogoliubov

Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 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. He actively works in the area of the theoretical particle physics and is a
referee for several APS and IOP journals. An experienced lecturer in Quantum
Field Theory and Quantum Chromodynamics, he has published two textbooks
based on his lecture course.

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
interaction processes entirely fall out of the scope of perturbative approach and
can be handled only within nonperturbative methods.
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
restrictions on a physical process on hand into the intrinsically nonperturbative
constraints on the related functions that should definitely be accounted for when
one goes beyond the scope of perturbation theory. Furthermore, the dispersion
relations provide the only way of the self-consistent description of the strong
interaction processes in the timelike domain.
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,
and delineates the dispersively improved perturbation theory. The latter merges
the nonperturbative constraints imposed by the pertinent dispersion relations
on the functions on hand with corresponding perturbative input in a self-
consistent way, gets rid of innate obstacles of the QCD perturbation theory, and
considerably extends its applicability range toward the low energies. The book
also elucidates the basics of the continuation of the spacelike perturbative results
into the timelike domain, which is essential for the study of electron-positron
annihilation into hadrons and the relevant strong interaction processes.
The book is based on a portion of a course of lectures on QCD, which was
delivered by the author at the Moscow State University and at the Moscow
Institute of Physics and Technology. The topics covered in the book, being of a

ix
x Preface

direct relevance to the numerous ongoing research programs in particle physics

and future collider projects, can be of benefit to graduate and postgraduate
students, academic lecturers, and scientific researchers working in the related
fields of theoretical and mathematical physics.

A.V. Nesterenko
Dubna, Russia
June 2016
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. De Fazio, 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, and H. Wittig for the interest, stimulating comments, valuable advices,
fruitful discussions, and continuous support.

xi
Introduction

The theoretical description of the strong interaction processes is utterly based

on the quantum non-Abelian gauge field theory, namely, Quantum Chromody-
namics (QCD). This theory originates in the notion of quarks and gluons. The
former are the constituents of all hadrons that were proposed in the mid-1960s
in the milestone works by Gell-Mann [1], Zweig [2, 3], and Petermann [4] (see
also paper [5]), whereas the latter are the quanta of a massless gauge vector field
providing interaction between quarks. Both quarks and gluons carry a specific
quantum number, which plays the crucial role in hadron physics. This quantum
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] (see also papers [12–14]).
Thorough theoretical and experimental investigations revealed that the
strong interactions possess two distinctive features. First, the strength of the
interaction between colored objects decreases when the characteristic energy of
the process on hand increases. In other words, the QCD invariant charge αs =
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
observed experimentally, which constitutes the so-called confinement of color.
These two phenomena are governed by the strong interactions at the opposite
energy scales. Specifically, the asymptotic freedom takes place at high energies,
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-
tensive employment of perturbation theory in the study of the strong interaction
processes at high energies. However, in practice the results of perturbative calcu-
lations in QCD appear to be of a limited applicability. This is primarily caused
by the fact that the perturbative approach in Quantum Field Theory entirely
relies on the assumption that the value of the corresponding invariant charge is
small enough. In particular, it is this assumption that allows one to approximate
an experimentally measurable physical observable by perturbative power series

xiii
xiv Introduction

in the respective coupling, which drastically simplifies the theoretical analysis

of the process on hand. In Quantum Electrodynamics (QED) the aforementioned
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
strong running coupling is valid for high and intermediate energies only. In the
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
remains beyond the scope of perturbation theory and can only be performed by
making use of nonperturbative methods. Additionally, it is necessary to outline
that the QCD perturbation theory is directly applicable to the study of the strong
interaction processes only in the spacelike (Euclidean) domain, whereas the
self-consistent description of hadron dynamics in the timelike (Minkowskian)
domain is based on pertinent dispersion relations.
In fact, the theoretical description of the strong interaction processes at low
energies remains one of the most challenging issues of the elementary particle
physics since the discovery of the QCD asymptotic freedom. In general, there is
a variety of methods that enable one to explore the tangled dynamics of colored
fields in the infrared domain and to effectively describe the nonperturbative
aspects of the strong interactions. In particular, over the past decades the
confinement of color has been addressed in the framework of such approaches
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], the lattice simulations [46–54], and many others.
Theoretical particle physics widely employs various methods originated in
the corresponding dispersion relations (see, e.g., books [55–59] and papers [60–
63]). In particular, these relations have proved their efficiency in such issues as
the extension of the applicability range of chiral perturbation theory [64–74],
the precise determination of parameters of resonances [75, 76], the assessment
of the hadronic light-by-light scattering [77–80], as well as many others. Basi-
cally, the dispersion relations provide an additional source of the information on
the pertinent physical processes, which does not rely on the perturbation theory.
Specifically, such relations convert the physical kinematic restrictions on the
process on hand into the intrinsically nonperturbative constraints on the related
functions that should definitely be accounted for when one comes out of the
applicability range of the perturbative approach. Furthermore, as noted earlier,
the relevant dispersion relations provide the only way to properly analyze the
strong interaction processes in the timelike domain.

1. In QED the perturbative approach fails in the ultraviolet asymptotic at the energy scale of the
order of the Planck mass (see also Section 2.2).
 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 (q2 ), the related function R(s), and the Adler func-
tion D(Q2 ). In particular, these functions govern, for example, such processes as
the electron-positron annihilation into hadrons, inclusive τ lepton and Z boson
hadronic decays, as well as the hadronic contributions to such observables of the
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
intrinsically nonperturbative aspects of hadron dynamics. It is also worthwhile
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, its FCC-ee part), the Circular Electron-
Positron Collider (CEPC) in China (specifically, the first phase of this project),
the International Linear Collider (ILC), the Compact Linear Collider (CLIC), as
well as the E989 experiment at Fermilab, the E34 experiment at Japan Proton
Accelerator Research Complex (J-PARC), and others.
The book mainly focuses on the description of the aforementioned hadronic
vacuum polarization function (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
presents the dispersively improved perturbation theory (DPT). The latter merges
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
hadrons and the related strong interaction processes.
The layout of the book is as follows. Chapter 1 presents the basics of the
dispersion relations for the functions on hand. Specifically, Section 1.1 describes
the function R(s), derives in the leading order of perturbation theory the total
cross sections of the electron-positron annihilation into a muon-antimuon pair as
well as into hadrons, discusses the kinematic restrictions for the process on hand,
and depicts the experimental data on R-ratio. Section 1.2 deals with the hadronic
vacuum polarization function (q2 ), discusses its properties in the complex q2 -
plane, derives the dispersion relation for (q2 ), and presents the experimental
xvi Introduction

