Hadron Physics and Dyson-Schwinger Equations

1.2 MB·English

by  A. Hoell

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HADRON PHYSICS AND
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DYSON-SCHWINGER EQUATIONS
n
a
J

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A.HO¨LL,†C.D.ROBERTS∗†ANDS.V.WRIGHT∗
2
∗Physics Division, Argonne National Laboratory,

1 Argonne IL 60439, USA
v Institut fu¨r Physik, Universit¨at Rostock,
†
1 D-18051 Rostock, Germany
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1 Detailedinvestigations ofthestructureofhadronsareessentialforunderstanding
0 how matter isconstructed fromthequarks andgluons of QCD,andamongst the
6 questions posed to modern hadron physics, three stand out. What is the rigor-
0 ous,quantitativemechanismresponsibleforconfinement? Whatistheconnection
/ between confinement and dynamical chiral symmetry breaking? And are these
h
phenomenatogether sufficienttoexplaintheoriginofmorethan98%ofthemass
t
- oftheobservableuniverse? Suchquestionsmayonlybeansweredusingthefullma-
l
c chinery of nonperturbative relativistic quantum field theory. These lecture notes
u provide an introduction to the application of Dyson-Schwinger equations in this
n context, andaperspectiveonprogresstowardansweringthesekeyquestions.
:
v
i
X
r
a
Table of Contents
Section 1 – Introduction ............................................. 2
Section 2 – Dyson-Schwinger Equation Primer .................. 10
2.1 - Photon Vacuum Polarisation ....................... 10
2.2 - Fermion Gap Equation .............................. 17
Section 3 – Hadron Physics ........................................ 25
3.1 - Aspects of QCD ...................................... 29
3.2 - Emergent Phenomena................................ 31
Section 4 – Nonperturbative Tool in the Continuum ............ 34
4.1 - Dynamical Mass Generation ........................ 36
4.2 - Dynamical Mass and Confinement ................. 39
Section 5 – Meson Properties ...................................... 43
5.1 - Coloured Two- and Three-point Functions ........ 43
5.2 - Colour-singlet Bound States ........................ 45
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Section 6 – Baryon Properties ..................................... 64
6.1 - Faddeev Equation .................................... 66
6.2 - Nucleon and ∆ Masses .............................. 74
6.3 - Nucleon Electromagnetic Form Factors ........... 79
Section 7 – Epilogue ................................................ 91
Appendix A – Euclidean Space .....................................94
References ..............................................97
1. Introduction
A theoretical understanding of the phenomena of Hadron Physics requires
the use of the full machinery of relativistic quantum field theory, which is
based on the relativistic quantum mechanics of Dirac, and is currently the
favoured way to reconcile quantum mechanics with special relativity.a
It is noteworthy that the unification of special relativity (viz., the re-
quirement that the equations of physics be Poincar´e covariant) and quan-
tummechanicstookquitesometime. Indeed,questionsstillremainastoa
practical implementation of an Hamiltonian formulation of the relativistic
quantum mechanics of interacting systems. The Poincar´e group has ten
generators: six associatedwith the Lorentz transformations (rotations and
boosts); and four associated with translations. Quantum mechanics de-
scribes the time evolutionof a system with interactions and that evolution
is generated by the Hamiltonian, or some generalisation thereof. However,
the Hamiltonian is one of the generators of the Poincar´e group, and it is
apparent from the Poincar´e algebra that boosts do not commute with the
Hamiltonian. Hence the state vector calculated in one momentum frame
will not be kinematically related to the state in another frame, a fact that
makes a new calculation necessaryin every momentum frame. The discus-
sionofscattering,whichtakesastateofmomentumptoanotherstatewith
momentum p, is therefore problematic.2,3
′
Moreover, relativistic quantum mechanics predicts the existence of an-
tiparticles; i.e., the equations ofrelativisticquantum mechanicsadmitneg-
ative energy solutions. However, once one allows for negative energy, then
particle number conservation is lost:
E =E +(E +E )+... ad infinitum, (1.1)
system system p1 p¯1
aInthefollowingweassumethatthereaderisfamiliarwiththenotationandconventions
ofrelativisticquantummechanics. Forthoseforwhomthatisnotthecasewerecommend
Ref.[1],inparticularChaps.1-6.
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where Ek¯ =−Ek. This poses a fundamental problem for relativistic quan-
tum mechanics: few particle systems can be studied, but the study of
(infinitely) many bodies is difficult and no general theory currently exists.
