Radiatively Induced Lorentz and CPT Violation in Electrodynamics

4 Pages·0.11 MB·English

by  Roman Jackiw

Radiatively induced Lorentz and CPT violation in electrodynamics
R. Jackiwa and V. Alan Kostelecky´b
aCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307
bPhysics Department, Indiana University, Bloomington, Indiana 47405
(preprint IUHET400, MIT CTP 2812, January 1999)
In a nonperturbative formulation, radiative corrections arising from Lorentz and CPT violation
in the fermion sector induce a definite and nonzero Chern-Simons addition to the electromagnetic
action. If instead a perturbative formulation is used, an infinite class of theories characterized by
the valueof the Chern-Simonscoefficient emerges at the quantumlevel.
LorentzandCPTsymmetryofMaxwellelectrodynam- The relevant quantity for deciding whether a Chern-
icsisdestroyedbyaddingtheterm = 1k εµαβγF A Simonstermis inducedis the vacuumpersistence ampli-
k 2 µ αβ γ
L
9 to the Lagrange density [1–3]. Here, kµ is a prescribed, tude, or equivalently from Eq. (1) the fermion determi-
9 constant 4-vector. The term k is of the Chern-Simons nantdet(i/∂ A/ m /bγ5), computed to secondorderin
L − − −
9 form [4]: it changes by a total derivative when the the photon variables. We are thus led to examining the
1 gaugepotentialundergoesagaugetransformationA standard one-loop vacuum-polarization amplitude Πµν,
µ
→
n A + ∂ Λ. Consequently, the action and equations of butwiththe usualfree-fermionpropagatorS(l)replaced
µ µ
a motion are gauge invariant, but the Lagrange density is by the b -exact propagator from Eq. (1):
µ
J
not. The modified theory predicts birefringence of light
0 in vacuo [1,3]. Observation of distant galaxies puts a i
2 G(l)= . (2)
stringent bound on kµ: it should effectively vanish [1,5]. /l−m−/bγ5
1 A natural question is whether such a term would be
This may also be presented as
v induced through radiative corrections when Lorentz and
8 CPT symmetries are violatedin other sectors of a larger
5 theory. If so, then the stringent experimental limits on G(l)=S(l)+Gb(l) , (3a)
3
would severely restrict the viability of models with
1 Lk where
0 Lorentz and CPT breaking [6].
9 To study explicitly this issue, one may consider ex- 1
9 tending the quantum-electrodynamics (QED) action of Gb(l)= /l m /bγ5/bγ5S(l) . (3b)
/ − −
h a single Fermi field by including a Lorentz- and CPT-
p violating axial-vector term [2,3]: With this decomposition, Πµν splits into three terms:
-
ep I d4xψ¯(i/∂ A/ m /bγ5)ψ . (1) Πµν =Πµ0ν +Πµbν +Πµbbν . (4)
h ⊃Z − − −
: The term Πµ0ν is the usual lowest-order vacuum-
v Here, bµ is a constant, prescribed 4-vector, and our γ5 polarization tensor of QED, which we shall not discuss
Xi is Hermitian with trγ5γαγβγγγδ = 4iεαβγδ. The only further. The term Πµν is at least quadratic in b; it is
otherpossible nonderivativeCPT-andLorentz-violating bb
r atmostlogarithmicallydivergentandsuffersnoambigu-
a term in the fermion sector is uninteresting here because
ity in routing the internal momenta [10]. The b -linear
µ
its properties under charge conjugation prevent it con- contribution to the Chern-Simons term arises from Πµν,
tributing to the Chern-Simons term [3]. b
which is given explicitly by
Severalcalculationshavebeenperformedtodetermine
