Dyson schwinger equations: Connecting small and large length-scales

8 Pages·0.12 MB·English

by  Craig Roberts

DYSON SCHWINGER EQUATIONS:
CONNECTING SMALL AND LARGE LENGTH-SCALES
CRAIG ROBERTS
Physics Division, 203, Argonne National Laboratory, Argonne IL 60439, USA
I illustrate the phenomenologicalapplication of Dyson-Schwinger equations to the
9 calculationof mesonproperties observable at TJNAF. Particularemphasis is given
9 to this framework’s ability to unify long-range effects constrained by chiral sym-
9 metry with short-range effects prescribed by perturbation theory, and interpolate
1
between them.

n PACS numbers: 12.38.Lg, 13.40.Gp, 14.40.Aq, 24.85.+p
a
J Keywords: Dyson-SchwingerEquations,Nonperturbativemethods,ContinuumQCD,
Confinement,Dynamicalchiralsymmetrybreaking,Pionelectromagneticandanoma-
7
2 lous transition form factors

1
1. Dressed Quarks
v
1
The Dyson-Schwingerequations(DSEs)[1]provideanonperturbativeapproach
9
0 tostudyingthecontinuumformulationofQCD,makingaccessiblephenomenasuch
1 as confinement, dynamical chiral symmetry breaking (DCSB) and bound state
0 structure. However, they also provide a generating tool for perturbation theory
9
andhence their phenomenologicalapplicationis tightly constrainedathigh-energy.
9
This is the particular feature of the phenomenological application of DSEs: their
/
h abilityto furnishaunifieddescriptionofhigh-andlow-energyphenomenainQCD.
t
- It is elucidated in Refs. 2,3 and here I only illustrate this aspect, using as primary
l ∗
c exemplars the electromagnetic pion form factor and the γ π → γ transition form
u factor, which are particularly relevant to the TJNAF community.
n
A key element in the description of hadronic observables is the dressing of the
:
v quark propagator,which is described by the quark DSE:
i
X S (p)−1 := iγ·pA (p2)+B (p2)=A (p2) iγ·p+M (p2) (1)
f f f f f
r
a Λ (cid:0) λa (cid:1)
= Z (iγ·p+mbm)+ Z g2D (p−q) γ S (q)Γfa(q,p).(2)
2 f 1 Z µν 2 µ f ν
q
Here f is a flavour label, D (k) is the dressed-gluon propagator [4,5], Γfa(q,p) is
µν ν
the dressed-quark-gluon vertex, mbm is the Λ-dependent current-quark bare mass
f
and Λ := Λd4q/(2π)4 represents mnemonically a translationally-invariant regu-
q
larisaRtion oRf the integral, with Λ the regularisation mass-scale. The quark-gluon-
vertex and quark wave function renormalisation constants, Z and Z , depend on
1 2
the renormalisationpoint, ζ, and the regularisationmass-scale.
The qualitative features of the solution of Eq. (2) are known. In QCD the
chiral limit is defined by mˆ =0, where mˆ is the renormalisation-point-independent
current-quarkmass. Formˆ =0 there is no mass-likedivergenceinthe perturbative
1
101
100
V)
Ge10−1
2p) ( b−quark
M( c−quark
10−2 s−quark
u,d−quark
chiral limit
M2(p2) = p2
10−3
10−2 10−1 100 101 102
p2 (GeV2)
Figure 1: Quark mass function obtained as a solution of Eq. (2) using D (k) and
µν
Γfa(q,p) from Ref. 6, and mζ = 3.7MeV, mζ = 82MeV, mζ = 0.58GeV and
ν u,d s c
mζ = 3.8GeV (ζ = 19GeV). The indicated solutions of M2(p2) = p2 define the
b
Euclidean constituent-quark mass, ME, which takes the values: ME =0.56GeV,
f u,d
ME =0.70GeV, ME =1.3GeV, ME =4.6GeV.
s c b
evaluation of the quark self energy and hence for p2 > 20GeV2 the solution of
Eq. (2) is [6]
M (p2)larg=e−p2 2π2γm −hq¯qi0 , (3)
0 3 (cid:0) (cid:1) 1−γm
p2 1ln p2/Λ2
2 QCD
(cid:16) h i(cid:17)
where γ = 12/(33−2N ) is the gauge-independent mass anomalous dimension
m f
andhq¯qi0 is the renormalisation-point-independentvacuum quarkcondensate. The
momentum-dependence is a model-independent result. The existence of DCSB
meansthathq¯qi0 6=0,however,itsactualvaluedependsonthelong-rangebehaviour
ofD (k)andΓ0a(q,p),whichismodelledincontemporaryDSEstudies. Requiring
µν ν
a good description of light-meson observables necessitates hq¯qi0 ≈−(0.24GeV)3.