Relativisticquantumfieldtheoryprovidesawayforward. Inthisframe-
work the fundamental entities are fields, which can simultaneously repre-
sent infinitely many particles. The neutral scalar field, φ(x), provides an
example. One may write
d3k 1
φ(x)= a(k)e ikx+a (k)eikx , (1.2)
− · † ·
(2π)32ω
Z k
(cid:2) (cid:3)
where: ω = ~k 2+m2 istherelativisticdispersionrelationforamassive
k
| |
particle; the qfour-vector (kµ) = (ω ,~k); a(k) is an annihilation (creation)
k
operatorforaparticle(antiparticle)withfour-momentumk( k);anda (k)
†
−
is a creation (annihilation) operator for a particle (antiparticle) with four-
momentum k ( k). With this plane-wave expansion of the field one may
−
proceed to develop a framework in which the nonconservation of particle
numberisnotaproblem. Thatiscrucialbecausekeyobservablephenomena
in hadron physics are essentially connected with the existence of virtual
particles.
Relativistic quantum field theory has its own problems, however. For
example, the question of whether a given quantum field theory is rigor-
ously well defined is an unsolved mathematical problem. All relativistic
quantum field theories admit analysis via perturbation theory, and per-
turbative renormalisation is a well-defined procedure that has long been
usedinQuantumElectrodynamics(QED)andQuantumChromodynamics
(QCD). However, a rigorous definition of a theory means proving that the
theorymakessensenonperturbatively. Thisisequivalenttoprovingthatall
thetheory’srenormalisationconstantsarenonperturbativelywell-behaved.
An understanding of the properties of hadrons; viz., Hadron Physics,
involvesQCD.Thistheorymakesexcellentsenseperturbatively,asdemon-
strated in the Nobel Prize winning work on asymptotic freedom by Gross,
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PolitzerandWilczek. However,QCDisnotknowntobearigorouslywell-
defined theory and hence it cannot yet truly be described as the theory of
the strong interaction.
Nevertheless, the development of an understanding of observable phe-
nomenacannotwaitonmathematics. Assumptionsmustbemadeandtheir
consequencesexplored. Practitionersthereforeassumethat QCDis (some-
how) well-defined and follow where it may lead. In experiment that means
exploringandmapping the hadronphysicslandscape withwell-understood
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probes, such as the electron at JLab; while in theory one employs estab-
lished mathematical tools, and refines and invents others in order to use
the Lagrangian of QCD to predict what should be observable real-world
phenomena.
A primary aim of the world’s current hadron physics programmes in
experimentandtheoryistodeterminewhetherthereareanycontradictions
withwhatwecanactuallyprove inQCD.Hitherto,therearenonethatare
uncontroversial.b Inthisfieldtheinterplaybetweenexperimentandtheory
is the engine of discoveryandprogress,and the discovery potential of both
is high. Much has been learnt in the last five years and one can safely
expect that many surprises remain in Hadron Physics.
QCD is a local gauge theory, and such theories are the keystone of
contemporary hadron and high-energy physics. They are difficult to quan-
tise because one must deal with the gauge dependence, which is an extra
non-dynamical degree of freedom. The modern approach is to quantise
thesetheoriesusingthe methodoffunctionalintegrals,andRefs.[7,8]pro-
vide excellent descriptions. The method of functional integration replaces
canonical second-quantisation. One may view this approach as originating
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in the path integral formulation of quantum mechanics. NB. In general,
mathematicians do not regard local gauge theory functional integrals as
well-defined.
In quantum field theory all physical amplitudes can be obtained from
Green functions, which are expectation values of time-ordered products
of fields measured with respect to the physical vacuum.c They describe
all the characteristics of an interacting system. The Green functions are
obtainedfromgeneratingfunctionals,thespecificationofwhichbeginswith
thetheory’sactionexpressedintermsofthePoincar´einvariantLagrangian
density.
Ananalysisofthegeneratingfunctionalforinteractingbosonsproceeds
almost classically. The field variables and functional derivatives can be
treated as “c-numbers”, and a perturbative truncation of any Green func-
tioncanbeobtainedinastraightforwardmanner. Ameasureofclarityand
rigour may be introduced by interpreting spacetime as a discrete lattice of
points and recovering the continuum via a limiting procedure. On these
aspects, Appendix B of Ref.[8] is instructive. Following this route it is
bThepion’svalence-quarkdistributionisonesuchcontentious example.5,6
cThephysicalorinteractingvacuumistheanalogueofthetruegroundstateinquantum
mechanics.
