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Reparameterisation Invariance Constraints

arXiv:hep-ph/9205228v1 21 May 1992Reparameterisation Invariance Constraintson Heavy Particle Effective Field TheoriesMichael Luke and Aneesh V. ManoharDepartment of Physics 0319University of California, San Diego9500 Gilman DriveLa Jolla, CA 92093-0319AbstractSince fields in the heavy quark effective theory are described by both a velocity anda residual momentum, there is redundancy in the theory: small shifts in velocity maybe absorbed into a redefinition of the residual momentum.We demonstrate that thistrivial reparameterisation invariance has non-trivial consequences: it relates coefficients ofterms of different orders in the 1/m expansion and requires linear combinations of theseoperators to be multiplicatively renormalised. For example, the operator −D2/2m in theeffective lagrangian has zero anomalous dimension, coefficient one, and does not receive anynon-perturbative contributions from matching conditions.

We also demonstrate that thisinvariance severely restricts the forms of operators which may appear in chiral lagrangiansfor heavy particles.UCSD/PTH 92-15May 19921

1. IntroductionThe dynamics of heavy particles at low energies may be described by a heavy particleeffective field theory, in which the effective lagrangian is expanded in inverse powers ofthe heavy particle mass [1]–[7].The particles in the effective theory are described byvelocity dependent fields [6] with velocity v, residual momentum k, and total momentump = mv+k.

There is an ambiguity in assigning a velocity and momentum to a particle whenone considers 1/m corrections to the effective field theory. The same physical momentummay be parameterised by(v, k) ↔v + qm, k −q,v2 =v + qm2= 1,(1.1)where q is an arbitrary four-vector which satisfies (v+q/m)2 = 1.

The effective field theorymust be invariant under the reparameterisation of the velocity and momentum, Eq. (1.1).This invariance has long been recognised (see, for example, [8]), however what is less wellknown is that it places constraints on the effective lagrangian, and relates coefficients ofterms which are of different order in the 1/m expansion.We will first discuss the consequences of reparameterisation invariance for the simplecase of a spin-0 field in Sec.

2, and generalise the result to the somewhat more complicatedcase of particles with spin in Sec. 3.

A few sample applications to matching conditions,anomalous dimensions and chiral lagrangians are discussed in Sec. 4.

One important resultthat is obtained in Sec. 4 is that the coefficients of certain 1/m operators in the effectivetheory are exactly fixed, and cannot be modified by non-perturbative corrections.

Section 5discusses the consequences of reparameterisation invariance for matrix elements.2. Reparameterisation Invariance for Scalar FieldsConsider a coloured scalar field [9] with mass m coupled to gluons, with lagrangianL = Dµφ∗Dµφ −m2φ∗φ.

(2.1)The low energy effective lagrangian is given in terms of a velocity dependent effective field[6]φv(x) =√2m eimv·xφ(x),(2.2)2

where v is a velocity four-vector of unit length, v2 = 1. The field φv creates and annihilatesscalars with definite velocity v, which is a good quantum number in the m →∞limit.

Theeffective lagrangian which describes the low-energy dynamics of the full theory Eq. (2.1) isLeff=Xvφ∗v (iv · D) φv + O 1m.

(2.3)The reparameterisation transformation corresponding to Eq. (1.1) for the velocity depen-dent fields isφw(x) = eiq·xφv(x),w = v + qm,(2.4)under which the effective lagrangian must remain invariant.

We will explicitly work outthe consequences of reparameterisation invariance up to order 1/m. The most generaleffective lagrangian for the scalar field theory up to terms of order 1/m isLeff=Xvφ∗v (iv · D) φv −A2m φ∗v D2φv,(2.5)where A is a constant, and we have used the lowest order equation of motion to eliminate aterm of the form φ∗v (v · D)2 φv.

Substituting the reparameterisation transformation (2.4)givesLeff=Xvφ∗w {v · (iD + q)} φw −A2m φ∗w (Dµ −iqµ)2φw. (2.6)Relabelling the dummy variable w in Eq.