prediction for the latter. Section 1.3 recounts the Adler function D(Q2 ), derives
the corresponding dispersion relations, discusses the nonperturbative constraints
imposed by the latter, and portrays the experimental prediction for D(Q2 ).
Chapter 2 delineates the perturbative strong running coupling and its basic
features. In particular, Section 2.1 describes the renormalization group equation
for the QCD invariant charge, discusses the perturbative calculation of the
corresponding β function up to the five-loop level, and briefly recaps the
estimations of its coefficients, which have not been explicitly calculated yet.
Section 2.2 derives the solution of the renormalization group equation for the
strong running coupling at the one-loop level, examines its properties, and
describes the so-called matching procedure. Section 2.3 presents the solutions
of the renormalization group equation for the QCD invariant charge at the higher
loop levels and discusses their peculiarities. Chapter 3 elucidates the calculation
of the functions on hand within perturbative approach. Specifically, Section 3.1
derives the hadronic vacuum polarization function (q2 ) in the leading order in
the strong coupling, discusses the corresponding perturbative corrections up to
the four-loop level, and recounts the general features of the obtained expressions
for (q2 ). Section 3.2 presents the perturbative approximation for the Adler
function D(Q2 ) at the first four loop levels, describes its basic peculiarities,
and briefly discusses the estimations of the uncalculated yet higher-order
corrections. Section 3.3 derives the R-ratio of electron-positron annihilation into
hadrons in the leading order of perturbation theory, concisely recaps the most
salient features of the perturbative approximation of the function R(s) at the
higher loop levels, and expounds its general peculiarities. Chapter 4 presents
the essentials of the dispersive approach to QCD. Specifically, Section 4.1
derives the unified integral representations for (q2 ), R(s), and D(Q2 ), which
embody all the nonperturbative constraints imposed by the pertinent dispersion
relations and express the functions on hand in terms of the common spectral
density. Section 4.2 provides the explicit expression for the perturbative spectral
density at an arbitrary loop level and describes its basic features. Section 4.3
discusses the obtained integral representations for the functions on hand in
the massless limit and recounts its implications. Chapter 5 delineates the
calculation of the functions (q2 ), R(s), and D(Q2 ) within dispersively im-
proved perturbation theory and elucidates their salient distinctions. Specifically,
Section 5.1 studies the hadronic vacuum polarization function (q2 ) at the
one-loop level, discusses the pertinent higher loop corrections, compares the
function (q2 ) with its experimental prediction, and recounts the scheme
stability of the obtained results. Section 5.2 expounds the function R(s) at
the one-loop level, studies the higher loop corrections to R(s), examines its
scheme stability, and juxtaposes the obtained results with experimental data
on R-ratio by making use of the smearing method. Section 5.3 recounts the
Adler function D(Q2 ) at the one-loop level, examines the relevant higher loop
corrections, compares the function D(Q2 ) with its experimental prediction, and
expounds the scheme stability of the obtained results. Chapter 6 elucidates
 Introduction xvii

the perturbative approximation of the R-ratio of electron-positron annihilation

into hadrons and discusses its basic peculiarities. Specifically, Section 6.1
studies the reexpansion of the function R(s) at high energies and describes the
appearance of the π 2 -terms. Section 6.2 delineates the perturbative expression
for the R-ratio beyond the three-loop level, recounts its convergence range, and
discusses the impact of the π 2 -terms. Section 6.3 expounds the estimation of
the uncalculated yet coefficients of the perturbative approximation of R-ratio.
The technical auxiliary materials are gathered in the Appendices. Specifically,
Appendix A supplies the explicit expressions for the perturbative strong running
coupling at the higher loop levels. Appendix B presents the explicit expressions
for the aforementioned spectral function at the higher orders of perturbation
theory and discusses the numerical evaluation of the functions (q2 ), R(s),
and D(Q2 ) within DPT. Appendix C recaps the perturbative approximation
of the R-ratio of electron-positron annihilation into hadrons and provides the
explicit expressions for the π 2 -terms at the higher-loop levels.
Chapter 1

Basic Dispersion Relations

The hadronic vacuum polarization function, as well as the related Adler function
and the R-ratio of electron-positron annihilation into hadrons, plays a significant
role in various issues of contemporary elementary particle physics. The theoreti-
cal description of these functions is inherently based on the perturbation theory,
which supplies the corresponding high-energy input, and dispersion relations,
which embody the pertinent low-energy kinematic constraints and enable one
to study the hadron dynamics in the timelike domain in a self-consistent way.
The functions on hand form the basis of the theoretical analysis of a variety
of the strong interaction processes, including the hadronic contributions to
precise electroweak observables, that, in turn, constitutes a decisive test of
Quantum Chromodynamics (QCD) and entire Standard Model and imposes
strict restrictions on a possible new fundamental physics beyond the latter.

1.1 R-RATIO OF ELECTRON-POSITRON ANNIHILATION

INTO HADRONS
The process of electron-positron annihilation into hadrons plays a distinctive
role in elementary particle physics. This is primarily caused by a remarkable
fact that its theoretical description requires no phenomenological models of the
process of hadronization, which forms the experimentally detected final-state
particles. In turn, this feature makes it possible to extract the key parameters of
the theory from pertinent data in the model-independent way, that is certainly
essential for the experimental verification of the latter.
The analysis of the process on hand usually involves the so-called R-ratio of
electron-positron annihilation into hadrons, which is defined as the ratio1 of two
experimentally measurable total cross sections
 
σ e+ e− → hadrons; s
R(s) =  + − . (1.1)
σ e e → μ+ μ− ; s

1. It is assumed that both total cross sections on the right-hand side of Eq. (1.1) correspond to the
leading order in the electromagnetic fine structure constant αem = e2 /(4π ).

Strong Interactions in Spacelike and Timelike Domains. http://dx.doi.org/10.1016/B978-0-12-803439-2.00001-6

© 2017 Elsevier Inc. All rights reserved. 1
2 Strong Interactions in Spacelike and Timelike Domains

e− p1 k1 μ−
q

γ, Z
e+ p2 k2 μ+
FIG. 1.1 The process of the electron-positron annihilation into muon-antimuon pair in the lowest
order of perturbation theory.

Such normalization proves to be particularly convenient for the studies of the

hadronic vacuum polarization. Specifically, the leptonic parts of the processes
contributing to Eq. (1.1) are identical to each other that makes the R-ratio strictly
focused on the effects due to the strong interactions.
Let us first address the process of the electron-positron annihilation into
a muon-antimuon pair (see Fig. 1.1). The matrix element corresponding to
the diagram with an intermediate photon takes the form (see also papers by
Bhabha [81] and Feynman [82, 83]):
 
e2 qμ qν [μ]
Mfi = iv̄ (p2 , σ2 )γ u (p1 , σ1) 2 gμν −(1−ξ ) 2 ū (k1 , λ1 )γ ν v[μ] (k2 , λ2 ).
[e] μ [e]
q q
(1.2)
In this equation e is the charge of the electron, gμν = diag(1, −1, −1, −1) stands
for the metric tensor, γμ denotes the Dirac matrix, q = p1 + p2 is the timelike
kinematic variable, and ξ stands for the gauge parameter (a real constant),
whereas u[e] (p1 , σ1 ), v̄[e] (p2 , σ2 ), ū[μ] (k1 , λ1 ), and v[μ] (k2 , λ2 ) denote the Dirac
spinors of incoming electron with four-momentum p1 and spin σ1 , the incoming
positron with four-momentum p2 and spin σ2 , the outgoing muon with four-
momentum k1 and spin λ1 , and the outgoing antimuon with four-momentum k2
and spin λ2 , respectively. In what follows the Feynman gauge (ξ = 1) will
be assumed, which makes the expression for the photon propagator quite
simple. The cross section of the process on hand is proportional to the matrix
element (1.2) squared, which can be represented in the following way:
e4 [e]
|Mfi |2 = ū (p1 , σ1 )γν v[e] (p2 , σ2 )v̄[e] (p2 , σ2 )γμ u[e] (p1 , σ1 )
s2
× v̄[μ] (k2 , λ2 )γ ν u[μ] (k1 , λ1 )ū[μ] (k1 , λ1 )γ μ v[μ] (k2 , λ2 ), (1.3)
where s = q2 > 0. Assuming that the spins of the final state particles are unknown
and that the colliding beams are not polarized, one has to sum Eq. (1.3) over the
spins of muon and antimuon and average the result over the spins of electron
and positron. This eventually yields
 Basic Dispersion Relations Chapter | 1 3