In contrast, for mˆ 6=0,
f
M (p2)larg=e−p2 mˆf . (4)
f γm
1ln p2/Λ2
2 QCD
(cid:16) h i(cid:17)
Anobviousqualitativedifferenceisthat,relativetoEq.(4),thechiral-limitsolution
is 1/p2-suppressed in the ultraviolet.
There is some quantitative model-dependence in the p2-evolution of M (p2)
f
into the infrared. However,for any forms of D and Γfa that provide an accurate
µν ν
description of f and m , one obtains profiles like those illustrated in Fig. 1.
π,K π,K
2
Theevolutiontocoincidencebetweenthechiral-limitandu,d-quarkmassfunctions,
apparent in this figure, makes clear the transition from the perturbative to the
nonperturbative domain. The chiral limit mass-function is nonzero only because
of the nonperturbative DCSB mechanism whereas the u,d-quark mass function
is purely perturbative at p2 > 20GeV2, where Eq. (4) is accurate. The DCSB
mechanismthus has asignificanteffect onthe propagationcharacteristicsofu,d,s-
quarks,andthisisfundamentallyimportantinQCDwithobservableconsequences.
2. Bound States
A meson is a bound state of a dressed-quark and -antiquark, and its internal
structure is described by a Bethe-Salpeter amplitude obtained as the solution of
Λ
[Γ (k;Q)] = [χ (q;Q)] Krs(q,k;Q), (5)
P tu Z P sr tu
q
where χP(q;Q) = S(q+)ΓP(q;Q)S(q−); S(q) = diag(Su(q),Sd(q),Ss(q),...); q+ =
q+ηQQ, q− =q−(1−ηQ)Q, with Q the totalmomentum of the bound state and
η ∈ [0,1] the relative-momentum partitioning parameter; and r,...,u represent
Q
colour-, Dirac- and flavour-matrix indices. For a pseudoscalar meson, such as the
pion, the solution has the general form
Γ (k;Q) = TPγ iE (k;Q)+γ·QF (k;Q) (6)
P 5 P P
(cid:20)
+γ·kk·QG (k;Q)+σ k Q H (k;Q) ,
P µν µ ν P
(cid:21)
where TP is a flavour matrix identifying the meson; e.g., Tπ+ = 1 λ1+iλ2 , with
2
{λj,j =1...Nf2−1} the Gell-Mann matrices of SU(Nf). (cid:0) (cid:1)
In Eq. (5), K is the renormalised, fully-amputated, quark-antiquark scattering
kernel. Important in the successful application of DSEs is that K has a systematic
skeletonexpansionintermsofthe elementarydressed-particleSchwingerfunctions;
e.g.,the dressed-quarkand-gluonpropagators. TheexpansionintroducedinRef.7
providesameansofconstructingakernelthat,order-by-orderinthenumberofver-
tices,ensuresthepreservationofvectorandaxial-vectorWard-Takahashiidentities;
i.e., current conservation. Only with such a truncation is an accurate description
of the light-quark mesons possible.
In QCD the leptonic decay constant of a pseudoscalar meson is [6]
Λ
f Q =trZ (TP)Tγ γ χ (k;Q), (7)
P µ 2 5 µ P
Z
k
where the traceis overcolour,Dirac and flavourindices. Equation(7)is exact: the
Λ-dependenceofZ ensuresthattheright-hand-side(r.h.s.) isfiniteasΛ→∞,and
2
its ζ- and gauge-dependence is just that necessaryto compensate that of χ (k;Q).
P
3
In the chiral limit the axial-vector current is conserved, and employing any
Ward-Takahashi identity preserving truncation of K one obtains
f E (k;0)=B (k2), F (k;0)+2f F (k;0)=A (k2),
GP (kP;0)+2f G0 (k;0)=2A′(k2), HR(k;0)+2fP HP (k;0)=00, (8)
R P P 0 R P P
where F , G and H are calculable functions in the dressed axial-vector ver-
R R R
tex, ΓH(k;Q). These identities are associated with Goldstone’s theorem and in
5µ
fact one can show [6] that when chiral symmetry is dynamically broken: 1) the
flavour-nonsinglet,pseudoscalarBSEhasamasslesssolution;2)theBethe-Salpeter
amplitude for the massless bound state has a term proportional to γ alone, with
5
the momentum-dependence of E (k;0) completely determined by that of B (k2),
P 0
in addition to terms proportional to other pseudoscalar Dirac structures that are
nonzero; and 3) ΓP (k;Q) is dominated by the pseudoscalar bound state pole for
5µ
Q2 ≃ 0. The converse is also true. Hence, in the chiral limit, the pion is a mass-
less composite of a quark and an antiquark, each of which has an effective mass
ME ∼0.5GeV.
