Table Of ContentSome remarks on the use of effective Lagrangians in
QED and QCD
Walter Dittrich
Institute for Theoretical Physics
University of Tu¨bingen
5 Auf der Morgenstelle 14
1 D-72076 Tu¨bingen
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Germany
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[email protected]
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Abstract
We discuss in this article the usefulness of the effective Lagrangians (Leff)
of QED and QCD within the one-loop approximation. Instead of calculating
Leff via complicated computations with Schwinger’s proper-time technique
or Feynman graphs, we prefer to employ the energy-momentum tensor and
the leading-log model. The advantage is that we do not have to demand the
external electromagnetic or color field to be constant. There are also some
critical remarks added which cast doubt on the use of L with covariant
QCD
constant fields in explaining the nature of the QCD vacuum.
0.1 Introduction
In the first chapter we compile the most important results with regard to the
effectiveLagrangianinaconstantelectromagneticfield. Ourobjectiveistofind
the Green’s function of a spin-1 particle in an external constant magnetic field
2
that points in the zˆ direction. This can be achieved with Schwinger’s proper-
time technique. With theresultwecancomputethe entireeffectiveLagrangian
as a function of the constant (E,H) field. In this way we obtain the famous
Heisenberg-Euler effective Lagrangian. We will then set up a relation between
the effective Lagrangianand the trace of the energy-momentumtensor for con-
stant magnetic and electric fields. Thereafter we give up the requirement that
thefieldsbeconstantandallowforarbitraryvaryingfields. Thisisdoneincon-
nection with the effective action for Yang-Mills fields. Rather than attempting
to compute Leff, we will make an ansatz motivated by the requirement that
Leff give the correct trace anomaly for the energy-momentum tensor. In this
way we are able to construct the leading-log effective Lagrangian. Similar con-
siderations are used to investigate the effective Lagrangianin QED. Finally we
briefly study Adler’s leading-log model in QCD and state his result concerning
thestaticpotentialbetweenaquark-antiquarkpairforlongandshortdistances.
Althoughthecalculationsarehighlynon-trivial,theresultsofthelinearlyrising
potential for large quark separation and the Coulombic r 1 potential for small
−
distances are very encouraging.
0.2 Compendium of Useful Formulae[DR85]
We start with the Green’s function of a spin-1 particle in an external electro-
2
magnetic field:
1
γµ ∂µ eAµ +m G+(x,x′;A)=δ(x x′). (1)
i − −
(cid:20) (cid:18) (cid:19) (cid:21)
IfwepickaspecialgaugefieldsothatF isconstant,weobtaintheclosed-form
µν
solution
G (x,x;A)=φ(x,x) ∞ 1 m 1γµ f(s)+eF (x x)ν
+ ′ ′ s2 − 2 { }µν − ′
Z0 (cid:20) (cid:21)
e−im2s−L(s)+4i(x−x′)f(s)(x−x′)e2iσFsds ,
×
with
f(s)=eFcoth(eFs)
1 sinh(eFs)
L(s)= trln ,
2 eFs
(cid:20) (cid:21)
and φ(x,x′)=eie xx′Aµ(ξ)dξµ with a straight path between x and x′.
Our central suRbject of interest is the vacuum amplitude in the presence of
an external field which, in the framework of a one-loop approximation for the
effective Lagrangian,can be written as
0 0 A =eiW(1)[A] =ei L(1)(x)d4x , (2)
+
h | −i
R
2
with
1 G [A]
iW(1)[A]= Trln = Trln + . (3)
− 1 eγAG − G [0]
(cid:18) − +(cid:19) (cid:18) + (cid:19)
Here G = G [0] is the electron propagator in the field-free case, connected
+ +
with G [A] by
+
G [A]=G (1 eγAG ) 1 . (4)
+ + + −
−
Furthermore, Tr indicates the trace both in spinor and configurationspace.