1 e4  
|Mfi |2 = 2 Tr (p̂1 + me )γν (p̂2 − me )γμ
4 4s
spin

× Tr (k̂2 − mμ )γ ν (k̂1 + mμ )γ μ , (1.4)

with
 
uα (p, σ )ūβ (p, σ ) = (p̂ + m)βα , vα (p, σ )v̄β (p, σ ) = (p̂ − m)βα (1.5)
σ σ
being employed. In Eq. (1.4) me  0.511 MeV and mμ  105.66 MeV stand
for the masses of electrons and muons, respectively [84]. Then by making use
of the relations
 
Tr γμ γν = 4gμν , (1.6)

2r+1
Tr γμj = 0, r = 0, 1, 2, 3, . . . , (1.7)
j=1

   
Tr γμ γν γρ γσ = 4 gμν gρσ − gμρ gνσ + gμσ gνρ , (1.8)
one arrives at
 
Tr (p̂1 + me )γν (p̂2 − me )γμ = 4 p1μ p2ν + p2μ p1ν − (p1 p2 )gμν − m2e gμν
(1.9)
and
μ μ
Tr (k̂2 − mμ )γ ν (k̂1 + mμ )γ μ = 4 k1 k2ν + k2 k1ν − (k1 k2 )gμν − m2μ gμν .
(1.10)
Thus the expression (1.4) acquires the following form:
1 8e4
|Mfi |2 = 2 (p1 k1 )(p2 k2 ) + (p1 k2 )(p2 k1 ) + m2e (k1 k2 )
4 s
spin

+ m2μ (p1 p2 ) + 2m2e m2μ . (1.11)

In what follows, it is convenient to study the process on hand in the

center-of-mass reference frame. In this case the four-momenta of the involved
particles read
p1 = (E, p), p2 = (E, −p), 
k1 = (E, k), 
k2 = (E, −k). (1.12)
In turn, the center-of-mass energy squared s is
s = (p1 + p2 )2 = q2 = (k1 + k2 )2 = 4E2 , (1.13)
the pertinent three-momenta are
4 Strong Interactions in Spacelike and Timelike Domains

s s
p2 = − m2e , k2 = − m2μ , (1.14)
4 4
and
⎛   ⎞
s⎝ 4m2e 4m2μ
(p1 k1 ) = (p2 k2 ) = 1 − cos ϕ 1 − 1− ⎠, (1.15)
4 s s

⎛   ⎞
s⎝ 4m2e 4m2μ
(p1 k2 ) = (p2 k1 ) = 1 + cos ϕ 1 − 1− ⎠, (1.16)
4 s s

s s
(p1 p2 ) =
− m2e , (k1 k2 ) = − m2μ , (1.17)
2 2
where ϕ denotes the angle between the three-momentum of electron p and
the three-momentum of muon k.  Hence, Eq. (1.11) can be represented in the
following way:
    
1 4m 2 4m 2
4  
e μ
|Mfi |2 = e4 1 + cos2 ϕ 1 − 1− + m2e + m2μ .
4 s s s
spin
(1.18)

Thus in the center-of-mass reference frame the differential cross section of

the process on hand can be represented as
   1/2
dσ e+ e− → μ+ μ− ; s 1 1 |k|  1 e4 1 s − 4m2μ
= |Mfi | =
2
d (4π )2 s |p| 4 (4π )2 4s s − 4m2e
spin
    
4m 2 4m 2
4  
e μ
× 1 + cos2 ϕ 1 − 1− + m2e + m2μ .
s s s
(1.19)
In the limit of a massless electron (me = 0) and muon (mμ = 0) this equation
acquires the form
 
dσ e+ e− → μ+ μ− ; s α2  
= em 1 + cos2 ϕ , (1.20)
d 4s
where αem = e2 /(4π ) stands for the electromagnetic fine structure constant.
Therefore, the total cross section of the electron-positron annihilation into
muon-antimuon pair reads
2π 1  
 + − α2+ −
 2
4π αem
σ e e → μ μ ; s = em dφ 1 + cos2 ϕ d cos ϕ = .
4s 3 s
0 −1
(1.21)
 Basic Dispersion Relations Chapter | 1 5

p1 h
e− qf
q a
d
r
γ,Z o
e+ p2 qf n
s
FIG. 1.2 The process of the electron-positron annihilation into hadrons in the lowest order of
perturbation theory.

The obtained expression for the cross section of the process on hand (1.21) is a
rapidly decreasing function of the center-of-mass energy squared s. The electro-
magnetic running coupling αem entering Eq. (1.21), as well as its dependence
on the energy scale, was thoroughly studied in the works by Burkhardt and
Pietrzyk [85–89], Jegerlehner [90–93], and Hagiwara et al. [94–97], see also
review [98] and references therein. Note that the process of electron-positron
annihilation into muon-antimuon pair also involves intermediate Z boson (MZ 
91.188 GeV [84]), which has to be accounted for separately (see Fig. 1.1).
Let us turn now to the process of the electron-positron annihilation into
hadrons. In the leading order of perturbation theory the latter is governed by the
diagram displayed in Fig. 1.2. The matrix element corresponding to the diagram
with intermediate photon takes the form
gμν
Mfi = iv̄[e] (p2 , σ2 )eγ μ u[e] (p1 , σ1 ) 2 |J ν (q)|0. (1.22)
q
As before, in this equation u[e] (p1 , σ1 ) and v̄[e] (p2 , σ2 ) stand for the Dirac
spinors of incoming electron with four-momentum p1 and spin σ1 and in-
coming
nf positron with four-momentum p2 and spin σ2 , respectively, Jν =
f =1 Q f :q̄f γν qf : denotes the electromagnetic quark current, nf is the number
of active flavors,2 Qf stands for the electric charge of the quark of f th flavor (in
units of the elementary charge e), and  is a hadronic final state. The photon
propagator in Eq. (1.22) corresponds to the Feynman gauge, whereas q2 = s =
(p1 + p2 )2 > 0 denotes the respective timelike kinematic variable, namely, the
center-of-mass energy squared.
Following the very same lines as earlier, it is quite straightforward to
demonstrate that the total cross section of electron-positron annihilation into
hadrons can be represented as
  α2
σ e+ e− → hadrons; s = 8π 2 em Lμν Hμν . (1.23)
s3

2. The number of active flavors is the number of quark flavors whose masses are less than the
characteristic energy of a process on hand, see Table 1.1 and Section 2.1.
6 Strong Interactions in Spacelike and Timelike Domains

TABLE 1.1 The Electric Charges (in Units of the

Elementary Charge) and Masses of Quarks [84]
f Qf mf
u 2/3 2.3+0.7
−0.5 MeV
a

d −1/3 4.8+0.5
−0.3 MeV
a

s −1/3 95 ± 5 MeVa
c 2/3 1.275 ± 0.025 GeVb
b −1/3 4.18 ± 0.03 GeVb
t 2/3 173.21 ± 0.87 GeVc
a Current quark mass in a mass-independent subtraction scheme at
μ = 2 GeV.
b Running mass in the MS scheme.
c Direct measurement.