whether radiative corrections induce the Chern-Simons d4l
term with kµ ∝ bµ. At leading order in bµ and the fine- Πµbν(p)=Z (2π)4 trnγµS(l)γνGb(l+p)
structure constant, a perturbative treatment ofthe term
in Eq. (1) has been shown to generate an ambiguous re- +γµGb(l)γνS(l+p) . (5)
o
sult: thecoefficientk oftheinducedChern-Simonsterm
µ
isregularizationdependentandcanbefreelyselected[3]. There are several important features of this expres-
Other claims include both a definite zerovalue for k [7] sion. Each of the two integrals is (superficially) linearly
µ
and a definite nonzero value [8]. divergent. However,thedivergencescancelwhenthetwo
Thepurposeofthisworkistoclarifythissituationand termsaretakentogetherandthetracesareevaluated. As
callattentiontosubtleissues,relatedtochiralanomalies aconsequence,thereisno momentum-routingambiguity
[9], underlying the discrepancies between these various in the summed integrand: when the integration momen-
results. First, adirectcalculationofk is presentedthat tumisshiftedbythesameamountinbothintegrandsthe
µ
is nonperturbative in b . The various issues are then value of the integral does not change, even though shift-
µ
disentangled. ing separately by different amounts in each of the two
1
integrands changes the value of the integral by a sur- I˜µνα(p)=Iνµα( p)+∆µνα(p) , (8)
−
face term. It follows that different momentum routings
leave unchanged the value of Πµν because they produce where
b
a simultaneous shift of integration variable by the same 4
d l
aamuonuiqnuteinvaeluacehcaonf tbheecianltceuglraatneddsfoinr ΠEµqν.,(w5)h.icThhweeresfhoraell, ∆µνα(p)=−iZ (2π)4 trnγµS(l)γαγ5S(l)γνS(l+p)
b
show leads to a finite Chern-Simons term. γνS(l)γµS(l p)γαγ5S(l p) . (9)
WenextevaluateΠµν tolowestorderinb,byreplacing − − − o
b
Gb(l) with iS(l)b/γ5S(l). This gives Since the tensorial form of Iµνα must be εµναβp , the
− β
Πµν Πµναb , (6) crossed term coincides with the uncrossed term. The
b ≃ α variable shift produces a surface term, and ∆µνα(p) is
where evaluated as
4 1
Πµνα(p)=−iZ (2dπ)l4 trnγµS(l)γνS(l+p)γαγ5S(l+p) ∆µνα(p)=−8π2εµναβpβ . (10)
+γµS(l)γαγ5S(l)γνS(l+p) Thus, we have
o
Iµνα(p)+I˜µνα(p) . (7) Πµνα(p)= 1 εµναβp
≡ −8π2 β
A shift of integration variables in the second term re- d4l 1 1 1
duces it to a crossedformof the first plus a contribution −2Z (2π)4 trγµ/l mγν/l+p/ mγαγ5/l+p/ m. (11)
arising from shifting variables in a linearly divergent in- − − −
tegral: To evaluate the integral, note first that
1 1 1 1 2m
/l+p/ mγαγ5/l+p/ m = /l+p/ mγah/l+−p/ m + (l+p)2 m2iγ5
− − − − −
∂ 1 2m 1
= ∂p /l+p/ mγ5+ /l+p/ mγαγ5(l+p)2 m2 . (12)
α
− − −
The p -derivative contributes a term
α
4
∂ d l 1 1
−2∂pα Z (2π)4 tr(cid:16)γµ/l mγν/l+p/ mγ5(cid:17) .
− −
However,the aboveintegralmustvanish: no two-indexpseudotensorexists involvingthe antisymmetricpseudotensor
and depending only on a single variable p. Therefore, one is left with an entirely finite integral. We find:
4
1 d l 1 1 1
Πµνα(p)=−8π2εµναβpβ −4mZ (2π)4 trγµ/l mγν/l+p/ mγαγ5(l+p)2 m2
− − −
= 1 εµναβp + im2εµναβp d4l 1 1
−8π2 β π4 βZ l2 m2((l+p)2 m2)2
− −
1 2 ∞ m2 1
= εµναβp + da (13)
− β(cid:16)8π2 π2 Z2m √a2−4m2p2−a2+iε(cid:17)
The final result is: and
p θ 1
Πµνα(p)=εµναβ β , (14) 3
2π2(cid:16)sinθ − 4(cid:17) kµ = 16π2bµ . (16)
where θ 2sin−1( p2/2m) and p2 <4m2.