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plain that in perturbation theory the vacuum is trivial; viz., features such
as dynamical symmetry breaking are impossible.
A complication is encountered in dealing with fermions; namely,
fermionic fields do not have a classical analogue because classical physics
does not contain anticommuting field variables. In order to treat fermions
usingfunctionalintegralsonemustemployGrassmannvariables. Reference
[10]is thestandardsourceforarigorousdiscussionofGrassmannalgebras,
and Appendix B of Ref.[8] is again instructive in this connection.
Inordertoillustrate someofthe conceptsdescribedabovewewillwork
throughanexample: thecaseofanoninteractingDiracquantumfield. The
Lagrangiandensity for the free Dirac field is
Lψ(x)=ψ¯(x)(i∂/ m)ψ(x). (1.3)
0 −
Consider therefore the functional integral
i d4xψ¯(x) i∂/ m+iη+ ψ(x)
W[Ξ¯,Ξ]= [Dψ¯(x)][Dψ(x)]e −
Z
(cid:0) (cid:1)
Z
i d4x ψ¯(x)Ξ(x)+Ξ¯(x)ψ(x)
e , (1.4)
Z
× (cid:0) (cid:1)
where η 0+ as the last step in any calculation.d This is the generating
→
functionalforcompleten-pointGreenfunctionsinthequantumfieldtheory.
Here“complete”meansthatthen-pointGreenfunction,G(x ,x ,...,x ),
1 2 n
will include contributions from products of lower-order Green functions
(m ,m ,etc.<n); i.e., disconnected diagrams. In Eq.(1.4), ψ¯(x), ψ(x) are
i j
identified with the generators of G, a Grassmann algebra with involution:
thelattermeansthataninner-productofsortsisdefined. Thereisaminor
additionalcomplicationhere–the spinordegree-of-freedomisimplicit; i.e.,
to be explicit, one should write
4 4
[Dψ¯ (x)] [ψ (x)]. (1.5)
r s
r=1 s=1
Y Y
However, that only adds a finite matrix degree-of-freedom to the problem,
whichmayeasilybehandled. InEq.(1.4)wehaveintroducedanticommut-
ing sources: Ξ¯(x), Ξ(x), which are also elements in G.
dη is a convergence factor, which is necessary to define the integral. It subsequently
appears inpropagators and thereby implements Feynman boundary conditions, as dis-
cussed,e.g.,inRef.[8],App.B.
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To evaluate the free-field functional integral, which is Gaussian, one
writes
O(x,y)=(i∂/ m+iη+)δ4(x y) (1.6)
− −
and observes that the solution of
d4wO(x,w)P(w,y) =I δ4(x y); (1.7)
D
−
Z
i.e., the inverse of the operator O(x,y), is precisely the free-fermion prop-
agator:
d4p
P(x,y)=S (x y)= e ip(x y)S (p), (1.8)
0 − (2π)4 − · − 0
Z
with
p/ +m
S (p)= . (1.9)
0 p2 m2+iη+
−
NB. This can be verified by substitution, using γ ,γ = 2g I . It is
µ ν µν D
{ }
true in general that in the absence of external sources, n-point functions
are translationally invariant.
One can now rewrite Eq. (1.4) in the form
i d4xd4yψ¯(x)O(x,y)ψ (y)
′ ′
W[Ξ¯,Ξ]= [Dψ¯(x)][Dψ(x)]e
Z
Z
i d4xd4yΞ¯(x)S (x y)Ξ(y)
0
e− − , (1.10)
Z
×
wherein
ψ¯(x) := ψ¯(x)+ d4wΞ¯(w)S (w x),
′ 0 − (1.11)
ψ (x) := ψ(x)+ d4wS (x w)Ξ(w).
′ R 0 −
The new fields ψ¯(x) and ψ (x) areRstill in G, and are related to the orig-
′ ′
inal variables by a unitary transformation. Thus the change of variables
introduces only a unit Jacobian and hence
i d4xd4yΞ¯(x)S (x y)Ξ(y)
0
W[Ξ¯,Ξ]=e− −
Z
i d4xd4yψ¯(x)O(x,y)ψ (y)
′ ′
[Dψ¯(x)][Dψ (x)]e . (1.12)
′ ′ Z
×
Z
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The expression in the second line of this equation is a standard functional
integral:
i d4xd4yψ¯′(x)O(x,y)ψ′(y)
e =Det[O(x y)], (1.13)
Z
−
where “Det” is a generalisation of the concept of a matrix determinant.