In this equation Lμν denotes the leptonic tensor (the electron is assumed to be
massless hereinafter)
1  [e] ∗
Lμν = v̄ (p2 , σ2 )γμ u[e] (p1 , σ1 ) v̄[e] (p2 , σ2 )γν u[e] (p1 , σ1 )
4
spin
1
= qμ qν − gμν q2 − (p1 − p2 )μ (p1 − p2 )ν , (1.24)
2
and Hμν stands for the hadronic tensor

Hμν = (2π )4 δ (p1 + p2 − p ) 0|Jμ (−q)||Jν (q)|0


= d x eiqx 0|[Jμ (x), Jν (0)]|0,
4
q2 ≥ 4m2π , (1.25)

where p denotes the four-momentum of a final hadronic state  and mπ 

139.57 MeV stands for the mass of π meson [84]. Eq. (1.25) employs the
completeness of set of hadronic states

|| = θ (q2 − 4m2π ), (1.26)

with θ (x) being the Heaviside unit step function:

1, x ≥ 0,
θ (x) = (1.27)
0, x < 0
(note that θ (x) is a dimensionless function, though its argument may be of the
dimension of, e.g., mass squared). Basically, Eq. (1.26) explicitly expresses
 Basic Dispersion Relations Chapter | 1 7

the physical fact that the hadrons can be produced √ in the process of electron-
positron annihilation only if the respective energy s is high enough, otherwise
the matrix element of the process on hand (1.22) and its square vanish (see also
discussion of this issue in, e.g., book [99] and papers [100, 101]). It is also
worthwhile to mention that the processes of creating the quark-antiquark pair
and its subsequent hadronization are commonly assumed to be well separated in
time and, hence, independent of each other.
Then it proves to be convenient to define the hadronic vacuum polarization
tensor (or the two-point correlation function of quark currents, as it is also
called)

i
μν (q ) = i d4 x eiqx 0|T{Jμ (x)Jν (0)}|0 =
2
(qμ qν − gμν q2 )(q2 )
12π 2
(1.28)
and isolate its scalar part (q2 ), the latter being the hadronic vacuum polariza-
tion function. By virtue of the optical theorem, the hadronic tensor Hμν (1.25) is
proportional to the discontinuity of the function μν (q2 ) across the physical cut
that eventually makes it possible to express the total cross section of the process
on hand in terms of (q2 ), specifically
  4 αem
2
σ e+ e− → hadrons; s = Im lim (s + iε). (1.29)
3 s ε→0+

In this equation αem stands for the electromagnetic fine structure constant and
s denotes the center-of-mass energy squared. Therefore the R-ratio of electron-
positron annihilation into hadrons (1.1) acquires the form
1
R(s) = Im lim (s + iε). (1.30)
π ε→0+

As discussed earlier, the experiments on electron-positron annihilation

into hadrons play a decisive role in the precise self-consistency tests of
QCD and entire Standard Model that puts strong limits on a possible new
fundamental physics beyond the latter. This is primarily caused by the fact
that the theoretical description of the process on hand, contrary to some
other strong interaction processes, does not require the phenomenological
models of the process of hadronization. In particular, the summation over
hadronic final states in Eq. (1.25) and the completeness of the set of hadronic
states (1.26) enable one to eliminate the process of hadronization, which still
has no reliable theoretical description, from the analysis. In turn, this opens
the way for a model-independent evaluation of the key parameters of the
theory.
At the same time, the physical fact that the matrix element of the process on
hand (1.22) vanishes identically for the energies below the hadronic production
threshold imposes intrinsically nonperturbative constraints on the hadronic
tensor Hμν (1.25), as well as on all the related functions. Specifically, this
kinematic restriction implies that the hadronic tensor Hμν assumes nonzero
8 Strong Interactions in Spacelike and Timelike Domains

103

102 f
w
101
r⬘

100
r

10–1
100 101 102
FIG. 1.3 The experimental data on R-ratio of electron-positron annihilation into hadrons (1.1).
(0)
A dashed line corresponds to the parton model prediction Rpert (s) (1.32). The displayed data are
available at http://pdg.lbl.gov/2015/hadronic-xsections.

values only for the energies exceeding the hadronic production threshold (i.e.,
for q2 ≥ 4m2π ), whereas Hμν ≡ 0 for q2 < 4m2π , and so does the function
R(s) (1.1), namely
R(s) ≡ 0, 0 ≤ s < 4m2π . (1.31)
Additionally, the aforementioned kinematic restriction implies that the hadronic
vacuum polarization function (q2 ) (1.28) has the only cut in the complex q2 -
plane, which starts at the hadronic production threshold q2 = 4m2π and goes
along the positive semiaxis of real q2 , see Fig. 1.4. Basically, this feature
also leads to a number of nonperturbative constraints on the related Adler
function D(Q2 ) (1.46), see Section 1.3 for details. In turn, as it will be discussed
in Chapter 4, all these restrictions play a substantial role in the study of the
functions on hand at low energies.
The experimental data on R-ratio of electron-positron annihilation into
hadrons are displayed in Fig. 1.3. As one can infer from this figure, a distinctive
feature of the function R(s) (1.1) is the presence of the resonance peaks, which
correspond to the bound states of quarks. The formation of the latter, which
involves all the tangled nonperturbative dynamics of colored fields, still has no
rigorous unabridged theoretical description. In particular, this fact implies that
the direct comparison of the experimental data on R-ratio with its perturbative
approximation Rpert (s) is sensible only in the energy intervals, which are located
far enough from the hadronic resonances. Alternatively, one can employ the
commonly used method of “smearing” the function R(s) [102], which averages
out the resonance structure of the R-ratio, see Section 5.2 for details.
 Basic Dispersion Relations Chapter | 1 9

TABLE 1.2 Numerical Values of the Factors Appearing in

Front of Nonsinglet and Singlet Parts of the Hadronic
Vacuum Polarization Function (Second and Third
Columns, Respectively)
nf  2
nf
nf Nc f =1
Q2f Q
f =1 f
1 4/3 4/9
2 5/3 1/9
3 2 0
4 10/3 4/9
5 11/3 1/9
6 5 1