≡
The bµ-linear cpontribution to the induced Chern- This completes our calculation.
Simons term is determined by this expression at p2 =0. We have chosen to extract the leading-order result in
One finds a definite, nonzero, and finite result [8,11]: b . However,ourcalculationisinfactnonperturbativein
µ
3 bµ in the sense that it has been performed keeping care-
Πµνα(p)|p2=0 = 8π2εµναβpβ , (15) ful track of contributions from the bµ-exact propagator
2
in Eq.(3a). Thus, in this calculation,we are choosingto limit of zero axial-vector momentum it corresponds to
define the theory of Eq. (1) in a nonperturbative way. the result of the Coleman-Glashow assumption, giving
If instead the theory in Eq. (1) is defined through its uniquely
perturbation series in b , the same θ dependence as in
µ 4
Efoqre. (t1h4e)neemterregseusltbfuotrtthheeainddduitcieodnaClhceornns-tSainmtoannsdtetrhmereis- ΠµCνGα(p)=−iZ (2dπ)l4 trnγµS(l)γνS(l+p)γαγ5S(l+p)
different [3]. At first order, one finds that the two inte-
+γµS(l+3p)γαγ5S(l+3p)γνS(l+4p)
grals(7)arisefromatriangleVVAgraphandits crossed o
expression with zero axial-vector momentum. In per-
after an inocuous shift in both integrands. The integra-
turbation theory, no correlation is determined a priori
tion momentum in the second expression must be de-
between the momentum routings in the two graphs. If
creased by 3p to bring this result into conformity with
the relative routings are as in (7), the resulting expres-
Eq. (7). Therefore, from (10) it follows that
sioncoincideswith(14). Otherwise,ashiftofintegration
variablesproducestheconfiguration(7),butgeneratesan 3
additionalcontribution. Takingthe shift asproportional ΠµCνGα(p)=Πµνα(p)− 8π2εµναβpβ . (17)
totheexternalmomentumgivesrisetoanarbitrarymul-
tiple of ∆µνα εµναβp , leaving the Chern-Simons co- This result vanishes at p2 = 0, in agreement with the
β
∝
efficient kµ proportionalto bµ but with anundetermined Coleman-Glashow claim. However, other than the de-
proportionality constant. mand that gauge invariance be maintained not only for
Coleman and Glashow have recently argued that k the action but also for the induced Lagrange density,
µ
must unambiguously vanish to first order in b for any there is no reason to prefer this result over any other.
µ
gauge-invariant CPT-odd interaction [7]. Their result is A further degree of arbitrariness in the induced term
basedontheirhypothesisthatonedefinetheaxialvector appears according to the choice of regularizationscheme
j5µ(x) ψ¯(x)γµγ5ψ(x)tobegaugeinvariantinthequan- used in the calculation. This can be true even within a
≡
tum theory at arbitrary 4-momentum, that is, at every nonperturbative formulation.
point in x. However,a weaker condition is true: if jµ(x) One possible choice is Pauli-Villars regularization,
5
does not couple to any other field, then physical gauge which enforces gauge invariance for all axial momenta.
invariance is maintained provided jµ(x) is gauge invari- The various values of the induced Chern-Simons coeffi-
5
ant at zero 4-momentum. Equivalently, physical gauge cient are mass-independent, so they are subtracted and
invarianceis maintainedprovided d4xjµ(x) andthere- vanish in Pauli-Villars regularization [3]. This would be
5
fore the action are gauge invariantR, without the require- true whether or not the theory is formulated perturba-
ment that the Lagrange density also be gauge invariant. tively or nonperturbatively. Another possibility is di-
ThisbehaviorcharacterizestheChern-Simonsterm,soit mensional regularization. This is problematic with the
is unsurprising that demanding gauge invariance of the γ5-matrix, and a variety of answers for kµ can be ob-
Lagrange density can prevent generating the noninvari- tained depending on how γ5 is generalized to arbitrary
ant Chern-Simons term. dimensions. In this sense, the physics of the theory (1)
The Coleman-Glashow argument is perturbative in b depends on the choice of regularization scheme.