Fornonzerovaluesofthe current-quarkmass,whethersmallorlarge,insteadof
Eqs. (8) one obtains [6]
f2 m2 = −M hq¯qiP, (9)
P P P ζ
where M =tr M TP, TP T e.g., for the π: M =mζ +mζ, and
P f (ζ) π+ u d
h n (cid:0) (cid:1) oi
Λ
−hq¯qiP = f trZ TP Tγ χ (q;Q). (10)
ζ P 4Z 5 P
q (cid:0) (cid:1)
Equation(9)isanexactmassformulaforflavournon-singletmesonsandInotethat
ther.h.s. doesnotinvolveadifferenceofmassivequarkpropagators: aphenomeno-
logicalassumptionoftenemployed. hq¯qiP inEq.(10)isan“in-hadron”condensate.
ζ
It is gauge-independent and its renormalisation point dependence is exactly that
required to ensure that the r.h.s. of Eq. (9) is renormalisation point independent.
Forsmallcurrent-quarkmasses,mˆ ∼0,Eq.(9)yieldswhatiscommonlycalled
q
the Gell-Mann–Oakes–Rennerrelation; i.e., m2 ∝mˆ , because
P q
limhq¯qiP =hq¯qi0. (11)
mˆ→0 ζ ζ
However, it also has an important corollary when the current-mass, mˆ , of one or
Q
both constituents becomes large, predicting [8]
m ∝mˆ , (12)
P Q
whichfollowsbecausehq¯qiP ismˆ -independentforlarge-mˆ andf ∝m−1/2. The
ζ Q Q P P
transition from the quadratic to the linear mass-relation occurs at mˆ ≈ 2mˆ [9],
q s
at which point explicit chiral symmetry breaking overwhelms DCSB.
4
Twootherimportantmodel-independentresultscanbeobtained[6]fromEq.(5)
and the systematic construction of K. The scalar functions in Eq. (6) depend on
threeinvariants;e.g.,E (k;Q)=E (k2,k·Q,Q2). ThezerothChebyshevmoment
P P
of these functions; e.g.,
2 π
0E (k2,Q2):= dx 1−x2E (k;Q), k·Q:=x k2Q2 (13)
P P
π Z
0 p p
are dominant in the description of bound state properties, and
0E (k2,Q2)larg∝e−k2 M (k2), (14)
P 0
with0F (k2,Q2),k20G (k2,Q2)andk20H (k2,Q2)behavinginpreciselythesame
P P P
way. Further
k20G (k2,Q2)larg=e−k2 20F (k2,Q2). (15)
P P
These results determine the asymptotic form of the electromagnetic pion form fac-
tor.
3. Electromagnetic Pion Form Factor
The impulse approximation to the electromagnetic pion form factor is [10]
(p +p ) F (q2):=Λ (p ,p ) (16)
1 2 µ π µ 1 2
Λ
=2NctrDZ Γ¯π(k;−p2)S(k++)iΓγµ(k++,k+−)S(k+−)Γπ(k0−;p1)S(k−−),
k
where Γ¯ (q;−P)T := C−1Γ (−q;−P)C with C = γ γ , the charge conjugation
π π 2 4
matrix, and k := k+αp /2+βq/2, p := p +q. No renormalisation constants
αβ 1 2 1
appear explicitly in Eq. (16) because the renormalised dressed-quark-photon ver-
tex, Γγ, satisfies the vector Ward-Takahashi identity. This also ensures current
µ
conservation: (p −p ) Λ (p ,p )=0.
1 2 µ µ 1 2
The calculation of F (q2) is simplified by using a model algebraic parametri-
π
sation of the dressed-quark propagator that efficiently characterises the essential
elements of the solution of the quark-DSE and determines the pion Bethe-Salpeter
amplitude via Eqs. (8) and (15), and an efficacious Ansatz [11] for Γγ in which the
µ
vertexis completely determined bythe dressed-quarkpropagator. The resultis de-
picted in Fig. 2. The current uncertainty in the experimental data at intermediate
q2 is apparentin the lowerpanel,as is the difference betweenthe results calculated
with or without the pseudovector components: F, G, of the pion Bethe-Salpeter
amplitude. ThesecomponentsprovidethedominantcontributiontoF (q2)atlarge
π
pion energy [10] because of the multiplicative factors: γ ·Q and γ ·kk·Q, which
contribute an additional power of q2 in the numerator of those terms involving F2,
FG and G2 relative to those proportionalto E. Including them one finds
F (q2)larg∝e−q2 α(q2) (−hq¯qi0q2)2 ; (17)
π q2 f4
π
5
1.0
0.8
0.6
2q)
F(π
0.4
0.2
0.0
0.0 0.2 0.4 0.6
q2 (GeV2)
0.40
0.30
2V)
e
G
2q) ( 0.20
F(π
2q
0.10
0.00
0.0 5.0 10.0 15.0
q2 (GeV2)
Figure2: Upperpanel: calculatedpionformfactorcomparedwithdataatsmall-q2.