Fig. 1.3 also displays the so-called naive parton model prediction (dashed
line)
nf
(0)

Rpert (s) = Nc Q2f , s → ∞, (1.32)
f =1

which represents a rather rough approximation of the function R(s) and is valid
at high energies only, see Refs. [99, 103]. In particular, in the deep ultraviolet
limit the effects due to the masses of all the involved particles, as well as
the strong corrections to the process on hand, can be safely neglected. In this
case the only distinction between the cross sections of two processes displayed
in Figs. 1.1 and 1.2 is accounted for by the product of two factors (1.32).
Namely, the factor Nc = 3 comes from the fact that every quark can be in
three color states, whereas the factors Q2f are due to the difference between the
electric charges of the muon and quark of f th flavor. The numerical values of
the parton model prediction (1.32), as well as the values of the factor entering the
calculation of the singlet part of the hadronic vacuum polarization function
at the higher loop levels (see Section 3.2), are listed in Table 1.2. Note that
since the factor (1.32) appears in front of all the functions on hand, it will
be omitted in what follows. It is also worthwhile to mention that an accurate
theoretical analysis of the R-ratio of electron-positron annihilation into hadrons
surely requires the proper account of the higher-order (in the strong coupling)
contributions to the hadronic vacuum polarization function (q2 ) (1.28). The
detailed description of this issue will be given in the next chapters.
It is necessary to emphasize that the results of the QCD perturbation theory
are not directly applicable to the study of the strong interaction processes
in the timelike domain (see papers [100, 101, 104–108]). In particular, the
10 Strong Interactions in Spacelike and Timelike Domains

self-consistent description of such processes (including the electron-positron

annihilation into hadrons) can be performed only by making use of the
corresponding dispersion relations. For example, the derivation of the parton
model prediction (1.32) includes the calculation of the hadronic vacuum
polarization function (0) (q2 ) in the zeroth order in the strong coupling (see
Eq. 3.32) and the subsequent continuation of the obtained result into the timelike
domain (1.40). Equivalently, Eq. (1.32) can also be obtained by making use of
the corresponding expression for the Adler function D(0) (Q2 ) (see Eqs. 1.46,
3.53) and the inverse relation (1.50). Basically, it appears that the effects due to
such continuation of the spacelike perturbative results into the timelike domain
substantially affect the function R(s) and thereby play an essential role in the
study of the R-ratio of electron-positron annihilation into hadrons and the related
quantities. A detailed discussion of this issue will be given in the next two
sections as well as in Chapters 4 and 6.

1.2 HADRONIC VACUUM POLARIZATION FUNCTION (q2 )

As noted in the previous section, the kinematics of the process on hand
determines the analytic properties of the hadronic vacuum polarization func-
tion (q2 ) (1.28) in the complex q2 -plane. Specifically, the physical fact that
the production of the final state hadrons in electron-positron annihilation is kine-
matically forbidden for the energies q2 < 4m2π implies that the function (q2 )
has the only cut q2 ≥ 4m2π along the positive semiaxis of real q2 .
In turn, once the location of the cut of the hadronic vacuum polarization
function (q2 ) in the complex q2 -plane is known, one can write down the
corresponding dispersion relation by making use of the Cauchy’s integral
formula (see, e.g., books [109–111] for details):

1 f (ξ )
f (x) = dξ . (1.33)
2πi ξ −x
C

Since the function (q2 ) logarithmically increases in the ultraviolet asymptotic

(see Eq. 3.32), it proves to be convenient to use the once-subtracted Cauchy’s
integral formula, which directly follows from Eq. (1.33)

1 f (ξ )
f (x) = f (x0 ) + (x − x0 ) dξ , (1.34)
2πi (ξ − x)(ξ − x0 )
C

with x0 being the subtraction point. Thus, for the function (q2 ) (1.28) the
Cauchy’s integral formula (1.34) subtracted at a point q20 reads

1 2 (ξ )
(q2 , q20 ) = (q − q20 ) dξ . (1.35)
2πi (ξ − q2 )(ξ − q20 )
C
 Basic Dispersion Relations Chapter | 1 11

FIG. 1.4 The integration contour C in the complex ξ -plane in the subtracted Cauchy’s integral
formula (1.35). The physical cut ξ ≥ 4m2π of the hadronic vacuum polarization function (ξ ) (1.28)
is shown along the positive semiaxis of real ξ .

In this equation (q2 , q20 ) = (q2 ) − (q20 ), whereas the closed integration
contour C, which encloses3 both points q2 and q20 (commonly, one chooses
q20 = 0) and goes counterclockwise along the circle of infinitely large radius,
is displayed in Fig. 1.4. Because the integration along the circles of infinitely
large (c∞ ) and infinitesimal (cm ) radii gives no contribution to Eq. (1.35), the
hadronic vacuum polarization function can be represented as
⎡

∞+iε
q − q0
2 2
⎢ (ξ ) dξ
(q , q0 ) =
2 2
lim ⎣
2πi ε→0+ (ξ − q2 )(ξ − q20 )
4m2π +iε
⎤
4m2π −iε

(ξ ) dξ ⎥
+ ⎦. (1.36)
(ξ − q2 )(ξ − q20 )
∞−iε

Then, in Eq. (1.36) it is convenient to change the integration variable

s = ξ − iε in the first term in the square brackets, which takes the form
∞
(s + iε)
ds + O(ε). (1.37)
(s − q2 )(s − q20 )
4m2π

3. Since both points q2 and q20 must be enclosed by the integration contour C in Eq. (1.35), for
real q2 and q20 only the values q2 < 4m2π and q20 < 4m2π can be chosen. The implications of this
fact for the massless limit will be discussed in Sections 4.3 and 5.1.
12 Strong Interactions in Spacelike and Timelike Domains

Similarly, after the change of the integration variable s = ξ + iε in the second

term, the latter reads

∞
(s − iε)
− ds + O(ε). (1.38)
(s − q2 )(s − q20 )
4m2π

Hence, the once-subtracted Cauchy’s integral formula for (q2 ) (1.35) can be
represented as

∞
q2 − q20 (s + iε) − (s − iε)
(q2 , q20 ) = lim ds. (1.39)
2πi ε→0+ (s − q2 )(s − q20 )
4m2π

Recall that for the function (ξ ), which satisfies the condition (ξ * ) = * (ξ ),
its discontinuity across the physical cut reads
1 1
R(s) = Im lim (s + iε) = lim [(s + iε) − (s − iε)] . (1.40)
π ε→0+ 2πi ε→0+
Thus, the dispersion relation for the hadronic vacuum polarization func-
tion (q2 ) subtracted at a point q20 acquires the well-known form

∞
R(s)
(q 2
, q20 ) = (q2
− q20 ) ds. (1.41)
(s − q2 )(s − q20 )
4m2π

It is worthwhile to mention that Eqs. (1.40), (1.41) enable one to express

the functions R(s) and (q2 ) in terms of each other. Also note that the
dispersion relation (1.41) inherently embodies the kinematic restrictions on the
process on hand. Namely, Eq. (1.41) explicitly implies that the hadronic vacuum
polarization function (q2 ) possesses the only cut4 along the positive semiaxis
of real q2 starting at the hadronic production threshold q2 ≥ 4m2π , so that its
discontinuity (1.40) vanishes everywhere in the complex q2 -plane except for this
cut thereby expressing the physical fact that the R-ratio (1.40) assumes nonzero
values only for the energies q2 ≥ 4m2π .
Basically, the hadronic vacuum polarization function (q2 ) (1.28), contrary
to the R-ratio of electron-positron annihilation into hadrons (1.1), cannot be
directly measured in the experiments. Nonetheless, the dispersion relation (1.41)
makes it possible to obtain the experimental prediction for the function (q2 ).