(2.6) as v and v as v −q/m, gives the modifiedlagrangianLeff=Xvφ∗vn(v −qm) · (iD + q)oφv −A2m φ∗v (D −iq)2φv. (2.7)Expanding to first order in the (infinitesimal) transformation parameter q gives the changein L,δLeff=Xvφ∗vv · q −iq · Dmφv + i Am φ∗v (q · D) φv= (A −1) φ∗vq · Dm φv + Oq2, 1m2,(2.8)using q · v = O(q2/m) from (1.1).

The lagrangian (2.5) is reparameterisation invariant upto order 1/m only if A = 1. Thus reparameterisation invariance has fixed the coefficientof one of the 1/m terms in the effective lagrangian.

The tree-level matching condition ofEq. (2.3) determined A = 1, but we now have the stronger result that A = 1 is exact.3

It is an elementary exercise to determine the most general possible reparameterisationinvariant scalar lagrangian. The most general possible lagrangian may be written in theformL =XvLv(φv(x), vµ, iDµ),(2.9)where Dµ represents a covariant derivative acting on the heavy field φv.

Substituting thefield reparameterisation Eq. (2.4), and replacing the dummy index w by v as before givesL =XvLv(φv(x),v −qmµ, iDµ + qµ).

(2.10)For the lagrangian to be reparameterisation invariant, it is necessary and sufficient thatfactors of v and D occur only in the combinationVµ = vµ + iDµm . (2.11)This linear combination is precisely pµ/m, where pµ is the total momentum of the particle,and is the only quantity which is unambiguously defined at order 1/m.The results just derived may be easily extended to include scalar fields coupled toan external source.The source is velocity independent, and in the effective theory, itmust couple only to reparameterisation invariant combinations of operators in the effectivetheory.

Thus a scalar source coupling J∗(x)φ(x) can couple to J∗(x)e−imv·xφv(x) as wellas higher dimension operators, a vector source can couple to e−imv·xVµφv(x), and so forth.3. Vector and Spinor FieldsThe preceding analysis also applies to particles with spin.The only complicationwhich arises is that the effective fields satisfy the velocity dependent constraints1 −v/2ψv = 0(3.1)for a heavy spinor ψv, andvµAµv = 0(3.2)for a heavy vector field Aµv[10], which must be preserved by the reparameterisation trans-formation.

This makes the transformation law for the fields somewhat more complicated.4

We first consider the case of a heavy vector field Av. The lagrangian must be invariantunder the transformationAµw(x) = eiq·xRµν(w, v)Aνv(x),w = v + qm,(3.3)where Rµν(w, v) is a Lorentz transformation whose form we must determine.

Define thematrix Λ(v′, v) to be a Lorentz transformation in the v −w plane which rotates v into v′,i.e. v′ = Λ(w, v) v. The Λ matrix may be written asΛ(w, v) = expiJαβv′αvβθ,(3.4)where θ is the boost angle, and[Jαβ]µν = −i (gαµgβν −gανgβµ)(3.5)are the Lorentz generators in the spin-1 representation.The Lorentz boost matrix iscomputed in Appendix A.

Consider an external state in the full theory with polarisationvector ǫ, satisfying p · ǫ = 0. In the effective theory, the polarisation vectors are given byǫv = Λ(v, p/m) ǫ,ǫw = Λ(w, p/m) ǫ,(3.6)so the appropriate reparameterisation transformation for spin-1 fields isǫw = Λ(p/m, w)−1Λ(p/m, v)ǫv.

(3.7)Note that because the Lorentz group is non-Abelian, this is not the same as the (incorrect)transformationǫw = Λ(w, v)ǫv.(3.8)Eqs. (3.7) and (3.8) differ by a Thomas precession term proportional to q[αkβ]/m2, thearea of the spherical triangle on S3 with vertices at v, w and p/m.

It is not possible tomake a reparameterisation invariant lagrangian using the transformation law of Eq. (3.8).The lagrangian may be made invariant at order 1/m using (3.8), but at order 1/m2, thereare terms which are antisymmetric in Dµqν, which cannot be cancelled by the variation ofany term of order 1/m2 in Leff.The transformation (3.7) is defined for polarisation vectors.