Theone-loopeffectiveactionW(1),i.e.,theeffectiveLagrangianL(1),isthe
formal expression for the effect which an arbitrary number of “external photon
lines” can have on a single Fermion loop.
The functional derivative with respect to the potential A (x) is given by
µ
δW(1)[A]
i = etr[γµG (x,x;A)] . (5)
+
δA (x) −
µ
This equation is fulfilled by the ansatz
iW(1) :=i L(1)d4x= 1 ∞ e−ism2Tr eis(γΠ)2 ds , (6)
·
−2 s
Z Z0 h i
where the proper-time representation of G [A] is given by
+
G+[A]· (γγ·ΠΠ)2−mm2 =(m−γ·Π)i ∞e−is[m2−(γ·Π)2]ds . (7)
· − Z0
We can then write for the unrenormalized Lagrangian
L(1)(x)= itr ∞ e−im2s xeis(γΠ)2 x ds , (8)
·
2 s h | | i
Z0
wherethetracerefersonlytothespinorindex. WiththisexpressionforL(1)(x)
we can show that
∂L(1)
i =tr G (x,x;A) . (9)
+
∂m
Withoutfurtherproofwealsofindforthetraceoftheenergy-momentumtensor
Tµ(x) = im tr G (x,x;A) . (10)
h µ i − +
This leads us to the equation
hTµµ(x)i=−imtrhx|(−γ·Π+m)i ∞e−i(m2−(γ·Π)2)sds|xi
Z0
=m2 xtr ∞e−i(m2−(γ·Π)2)sdsx .
h | | i
Z0
With the former expression for L(1) we obtain the useful equality
∂L(1)(x) ∂L(1)(x)
Tµ(x) =m = . (11)
h µ i ∂m ∂(lnm)
3
For a purely constant magnetic field the renormalized one-loop effective La-
grangianis known to be
L(1)(H)= 1 ∞ e−m2s (eHs)coth(eHs) 1(eHs)2 1 ds . (12)
−8π2 s2 − 3 −
Z0 (cid:20) (cid:21)
The integral can be explicitly calculated by dimensional or ζ-function regular-
ization. In the next chapter we will make the explicit expression for L(1)(H)
thestartingpointforourdetaileddiscussionofthetraceanomalyoftheenergy-
momentum tensor in QED.
0.3 The trace anomaly of the energy-momentum
tensor from the one-loop effective Lagrangian
in QED
We already mentioned the close connection between the effective Lagrangian
L(1) and the trace of the energy-momentum tensor:
∂L(1)(x)
Tµ(x) =m . (13)
h µ i ∂m
For constant fields we have the expression
Tµ(x) ( , )= m2 4 ∞ e−m2s e2s2 Recosh es√2(F +iG)1/2
h µ i F G 16π2 Z0 s2 " GImcosh(cid:0)es√2(F +iG)1/2(cid:1)
1 2e2s2 ds , (cid:0) (cid:1)
− − 3 F
(cid:21)
where
1 1
= F Fµν = H2 E2 ,
µν
F 4 2 −
= 1FµνF =E~(cid:0)H~ , (cid:1)
µν
G 4 ·
with
1
F = ǫ Fκλ, ǫ =1 .
µν µνκλ 0123
2
The closed-form expression L(1) for an external constant H-field only is
given by
1 4 m2
L(1)(H)= (2m4 4m2(eH)+ (eH)2) 1+ln
−32π2 − 3 2eH
(cid:20) (cid:20) (cid:18) (cid:19)(cid:21)
m2
+4m2(eH) 3m4 (4eH)2ζ 1, .
′
− − − 2eH
(cid:18) (cid:19)(cid:21)
4
The result of the mass-differentiation turns out to be
1 m4 m2 m2 m2
Tµ (H)= (eH)2 ln + (eH)ln
h µi −12π2 − 4π2 2eH 4π2 2eH
(cid:18) (cid:19) (cid:18) (cid:19)
m4 (eH)m2 m2 1
+ + lnΓ ln2π .