4. That signifies that the hadronic vacuum polarization function (q2 ) is free of the unphysical
singularities.
 Basic Dispersion Relations Chapter | 1 13

Specifically, for this purpose the function R(s) entering the integrand of
Eq. (1.41) can be approximated by the experimental data on the R-ratio (1.1)
at low and intermediate energies and by its perturbative expression5 at high
energies, namely
()
R(s)  θ (s0 − s)Rdata (s) + θ (s − s0 )Rpert (s). (1.42)
In this equation θ (x) stands for the Heaviside unit step function (1.27), whereas
the parameter s0 of the dimension of mass squared specifies the energy scale
()
of matching6 of the quantities Rdata (s) and Rpert (s), and has to be chosen
()
large enough to securely make the perturbative approximation Rpert (s) reliable
for all s ≥ s0 . Thus, the experimental prediction for the hadronic vacuum
polarization function (q2 ) can be represented in the following form:
s0 ∞
(q2 − q20 )Rdata (s) (q2 − q20 )R()
pert (s)
exp (q 2
, q20 ) = ds + ds. (1.43)
(s − q2 )(s − q20 ) (s − q2 )(s − q20 )
4m2π s0

As one might note, in general, the function exp (q2 , q20 ) (1.43) depends on the
value of parameter s0 , on the loop level , and (at the higher loop levels) on the
employed renormalization scheme.
In fact, the experimental predictions for all the quantities related to the
hadronic vacuum polarization function can be obtained in the very same
way as for (q2 ) itself. For example, the approximation (1.42) is commonly
employed for the evaluation of the Adler function D(Q2 ) in the infrared domain
(see Section 1.3), as well as for the assessment of the hadronic contributions
to such electroweak observables, as the muon anomalous magnetic moment
and the shift of the electromagnetic fine structure constant at the scale of
Z boson mass, see reviews [112–118] and recent papers [93, 96, 119–121] for
the details. It is also worthwhile to mention that for the energies below the mass
of τ lepton (i.e., for s < Mτ2 , Mτ  1.777 GeV [84]) the experimental data on R-
ratio of electron-positron annihilation into hadrons can be substituted by the so-
called inclusive τ lepton hadronic decay vector spectral function. The original
measurements of the latter by ALEPH and OPAL Collaborations, as well as its
recent updates can be found in papers [122–125] and [126–131], respectively.
However, to make such substitution justified one has to properly account for the
effects due to the breaking of the isospin symmetry, see papers [132–134] and
references therein for a detailed discussion of this issue.

5. The perturbative approximation of R-ratio of electron-positron annihilation into hadrons will be

discussed in Section 3.3 and Chapter 6.
6. Not to be confused with the matching condition for the strong running coupling, which will be
discussed in Section 2.2.
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Custard, milk Ordinary cup 122 4.29 26 56 18
Custard, tapioca Two-thirds ord. 69.5 2.45 9 12 79
[16]Doughnuts Half a doughn’t 23 .8 6 45 49
[16]Lady fingers Two 27 .95 10 12 78
[16]Macaroons Four 23 .82 6 33 61
[16]Pie, apple One-third piece 38 1.3 5 32 63
[16]Pie, cream One-fourth piece 30 1.1 5 32 63
[16]Pie, custard One-third piece 55 1.9 9 32 59
[16]Pie, lemon One-third piece 38 1.35 6 36 58
[16]Pie, mince One-fourth piece 35 1.2 8 38 54
[16]Pie, squash One-third piece 55 1.9 10 42 48
Pudding, apple sago 81 3.02 6 3 91
Pudding, brown betty Half ord. s’v’g. 56.6 2. 7 12 81
Pudding, cream rice Very small s’v’g. 75 2.65 8 13 79
Pudding, Indian meal Half ord. ser’g. 56.6 2. 12 25 63
Pudding, apple tapioca Small serving 79 2.8 1 1 98
Tapioca, cooked Ord. serving 108 3.85 1 1 98

SWEETS AND PICKLES

[16]Catsup, tomato, av. 170 6. 10 3 87
Candy, plain 26 .9 0 0 100
Candy, chocolate 30 1.1 1 4 95
[16]Honey Four teasp’ns 30 1.05 1 0 99
[16]Marmalade (orange) 28.3 1 .5 2.5 97
[16]Molasses, cane 35 1.2 .5 0 99.5
[16]Olives, green, edible
Five to seven 32 1.1 1 84 15
portion
[16]Olives, ripe, edible
Five to seven 38 1.3 2 91 7
portion
[16]Pickles, mixed 415 14.6 18 15 67
[16]Sugar, Three heaping
granulated 24 .86 0 0 100
tsp. or 1½ lumps
[16]Sugar, maple Four teaspoons 29 1.03 0 0 100
[16]Syrup, maple Four teaspoons 35 1.2 0 0 100

NUTS, EDIBLE PORTION

[16]Almonds, av. Eight to 15 15 .53 13 77 10
[16]Beechnuts 14.8 .52 13 79 8
[16]Brazil nuts Three ord. size 14 .49 10 86 4
[16]Butternuts 14 .50 16 82 2
[16]Cocoanuts 16 .57 4 77 19
[16]Chestnuts, fresh, av. 40 1.4 10 20 70
[16]Filberts, av. Ten nuts 14 .48 9 84 7
[16]Hickory nuts 13 .47 9 85 6
[16]Peanuts, av. Thirteen double 18 .62 20 63 17
[16]Pecans, polished About eight 13 .46 6 87 7
[16]Pine nuts, (pignolias) About eighty 16 .56 22 74 4
[16]Walnuts, California About six 14 .48 10 83 7