One therefore arrives at
1 i d4xd4yΞ¯(x)S0(x y)Ξ(y)
W[Ξ¯,Ξ]= e− − , (1.14)
Nψ Z
0
where
Nψ :=Det[iS (x y)]. (1.15)
0 0 −
Clearly, with the definition in Eq.(1.4), Nψ W[Ξ¯,Ξ] = 1. This is
0 Ξ¯=0=Ξ
not a convenient normalisation and it is therefore customary to redefine
(cid:12)
W[Ξ¯,Ξ] so that Nψ is included in the measure “[Dψ¯((cid:12)x)][Dψ (x)]” and
0 ′ ′
W[Ξ¯,Ξ] =1. (1.16)
Ξ¯=0=Ξ
Thetwo-pointGreenfunctionfo(cid:12)rthefree-fermionquantumfieldtheory
(cid:12)
is now easily obtained:
δ2W[Ξ¯,Ξ] 0T ψˆ(x)ψˆ¯(y) 0
=: h | { }| i
iδΞ¯(x)( i)δΞ(y) 00
− (cid:12)Ξ¯=0=Ξ h | i
(cid:12)
(cid:12)(cid:12) i d4xψ¯(x) i∂/ m+iη+ ψ(x)
= [Dψ¯(x)][Dψ(x)]ψ(x)ψ¯(y)e − (.1.17)
Z
(cid:0) (cid:1)
Z
The functional differentiation in Eq.(1.14) is straightforwardand yields
δ2W[Ξ¯,Ξ]
=iS (x y); (1.18)
iδΞ¯(x)( i)δΞ(y) 0 −
− (cid:12)Ξ¯=0=Ξ
(cid:12)
i.e., the inverse of the Dirac operato(cid:12)r.
(cid:12)
It is useful to have systematic procedure for the a priori elimination of
disconnected parts from n-point Green functions because the recalculation
ofm<n-pointGreenfunctionsateverystageisinefficient. Thegenerating
functional for “connected” n-pointGreen functions, Z[Ξ¯,Ξ], is defined via:
W[Ξ¯,Ξ]=:exp iZ[Ξ¯,Ξ] . (1.19)
It follows immediately from Eq.(1.14) t(cid:8)hat (cid:9)
Z[Ξ¯,Ξ]= d4xd4y Ξ¯(x)S (x y)Ξ(y). (1.20)
0
− −
Z
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This equation states that for a noninteracting field, there is one, and only
one,connectedGreenfunction; namely, the free particlepropagator,which
is the simplest possible Green function. NB. It is a property of theories
based on a Grassmann algebra with involution that one-point Green func-
tions for fermions are identically zero in the absence of external sources.
At this point it is useful to illustrate what is meant by the functional
determinantintroducedabove;i.e.,Det[O],whereOisanintegraloperator.
Thisisrelevantowingtotheimportanceofthisfermiondeterminantandits
absence,e.g.,innumericalsimulationsoflattice-regularisedQCD.Consider
a translationally invariant operator
d4p
O(x,y)=O(x y)= O(p)e ip(x y). (1.21)
− · −
− (2π)4
Z
Then, for any function f that may be expressed as a power series on some
domain:
∞
f(x)= f xi, (1.22)
i
i=0
X
we have
d4p
f[O(x y)]= f +f O(p)+f O(p)2+[...] e ip(x y)
− (2π)4 0 1 2 − · −
Z
d4p (cid:8) (cid:9)
= f(O(p))e ip(x y). (1.23)
− · −
(2π)4
Z
This expression may be applied directly to Nψ = Det[iS (x y)]. To
0 0 −
that end we proceed by noting
p/ 1
S (p)=m∆ (p2) 1+ , ∆ (p2)= ,(1.24)
0 0 m 0 p2 m2+iη+
(cid:20) (cid:21) −
S (x y)= d4wm∆ (x w)F(w y), (1.25)
0 0
⇒ − − −
Z
with ∆ (x y) the obvious Fourier transform of ∆ (p2), and
0 0
−
d4p p/
F(x y)= 1+ e−ip·(x−y). (1.26)
− (2π)4 m
Z (cid:20) (cid:21)
We now remark that for a bilocal operator P(x,y)
TrP := d4xtrP(x,x), (1.27)
Z
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where “tr” indicates a trace over whatever matrix structure is present.