4π2 2π2 2eH − 2
(cid:20) (cid:18) (cid:19) (cid:21)
This, by the way, is also the result of the calculation of the integral
Tµ (H)= ieHm2 ∞ e−2ihz zcotz 1+ 1z2 dz, h= m2 . (14)
h µi − 4π2 z2 − 3 2eH
Z0 (cid:20) (cid:21)
Now observe that for h 1 we can approximate lnΓ(h) lnh, such that
≪ ≈−
1
lim Tµ (H)= e2H2 , (15)
m 0h µi −12π2
→
which, when written covariantly,yields
1 2α1
m 0 : Tµ = e2F Fµν = F Fµν . (16)
→ h µi −24π2 µν −3π4 µν
We can also obtain the next-to-leading term,
1 2 α 1 α 2
Tµ = β(α) F Fµν, β(α)= + , (17)
h µi − 4 µν 3 π 2 π
(cid:16) (cid:17) (cid:16) (cid:17)
by incorporating results for the two-loop calculation L(2).
For large field strengths, eH 1, the dominant term is
m2 ≫
α
Tµ (H)= H2 , (18)
h µi −3π
while for small field strength, eH 1, we obtain, using Stirling’s (Moivre’s)
m2 ≪
formula for the logarithm of the Γ-function,
2α2H4 64 H6
Tµ (H)=4 + πα3 +... . (19)
h µi − 45 m4 315 m8
(cid:18) (cid:19)
Interestingly, the first term in this expansion agrees with Schwinger’s[Sch51]
from the Heisenberg-Euler Lagrangian:
16 1 2 1
T =TMaxwell 1 α2 δ α2 4 2+ 2 . (20)
µν µν − 45 m4F − µν45 m4 F FG
(cid:18) (cid:19)
(cid:0) (cid:1)
Inourpresentcaseweuse~=c=1,andforE~ =~0wehave = 1(H2 E2)=
F 2 −
1H2 and =E~ H~ =0.
2 G ·
Since we are interested in the trace of T , we obtain (in Schwinger’s nota-
µν
tion)
8α2 14α2
T = 4 2+ 2 ,
h µµi − 45m4F 45m4G
(cid:18) (cid:19)
which for =0 indeed yields
G
2α2H4
T (H)= 4 .
h µµi − 45 m4
(cid:18) (cid:19)
5
Let us prove Schwinger’s formula. He starts with
∂L
T =δ L 2 F . (21)
µν µν νλ
− ∂F
µλ
(Note that in Schwinger’s formula the factor 2 is missing!)
We need the following derivatives:
∂L( , ) ∂L ∂ ∂L ∂
F G = F + G ,
∂F ∂ ∂F ∂ ∂F
µλ µλ µλ
F G
∂ ∂ 1 1
F = F2 = F
∂F ∂F 4 ρσ 2 µλ
µλ µλ (cid:18) (cid:19)
∂ ∂ 1 1 ∂ i
∂FG = ∂F 4FρσFρ∗σ = 4∂F Fρσ2ǫρστωFτω
µλ µλ (cid:18) (cid:19) µλ (cid:18) (cid:19)
1
= 2Fλ∗µ .
Hence we can write
∂L( , ) ∂L 1 ∂L 1
∂FF G = ∂ 2Fµλ+ ∂ 2Fλ∗µ ,
µλ
F G
so that
∂L ∂L ∂L
2 F = F F F F
− ∂F νλ −∂ µλ νλ− ∂ λ∗µ νλ
µλ
F G
=Gδµν
∂L |∂L{z } ∂L ∂L
= F F δ + δ δ .
µλ νλ µν µν µν
− ∂ − G ∂ F ∂ − F ∂
F G (cid:18) F F (cid:19)
Putting everything together we obtain
∂L ∂L ∂L ∂L
T = F F +δ +δ L δ δ
µν µλ νλ µν µλ µν µν
− ∂ F ∂ − F ∂ − G ∂
F F F G
1 ∂L ∂L ∂L
= F F δ F2 +δ L ,
− µλ νλ− µν4 λκ ∂ µν −F ∂ −G ∂
(cid:18) (cid:19) F (cid:18) F G (cid:19)
=TM
µν
| {z }
which is a gauge-invariantexpression.