To find the correspondingfield redefinition, p/m should be replaced by the operatorpµ/m →Uµ,Uµ = Vµ/ |V|(3.9)5

in Eqs. (3.6) and (3.7), where V is defined in Eq.

(2.11). The reparameterisation transfor-mation Eq.

(3.7) can be written asAw(x) = eiq·x Λ vµ + iDµmvµ + iDµm, w!−1Λ vµ + iDµmvµ + iDµm, v!Av(x)= Λ wµ + iDµmwµ + iDµm, w!−1eiq·x Λ vµ + iDµmvµ + iDµm, v!Av(x),(3.10)sincewµ + iDµmeiq·x = eiq·xvµ + iDµm. (3.11)Thus the only operator transformation that is required is of the formΛ vµ + iDµmvµ + iDµm, v!

(3.12)where the same velocity v occurs in both arguments. There is an operator ordering ambi-guity in the transformation Eq.

(3.12) at order 1/m2, since[Vµ, Vν] = ig F µνm ,(3.13)which produces an ordering ambiguity in the reparameterisation transformation Eq. (3.7)at order 1/m3.

However, different orderings just differ by powers of the field strength F µνtimes Av, and correspond to field redefinitions in the effective theory. Thus one can picka particular ordering in the definition of Λ in Eq.

(3.12) and use it consistently. To order1/m, the field Av that appears in the effective lagrangian isAµv = Aµv −vµ iD · Avm+ O 1m2,(3.14)using Eq.

(A.6) and v · Av = 0.To construct the most general lagrangian invariant under (3.3), it is convenient tointroduce the fieldAµv(x) = Λµν(p/m, v)Aνv(x)(3.15)which simply picks up a phase under reparameterisationAµw(x) = eiq·x Aµv(x)(3.16)6

and satisfiespµAµ(x) = 0. (3.17)The most general reparameterisation invariant lagrangian may now be written in the formL =XvLv(Av(x), Vµ) =XvLv(Λµν(p/m, v)Aνv(x), Vµ),(3.18)using the same argument as for scalar fields.Heavy fermions in the effective theory are described by velocity dependent spinorfields ψv that satisfy the constraintv/ ψv = ψv(3.19)(we treat here only the case of fermions; the arguments are easily generalised to heavyanti-fermions, which satisfy v/ ψv = −ψv).

A consistent reparameterisation transformationfor spinor fields is defined by analogy with the vector transformation, Eq. (3.7),ψw(x) = eiq·x eΛ(w, p/m)eΛ(v, p/m)−1ψv(x),w = v + qm,(3.20)where eΛ are the Lorentz boosts in the spinor representation.

The spinor lagrangian maybe written in the formL =XvLv(Ψv(x), Vµ),(3.21)where the reparameterisation covariant spinor fieldΨv(x) ≡eΛ(p/m, v)ψv(x),(3.22)transforms asΨw(x) = eiq·x Ψv(x). (3.23)The field Ψ may be written using the explicit form for eΛ in Appendix A, and choosing aparticular operator ordering for the covariant derivatives.

At order 1/m,Ψv(x) =1 + i /D2mψv(x). (3.24)7

The terms in the effective lagrangian are bilinears in the Fermi fields. The reparameteri-sation invariant combinations of the standard fermion bilinears areΨvΨv = ψvψv,Ψvγ5Ψv = 0,ΨvγµΨv = ψvvµ + iDµmψv + O1/m2,Ψvγµγ5Ψv = ψvγµγ5 −vµ i /Dm γ5ψv + O1/m2,ΨvσαβΨv = ǫαβλσΨvγσγ5VλΨv.(3.25)4.

ApplicationsReparameterisation invariance constrains terms in the effective lagrangian. As a sim-ple example, we have already seen that the kinetic term in the effective theory must havethe formv · iD + (iD)22m ,(4.1)a result which was proved in Sec.

2 for scalar fields, but can also be seen to be true forvector and spinor fields using the results of Secs. 3–4.