Thus Eq.(1.22) entails that TrLn[O] = LnDet[O], in addition to the more
obvious result Ln[AB]=Ln[A]+Ln[B]. Hence
TrLniS (x y)=Tr Lnim∆ (x y)+Ln δ4(x y)+F(x y) .
0 0
− − − −
(1.28)
(cid:8) (cid:2) (cid:3)(cid:9)
Applying Eq.(1.23) to the second term above one obtains
d4p
TrLn δ4(x y)+F(x y) = d4x trln[1+F(p)]
− − (2π)4
Z Z
(cid:2) (cid:3) d4p p2
= d4x 2 ln 1 ,(1.29)
(2π)4 − m2
Z Z (cid:20) (cid:21)
while the first term givese
TrLnim∆ (x y)= d4x d4p 2 ln im∆ (p2) 2 . (1.30)
0 − (2π)4 0
Z Z
(cid:2) (cid:3)
Combining this result with Eq.(1.29) yields
d4p
LnNψ =TrLniS (x y)= d4x 2 ln∆ (p2), (1.31)
0 0 − (2π)4 0
Z Z
wherethefactor“2”reflectsthespin-degeneracyofthefree-fermion’seigen-
values. NB.Upontheinclusionofa“colour”degree-of-freedom,asinQCD,
this would become “2 N ,” where N is the number of colours.
c c
The interpretation of Eq.(1.31) is straightforward. As is the case for
finite dimensional matrices; viz.,
lndetM =ln λM = lnλM, (1.32)
i i
i i
Y X
the logarithm of the determinant of an operator is simply the logarithm of
the productoftheoperator’seigenvalues,whichisequivalenttothe sumof
the logarithms of these eigenvalues. In our particular instance, there is a
continuumofeigenvaluesfortheinverseofthefreeDiracoperator. Foreach
value ofthree-momentump~, we havetwospins anda positive andnegative
energy solution. The product of these four eigenvalues is described by the
function ∆ (p2)2. Finally, the integral over momentum in Eq.(1.31) is the
0
analogue of the sum in Eq.(1.32). This picture generalises to the case of
more complicated Dirac operators.
eIn both cases the multiplicative factor d4x simplymeasures the (infinite) volume of
spacetime. Thefactorposesnoproblemsinaproperlyregularisedtheory.
R
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Thusfarwehavemadelittlementionofgauge-bosonfields;namely,pho-
tons, gluons, etc. A generating functional for gauge field n-point functions
canbeconstructed. Theprimarydifficultyinthisinstanceistheproblemof
gauge fixing, which is not yet fully resolved. The so-called Faddeev-Popov
determinantisonepartofthesolution. Thatdeterminantcanbeexpressed
throughthe introductionofdynamicalghostfields. Wewillnotwritemore
on the issue herein.f The omission is not crucial for our development. At
this point,the basic qualitativeideas ofthe functionalintegralformulation
of relativistic quantum field theory have been presented.
2. Dyson-Schwinger Equation Primer
Ithaslongbeenknownthatfromthefieldequationsofquantumfieldtheory
one can derive a system of coupled integral equations interrelating all of
a theory’s Green functions 12,13. This collection of a countable infinity
of equations is called the complex of Dyson-Schwinger equations (DSEs).
It is an intrinsically nonperturbative structure, which is vitally important
in proving the renormalisability of quantum field theories. Moreover, at
its simplest level the complex provides a generating tool for perturbation
theory. Inthecontextofquantumelectrodynamics(QED)wewillillustrate
anonperturbativederivationoftwoDSEs. The derivationofothersfollows
the same pattern.
2.1. Photon Vacuum Polarisation
The vacuum polarisation is an essentially quantum field theoretical effect
andanimportantpartoftheLambshift. Itmaybederivedfromtheaction
for QED with N flavours of electromagnetically active fermions:
f
Nf
S[A ,ψ,ψ¯]= d4x ψ¯f(x) i ∂ mf +ef A ψf(x)
µ  6 − 0 0 6
Z fX=1 (cid:16) (cid:17)
1  1
F (x)Fµν(x) ∂µA (x)∂νA (x) . (2.1)
µν µ ν
−4 − 2ξ
0 (cid:21)
The action is manifestly Poincar´e covariant. ψ¯f(x), ψf(x) are elements of
a Grassmann algebra with involution that describe the fermion degrees of
fA pedagogical introduction is provided in Appendix B of Ref.[8] and a contemporary
perspectivemaybetracedfromRef.[11]andreferencestherein.