CEREALS
[16]Bread, brown,
Ord. thick slice 43 1.5 9 7 84
average
[16]Bread, corn
Small square 38 1.3 12 16 72
(johnnycake) av.
[16]Bread, white, home
Ord. thick slice 38 1.3 13 6 81
made
[16]Cookies, sugar Two 24 .83 7 22 71
Corn flakes, toasted Ord. cer. dish f’l. 27 .97 11 1 88
[16]Corn meal, granular,
2½ level tbsp. 27 .96 10 5 85
av.
Corn meal, unbolted, av. Three tbsp. 26 .92 9 11 80
[16]Crackers, graham Two crackers 23 .82 9.5 20.5 70
[16]Crackers, oatmeal Two crackers 23 .81 11 24 65
[16]Crackers, soda 3½ “Uneedas” 24 .83 9.4 20 70.6
[16]Hominy, cooked Large serving 120 4.2 11 2 87
[16]Macaroni, av. 27 .96 15 2 83
Macaroni, cooked Ord. serving 110 3.85 14 15 71
[16]Oatmeal, boiled 1½ serving 159 5.6 18 7 75
[16]Popcorn 24 .86 11 11 78
[16]Rice, uncooked 28 .98 9 1 90
[16]Rice, boiled Ord. cereal dish 87 3.1 10 1 89
[16]Rice, flakes Ord. cereal dish 27 .94 8 1 91
[16]Rolls, Vienna, av. One large roll 35 1.2 12 7 81
[16]Shredded wheat One biscuit 27 .94 13 4.5 82.5
[16]Spaghetti, average 28 .97 12 1 87
[16]Wafers, vanilla Four 24 .84 8 13 71
Wheat, flour, e’t’e w’h’t,
Four tbsp. 27 .96 15 5 80
av.
[16]Wheat, flour, graham,
4½ tbsp. 27 .96 15 5 80
av.
[16]Wheat, flour, patent,

family and straight grade Four tbsp. 27 .97 12 3 85

spring wheat, av.
[16]Zwieback Size of thick slice
23 .81 9 21 70
of bread

MISCELLANEOUS
[16]Eggs, hen’s boiled One large egg 59 2.1 32 68 00
[16]Eggs, hen’s whites Of six eggs 181 6.4 100 0 00
[16]Eggs, hen’s yolks Two yolks 27 .94 17 83 00
[16]Omelet 94 3.3 34 60 6
[16]Soup, beef, av. 380 13. 69 14 17
[16]Soup, bean, av. Very large plate 150 5.4 20 20 60
[16]Soup, cream of celery Two plates 180 6.3 16 47 37
[16]Consomme 830 29. 85 00 15
[16]Clam chowder Two plates 230 8.25 17 18 65
[16]Chocolate, bitter Half-a-square 16 .56 8 72 20
[16]Cocoa 20 .69 17 53 30
Ice cream (Phila) Half serving 45 1.6 5 57 38
Ice cream (New York) Half serving 48 1.7 7 47 46

Tables Showing Average Height, Weight, Skin Surface and Food

Units Required Daily With Very Light Exercise
 BOYS
Age Height in Weight in Surface in Calories or
Inches Pounds Square Feet Food Units
5 41.57 41.09 7.9 816.2
6 43.75 45.17 8.3 855.9
7 45.74 49.07 8.8 912.4
8 47.76 53.92 9.4 981.1
9 49.69 59.23 9.9 1,043.7
10 51.58 65.30 10.5 1,117.5
11 53.33 70.18 11.0 1,178.2
12 55.11 76.92 11.6 1,254.8
13 57.21 84.85 12.4 1,352.6
14 59.88 94.91 13.4 1,471.3

GIRLS
Age Height in Weight in Surface in Calories or
Inches Pounds Square Feet Food Units
5 41.29 39.66 7.7 784.5
6 43.35 43.28 8.1 831.9
7 45.52 47.46 8.5 881.7
8 47.58 52.04 9.2 957.1
9 49.37 57.07 9.7 1,018.5
10 51.34 62.35 10.2 1,081.0
11 53.42 68.84 10.7 1,148.5
12 55.88 78.31 11.8 1,276.8

MEN
Height Weight Surface in Calories or Food Units
in In. in Pounds Square Ft. Proteids Fats Carbohydrates Total
61 131 15.92 197 591 1,182 1,970
62 133 16.06 200 600 1,200 2,000
63 136 16.27 204 612 1,224 2,040
64 140 16.55 210 630 1,260 2,100
65 143 16.76 215 645 1,290 2,150
66 147 17.06 221 663 1,326 2,210
67 152 17.40 228 684 1,368 2,280
68 157 17.76 236 708 1,416 2,360
 69 162 18.12 243 729 1,458 2,430
70 167 18.48 251 753 1,506 2,510
71 173 18.91 260 780 1,560 2,600
72 179 19.34 269 807 1,614 2,690
73 185 19.89 278 834 1,668 2,780
74 192 20.33 288 864 1,728 2,880
75 200 20.88 300 900 1,800 3,000

WOMEN
Height Weight Surface in Calories or Food Units
in In. in Pounds Square Ft. Proteids Fats Carbohydrates Total
59 119 14.82 179 537 1,074 1,790
60 122 15.03 183 549 1,098 1,830
61 124 15.29 186 558 1,116 1,860
62 127 15.50 191 573 1,146 1,910
63 131 15.92 197 591 1,182 1,970
64 134 16.13 201 603 1,206 2,010
65 139 16.48 209 627 1,254 2,090
66 143 16.76 215 645 1,290 2,150
67 147 17.06 221 663 1,326 2,210
68 151 17.34 227 681 1,362 2,270
69 155 17.64 232 696 1,392 2,320
70 159 17.92 239 717 1,434 2,390

NOTE—With active exercise an increase of about 20

per cent total food units may be needed.

DIETARY CALCULATION WITH FOOD VALUES IN

CALORIES PER OUNCE
Breakfast Proteids Fats Carbohydrates Total
Gluten Gruel 5 oz. 23.5 1.0 30.0
Soft-Boiled Egg 26.3 41.9
Malt Honey 1 oz. 86.2
Creamed Potatoes 5 oz. 15.0 40.0 104.0
Zwieback 2 oz. 22.8 52.8 171.6
Pecans ¾ oz. 8.4 141.0 13.4
Apple 5 oz. 2.5 6.5 83.0
98.5 283.2 488.2 869.9
 DIETARY CALCULATION WITH FOOD SERVED IN 100
CALORIES PORTIONS
Portions
Dinner Proteins Fats Carbohydrates Total
in serving
French Soup ½ 10 20 20
Nut Sauce 1 29 55 16
Macaroni, Egg 1 15 59 26
Baked Potato 2 22 2 176
Cream Gravy ½ 5 33 12
Biscuit 1½ 20 2 128
Butter 1 1 99
Honey 2 200
Celery ¼ 4 21
Apple Juice ½ 50
10¼ 106 270 649 1,025

Hourly Outgo in Heat and Energy from the Human Body as

Determined in the Respiration Calorimeter by the
U. S. Dept. of Agriculture
Average (154 lbs) Calories
Man at rest (asleep) 65
Sitting up (awake) 100
Light exercise 170
Moderate exercise 190
Severe exercise 450
Very severe exercise 600