Now, from the Heisenberg-Euler effective Lagrangianwe are given
2α2
L = +C 4 2+7 2 , C = , ~=c=1 . (22)
−F F G 45m4
(cid:2) (cid:3)
From this expression we obtain the derivatives
∂L ∂L
= 1+8C , = +8C 2
∂ − F F ∂ −F F
∂LF ∂LF
=14C , =14C 2 .
∂ G G ∂ G
G G
6
Finally we end up with
16α2
T =TM 1 +δ L + 8C 2 14C 2 , L = +C(4 2+7 2)
µν µν − 45m4F µν F − F − G −F F G
(cid:18) (cid:19)
16α2 (cid:0)2α2 (cid:1) (cid:0) (cid:1)
=TM 1 δ 4 2+7 2 (cid:3)
µν − 45m4F − µν45m4 F G
(cid:18) (cid:19)
(cid:0) (cid:1)
Let us put things together. Besides Tµ (H), we can easily produce the cor-
h µi
responding result for a constant electric field by substituting H iE. The
→ −
result is
e2E2 m4 π m2 eEm2 π m2
Tµ (E)= i +ln i i +ln
h µi 12π2 − 4π2 2 2eE − 4π2 2 2eE
(cid:18) (cid:19) (cid:18) (cid:19)
m4 eEm2 im2 1
+ i lnΓ ln2π .
4π2 − 2π2 2eE − 2
(cid:20) (cid:18) (cid:19) (cid:21)
If we split this equation up into its real and imaginary part we obtain
e2E2 m4 m2 m2eE m4 eEm2 im2
Re Tµ (E)= ln + + + ImlnΓ
h µi 12π2 − 4π2 2eE 8π 4π2 2π2 2eE
(cid:18) (cid:19)
m4 m2 m2 eEm2 im2 1
Im Tµ (E)= eEln RelnΓ ln2π .
h µi −8π − 4π2 2eE − 2π2 2eE − 2
(cid:20) (cid:18) (cid:19) (cid:21)
The last expression can be simplified with the aid of
1 1 αsinh(πα)
RelnΓ(iα)=ln Γ(iα) = ln Γ(iα)2 = ln . (23)
| | 2 | | −2 π
(cid:18) (cid:19)
The result is
m4 eEm2 πm2
Im Tµ (E)= + ln 2sinh . (24)
h µi −8π 4π2 2eE
(cid:20) (cid:21)
Let us study these expressions in the limiting case m 0. To do this we
→
employ the asymptotic formulae (for z 1):
≪
lnΓ(z) lnz
≈
ImlnΓ(iz) Cz, (C 0.577216)
≈ ≈
lnsinhz lnz .
≈
We then obtain
1
lim Tµ (H)= e2H2
m 0h µi −12π2
→
1
lim Re Tµ (E)= e2E2
m 0 h µi 12π2
→
lim Im Tµ (E)=0 .
m 0 h µi
→
These three results are contained in
1
lim Tµ = e2F Fµν . (25)
m 0h µi −24π2 µν
→
7
Wethusobtainaconfirmationofthemoregeneralformula(inone-loopapprox-
imation)
1
Tµ(x) = m ψ¯(x)ψ(x) e2F (x)Fµν(x) , (26)
h µ i − h i− 24π2 µν
where lim m ψ¯(x)ψ(x) =0.
m 0
→ h i
Let us have a final look at Im Tµ (E), and write it in units of E2 := m4 ,
(cid:0) (cid:1) h µi cr 4πα
and the electric field in units of E := m2. Thus we obtain
cr e
α α π
Im Tµ (E)= + Eln 2sinh
h µi −2 π 2E
α α h π iπ
= + Eln e2E e−2E
−2 π −
α α
= + E ln(cid:2)e2πE +ln 1(cid:3) e−Eπ .