Lower panel: the large-q2 comparison, with the two solid lines showing the range
of model-dependent uncertainty. In both panels the dashed line [13] assumes that
F =0=G =H , and the data are taken from Refs. 12.
π π π
i.e., q2F (q2) ≈ const., up to calculable lnq2-corrections, in agreement with the
π
expectations raised by perturbative QCD. If the pseudovector components of Γ
π
are neglected, the additional numerator factor of q2 is missing and one obtains [13]
q4F (q2)≈const..
π
4. γ∗π → γ Transition Form Factor
The impulse approximation to this form factor is
1
T3 (k ,k ):= iε k k Tˆ(k2,k ·k ,k2) (18)
µν 1 2 4π2 µνρσ 1ρ 2ρ 1 1 2 2
Λ
=tr S(q )Γ (qˆ;−P)S(q )iQΓγ(q ,q )S(q )iQΓγ(q ,q ),
Z 1 π 2 µ 2 12 12 ν 12 1
q
6
where Q = (1I +τ3)/2 = diag(2/3,−1/3), and k , k are the photon momenta
3 1 2
[on-shell: k2 = 0 = k2, 2k ·k = P2], q is the loop-momentum, and q := q−k ,
1 2 1 2 1 1
q :=q+k , qˆ:= 1(q +q ), q :=q−k +k . The manner in whichthe “triangle
2 2 2 1 2 12 1 2
anomaly” is recovered with no dependence on model parameters is described in
Ref. [10], and this same mechanism applies to all anomalous pion and photopion
processes [14]. It is a unique feature of the DSE framework.
Usingthisexpressiononecancalculate[15,16]Tˆ(k2,k ·k ,k2)whenoneorboth
1 1 2 2
ofthephotonsisoffshellandalsodeterminetheasymptoticbehaviouranalytically.
The formal character of that derivation is presented in Ref. 16 but, in neglecting
essential aspects of renormalisation, it is imprecise. I remedy that here for the
illustrative case of one photon off-shell: k2 =Q2.
1
At large-Q2, k ·k ≈ −Q2/2, and in Eq. (18) the pion Bethe-Salpeter ampli-
1 2
tude focuses the integration support at qˆ = 0. As a consequence the asymptotic
behaviour of the integral can be determined using
1 1
S(q )≈ iγ·(k −k ), Γγ ≈Z γ , (19)
12 Z Q2 1 2 µ 1 µ
2
which follow from Eq. (2) and the DSE for the quark-photonvertex, so that
1 Λ
T3 (k ,k )≈i (k −k ) trZ 1τ3χ (qˆ;−P)γ γ γ , (20)
µν 1 2 Q2 1 2 σ 2Z 6 π µ σ ν
q
where I have used the Ward identity: Z =Z . Hence, the transition form factor
1 2
1
iε k k T(Q2):=T3 (k ,k )+T3 (k ,k )
4π2 µνρσ 1ρ 2σ µν 1 2 νµ 2 1
1 Λ
≈ − iε (k −k ) 2trZ 1τ3χ (qˆ;−P)γ γ (21)
Q2 µνρσ 1 2 ρ 3 2Z 2 π 5 σ
q
4 f
π
= iε k k , (22)
µνρσ 1ρ 2σ 3 Q2
where the last line follows from Eq. (7); i.e.,
T(Q2)larg≈e−Q2 4 4π2fπ , (23)
3 Q2
in agreement with the expectations raised by perturbative QCD. Thus, as with
Eq. (16), one equation unifies the small- and large-Q2 results and predicts the
evolution between them [15-17].
5. Epilogue
I have been necessarily brief. There are many other applications of interest
to this community, among them the diffractive electroproduction of neutral vector
7
mesons [18], the electromagnetic form factors of their charged states [19] and the
unificationoflight-andheavy-mesonobservables[8]. Themostpressingcontempo-
rary challenge relevant to this community is the extension of the framework to the
calculation of baryon observables, which is underway.
Acknowledgments
I would like to thank Dubravko Klabucar and the organisers for their assistance,
kindness and hospitality. This work was supported by the US Department of En-
ergy,Nuclear Physics Division, under contractnumber W-31-109-ENG-38,and the
National Science Foundation, under grant no. INT-9603385.
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