FOOTNOTES:
[14] The following receipts for fruit beverages are adapted from
Practical Diatetics by Alida Frances Pattee, Publisher, Mt. Vernon,
N. Y.
 [15] “Nutrition and Diatetics” by Dr. W. S. Hall, D. Appleton &
Co., New York.
[16] Chemical Composition of American Food Materials,
Atwater and Bryant, U. S. Department of Agricultural Bull. No. 28.
[17] Experiments on Losses in Cooking Meats. (1900-03),
Grindley, U. S. Department of Agricultural Bull. No. 141.
[18] Laboratory number of specimen, as per Experiments on
Losses in Cooking Meat.
 Contents
Abnormal Conditions, diet in, 233-282
Absorption of foods, 78-80
Achlochlorhydria Achlorhydria, 248
Adolescence, diet in, 228
Aged, diet for, 231
Albuminoids, 66, 122
Albuminized drinks, 287-289
Alcohol, 83-84
Alkali, 122
Anaemia, 49, 133, 210, 236
Appendix, 281
Appetite, 95-98
Apples, 125
Apricots, 125
Asparagus, 120

Bananas, 125-126
Barley water, 90, 175, 289
Beans, 120, 163
Beef, 128
Beef Extracts, 133
Beets, 116-118
Beverages, 183, 284-287
Bile, Influence on Digestion, 75
Biscuits, 146
Blackberries, 125
Bouillons, 133, 196
Bran, 141
Bread, 138, 143
Breathing, Effect of, 104-107
Breakfast foods, 150
Bright’s Disease, 270
Brussels Sprouts, 120
Butter, 55, 172
Buttermilk, 172

Carbo-nitrogenous foods, 137-139

Carbon, 20
Carbonaceous foods, 37, 28, 51, 115, 137
Candy, 61
Carrots, 116, 118
Cabbage, 120
Caffein, 184
Cancer of Stomach, 250
Casein, 172
Carbonized drinks, 186
Catarrh of Stomach, 242-250
Catarrh of Intestines, 254
Celery, 120
Cereals, 137-138
Cereals, Cooking of 199, 201
Cereal Coffee, 160
Cerealin, 140
Chemical Comp. of foods, etc., 20-24
Chemical action of plants, 21
Cherries, 125
Chard, 121
Chocolate, 161, 185
Cheese, 172
Citrates, 123, 125
Citrous fruits, 123, 125
Classification of foodstuffs, 37
Classification of foods, 115
Cod Liver Oil, 56
Cotton Seed Oil, 56
Corn, 138, 148
Cornstarch, 115-116
Coffee 161, 184
Coffee Eggnog, 287
Cocoa 161, 185
Condiments, 187
Cooking, 191
Constipation, 253
Cream, 55, 172
Cranberries, 123, 125
Creatin, 132-133
Crackers, 138
Cracked wheat, 155
Crust Coffee, 290
Cucumbers, 120
Currants, 123, 125
Custards (receipts), 295-296

Dates, 125
Dandelions, 121
Dextrin, 57, 153
Dextrose, 63, 74
Diabetes, 264
Diets, 207
Diets, Classification of, 284-292
Diet, vegetable, 219-222
Dilation of Stomach, 248
Digestion, 67-80
Intestines, 74-77
Stomach, 69-71
Drinking at meals, 43
Dyspepsia, 239
Dysentery, 256

Economy in food, 99-101

Eggs, 133
Eggs, ways of serving, 287-288
Egg broth, 287
Egg lemonade, 135, 285
Eggnog, 135, 287
Egg malted milk, 287
Enteritis, 254
Exercise, Effects of, 104-107
Extractives, 132

Farinaceous beverages, 289

Fat, 52, 75-76
Figs, 125
Fish, 129-130
Flours, 138, 141-143
Food Supply, 19
Food Values, 211-218, 299
Fruits, 122
Fruit juices, receipts for, 285-287
Fruit drinks, 286
Frequency of meals, 102-104

Gall stones, 75-76, 263

Gastric juice, 71-74
Gastritis, 242-250
Gelatin, 132
Gelatinoids, 132
Glycogen, 81-82
Gluten, 140-142
Gout, 273
Gooseberries, 123-125
Greens, 121
Green Vegetables, 119-122
Grapes, 125
Grape fruit, 123
Grape juice, 286
Grape nectar, 286
Grape lithia, 286
Grape yolk, 288
Gruels, receipts for, 297

Habit and regularity of eating, 101-102

Heat and energy, 24
Hydrochloric acid, 71, 123, 175
Hydrogen, 20
Hyperchlorhydria, 246
Hypochlorhydria, 247

Indigestion, 239-254
Intestinal disorders, 251-255
Intestines, Work of, 89-91
Intestines, Inflammation of, 256
Iron, 48

Jellies, Receipts for, 295

Junket, Receipts for, 296

Kidneys, 87-93
Kidneys, derangement of, 268
Legumes, 163
Lemons, 123-125
Lemonades, 186, 285-286
Lemon Whey, 286
Lentils, 168
Lettuce, 120
Limes, 123
Limewater, Proportions of, 176
Limewater, How to prepare, 175
Liver, Work of, 81-84, 92
Liver, Derangements of, 258
Lobsters, 129-130
Lungs, 87-93

Macaroni, 147
Magnesium, 122
Malted Milk, 178
Malates, 122
Maltose, 63, 69, 74
Meat, 127, 192, 199
Meat juice, how prepared, 290-291
Meat tea, how prepared, 291
Meat broth, how prepared, 291
Measures and Weights, 281-283, 299
Menus, 223-282
Milk, 170
Milk, clabbered, 180
Milk, skimmed, 161
Milk, condensed, 180
Milk, ways of serving, 287
Milk, tests, 176
Milk, sugar, 180
Milk junket, 186
Mind, Influence of, 110, 113
Mixed diet, 219-222
Mould, 146
Muscles, Work of, 84-86, 93
Mulberries, 125
Mutton, 128

Nerves, work of, 86-87, 93

Nervous disorders, 272
Neurasthenia, 273
Nephritis, 269
Nitrogen, 20
Nitrogenous foods, 38-39, 65-66, 127
Nutrition, 13, 14
Nutri-meal, 143
Nut Oil, 57
Nuts, 169

Oatmeals, 156
Oatmeal water, 175, 290
Oats, 138
Obesity, 278
Olive Oil, 57
Onions, 117
Oranges, 123-125
Orangeade, 286
Oxidation, 26
Oxygen, 20
Oysters, 129-130

Pancreatic trypsin, 259

Pancreatin, 245
Pastry, 201
Pasteurized milk, 177
Parsnips, 119
Peas, 120, 163-165-166
Peaches, 125
Peanuts, 163-164
Pears, 125
Peristalsis, 73, 76
Peptone, 72, 74
Pepsin, 71
Peptonized Milk, 244
Phosphorous, 20, 21
Pineapples, 123, 125
Pineapple juice, 286
Plums, 125
Pork, 128-131
Poultry, 128-131
Potatoes, 116-117
Potassium, 122
Predigested Foods, 153
Preservation of Foods, 189
Proteins, 21, 22, 39
Prunes, 125
Puffed Rice, 158
Puffed Wheat, 159
Purpose of Food, 19

Raisins, 125
Raspberries, 125
Rectal Feeding, 256
Rennin, 71
Rennet, 175
Rheumatism, 275
Rhubarb, 120, 123
Rice, 138, 147
Rice Water, 290
Roots and Tubers, 115-119
Rye, 138

Salt, 47
Sago, 115-116
Saliva, office of, 69, 91
Salivary Digestion, 69, 71