−2 π −
(cid:2) (cid:0) (cid:1)(cid:3)
Here we use ln(1 x)= ∞ xn, 1 x<1, which yields
− − n=1 n − ≤
P
ImhTµµi(E)=−α2 + α2 − απE ∞ n1e−Eπn =−απE ∞ n1e−Eπn . (27)
n=1 n=1
X X
For small values we find approximately (E 1):
≪
α
Im hTµµi(E)≈−πEe−Eπ , (28)
which goes to zero for E 0.
→
Ourresultcanalsobeobtainedbyusingthewell-knownformula(c.f. e+ e
−
−
pair production)
Im L(1)(E)= α E2 ∞ 1 e−πemE2n . (29)
2π2 n2
n=1
X
We only need to write
Im hTµµi(E)=m∂∂mIm L(1)(E)=m2απ2E2 ∞ n12 −2emEπn e−πemE2n
n=1 (cid:18) (cid:19)
X
= m2eE ∞ 1e−πemE2n ,
− 4π2 n
n=1
X
or, in our units,
ImhTµµi(E)=−απE ∞ n1e−Eπn (cid:3)
n=1
X
Uptonow,wehavealwaysrestrictedourcalculationstothecaseofconstant
electricormagneticfields. Itcanbe shown,however,thatthe leadingterms for
strongfields,i.e.,thoseoforderH2lnH orE2lnE,arethesameifthefieldsare
not constant. This will be demonstrated in the next chapter, where we extend
our discussion to the effective Yang-Mills field theory.
8
0.4 The Effective Action for Yang-Mills Theory
The effectiveactionofquantumchromodynamics(QCD)forcovariantconstant
color fields has been extensively treated in the literature by many researchers.
But they rarelypose the questionin how far their results arephysically reason-
able and applicable. If one assumes that the confinement hypothesis is correct,
then no constantcolor fields can exist. Thus it wouldbe physically senseless to
study the effective Lagrangian(or the effective potential) in an exact covariant
constant color field. If, however, we regard a color field that in an expanded,
but limited space, can be considered to be approximately covariant constant,
then one could suppose that in the space in question the effective Lagrangian
can be approximated by the effective Lagrangian of a covariant constant field.
Thus we try to extrapolate from the case of an unlimited, expanded covariant
color field to the case of a color field that is in an expanded but limited space,
approximately covariantconstant. Upon looking more closely, it turns out that
this procedure is physically unsatisfactory, because one first calculates the ef-
fective Lagrangian for the covariant constant field configuration, which is not
even theoretically feasible - this is forbidden by the confinement hypothesis -
and then tries to extrapolate to a physical situation. For such an extrapolation
from a nonphysical to a physical situation, one cannot expect that the result
is in any way physically acceptable. Thus the results obtained for the effective
potentials with covariant constant color fields should not be used to describe
the nature of the QCD vacuum, but rather a transitional phase in the search
for the true QCD vacuum.
After this prelude we will return to the role of the energy-momentum ten-
sor in Yang-Mills theory. Rather than attempt the difficult task of computing
Leff as done in [DR83], we will instead make an ansatz. Our ansatz will be
motivated by the requirements that Leff(x) gives the correct trace anomaly
for the energy-momentum tensor, and depends only on the algebraic invariant
F2 :=Fa Faµν[PT78].
µν
So we require
L(F2)
Θµν =2 ηµνL(F2)
∂η −
µν
β(g¯(t))
Θµ = Fa Faµν ,
µ 2g¯3(t) µν
sothatΘµ hastheusualformofthetraceanomaly. Nowwewanttoprovethat
µ
these requirements are satisfied by the ansatz
1 1 1 F2
Leff := F2 , t:= ln . (30)
−4g¯2(t) 4 µ2
9