µ
and is taken to first order. Only in the perturbative Referring to the dispersive representation, presented
framework does the axial vector arise as a distinct en- in Eq. (14), we see that the theory predicts a definite
tity: it is an insertion whose gauge variance can be dis- absorptive part, which is sufficiently well behaved to
cussed. In contrast, with the nonperturbative definition enter into an unsubtracted dispersion relation. Never-
ofthetheorytheaxialvectorhasnoseparateidentity,but theless, there remains the possibility of a real subtrac-
when the first-order contribution is extracted from our tion, which is perturbatively undetermined, eventhough
complete expression we find a nonzero result. Evidently we presented a nonperturbative argument for the value
2
the dynamics of the nonperturbative theory selects the 1/8π . This situation is familiar in quantum field the-
weaker option: gauge invariance only for d4xb jµ(x) ory. Parameters in a Lagrangian typically acquire in-
µ 5
but not for the unintegrated quantity. GauRge invariance finite radiative corrections that must be renormalized.
ispreservedinthesensethatp Πµν =0,andtheinduced They flow with the renormalization scale and are deter-
µ b
action is gauge invariant,but the induced Chern-Simons mined only by comparison with experiment. Here, the
Lagrange density is not. radiative corrections are finite, so infinite renormaliza-
Thegaugeanomalyvanishesforzeromomentuminthe tion is unnecessary, but nevertheless no definite value is
axial-vectorvertex, regardlessof the momentum routing determined in perturbation theory. Another instance of
in the two triangle graphs. However, for nonvanishing this phenomenon occurs in the chiral Schwinger model,
momentum in the axial vertex, only a special routing of whichgeneratesanundeterminedmassforitsvectorme-
the integrationmomenta gives a gauge-invariantanswer. son[13]. Theθ-angleofQCDprovidesafurtherexample
This special routing is known explicitly [12], and in the [14]. For all these, a finite parameter must be fixed by
3
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already been done [1,5]. S.Deser,R.Jackiw,andS.Templeton,Ann.Phys.(NY)
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17 (1996); S. Carroll and G. Field, Phys. Rev. Lett. 79,
in the theory (1)?” has no unique answer. We have
2394 (1997).
given a nonperturbative definition of the theory that in-
[6] Evenifanonzerotermisinducedbyaparticularsectorof
duces a definite and nonzero value of the Chern-Simons
alargertheory,theneteffectcanstillvanish.Thiswould
coefficient. If instead a perturbative definition is used,
occurinthestandard-modelextensionofRef.[3]through
an infinite class of theories characterized by the value
the anomaly-cancellation mechanism, for example.
of the Chern-Simons coefficient emerges at the quantum [7] S. Coleman and S. Glashow, hep-ph/9812418.
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ournonperturbativeformulationisthemostaesthetically (Princeton University Press/World Scientific, Prince-
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tary Particles and Field Theory, S. Deser, M. Grisaru,
one definition over another.
and H. Pendleton, eds. (MIT Press, Cambridge Mass.,
1970).
We thank Don Colladay, J.-M. Chung, and P. Oh for [10] Infact,thecontributionfromΠµν totheinducedChern-
bb
usefuldiscussions. This workis supportedinpartbythe Simonstermisatleastcubicinbandhencefinitebecause
United States Department of Energy under grant num- all terms quadratic in b are necessarily CPT even.
bers DE-FC02-94ER40818and DE-FG02-91ER40661. [11] Inthelimitofzerom,theCPT-violatingterminEq.(1)
can be removed altogether by a chiral field redefinition
[2].Theresulthereremainsnonzerointhislimitbecause
the chiral redefinition does not commute with quantiza-
tion (D. Colladay, privatecommunication).
[12] Jackiw, Ref. [9], pp. 118–22: Eqs. (4.26) show that the
routing ambiguity parameter must be a = −2, and in
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1231 (1990). carries zero momentum.
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4