The coefficient of the the (iD)2operator in the effective theory is fixed to be 1/2m, and is not renormalised. This agreeswith a one loop computation of the anomalous dimension [11].

More importantly, thisresult is a non-perturbative non-renormalisation theorem. It has recently been suggestedthat there may be non-perturbative corrections in the heavy quark theory [12] at order1/m that modify the matching condition for the operator D2/m.

This cannot be true ifthe effective theory is regulated to preserve reparameterisation invariance.⋆.As another example, the leading spin dependent term in the heavy quark effectivetheory isgC2m ψv σαβFαβ ψv = gC2m ǫαβλσψv vλγσγ5Fαβ ψv,(4.2)where C = 1 at tree level. This operator is not related to the kinetic term by reparam-eterisation invariance, so C is not protected from radiative corrections.

Using the results⋆We thank Mark Wise for discussions on this point.8

of Eq. (3.25), one finds that the reparameterisation invariant generalisation of Eq.

(4.2) toorder 1/m2 isgC2m ǫαβλσψv Fαβvλ + iDλmγσγ5 ψv= gC2m ψv σαβFαβ + 2FσαiDβm vσψv. (4.3)A similar analysis applies to external currents in the effective theory.

For example,the weak current Jµ = cvΓµbv′, where Γµ = γµ or γµγ5, and cv and bv′ are heavy c and bquark fields is written in reparameterisation-invariant form asJµ = cvΓµbv′ −12mccvi←D/Γµbv′ + O 1m2c+12mbcvΓµD/bv′ + O 1m2b,(4.4)This agrees with the results in [13], in which the O(αs) matching of the operators cvΓµbv′and (−i/2mc)cv←D/Γµbv′ were found to be identical. It also agrees with [14] where it wasfound that the operators cvΓµbv′ and cv←DΓµbv′ have the same anomalous dimension inthe effective theory.Furthermore, it extends this result to additional operators at allorders in 1/m.

Note that this does not mean that (4.4) is the complete expression for thecurrent in the effective theory. There will be other terms whose coefficients are unrelated tothe zeroth order coefficient by reparameterisation invariance, just as the quark magneticmoment operator is not determined from the zeroth order kinetic term in the effectivelagrangian.Finally, reparameterisation invariance also provides useful information for chiral per-turbation theory for heavy matter fields [15]–[20].

In this case, one cannot compute thematching conditions explicitly, so the operator coefficients are undetermined constants.Reparameterisation invariance eliminates a large number of operators in the chiral ex-pansion, or determines their coefficients, thus considerably reducing the number of freeparameters in the computation. As a simple example, consider a theory with a heavyscalar Tv and a heavy vector Bµv .

The effective lagrangian could contain a term of theformTv iDµ Bµv . (4.5)Under the reparameterisation transformation, this term has a variation of the formTv qµ Bµv(4.6)9

which cannot be cancelled by any term in the effective lagrangian which is of order one(or of higher order in 1/m). This is easily seen by writing the lagrangian in terms of thefields in (3.18), where (4.5) could only arise fromTv Vµ Bµvwhich is zero by (3.17).

Thus the term Tv iDµ Bµv cannot occur in the chiral lagrangian[20].5. Matrix ElementsThe discussion has focused on the applications of reparameterisation invariance tothe effective lagrangian; in this section we discuss some of the applications to matrixelements in the heavy particle effective field theory.As might be expected, the onlyconstraint it places on matrix elements is entirely trivial.Labelling states with bothvelocity and residual momentum increases the number of possible form factors allowed;imposing reparameterisation invariance simply reduces these back to the usual number ofform factors.States in the effective theory have a velocity v and a residual momentum k, with totalmomentum p = mv +k.

Thus there is also a reparameterisation invariance transformationon the physical states which redefines v and k, but keeps p fixed. Consider the matrixelement of the vector current between two spinless particles,⟨v, k′| jµ |v, k⟩= f1vµ + f2 (kµ + k′µ) + f3 (kµ −k′µ) ,(5.1)where fi are three independent form factors, andjµ = ψvvµ + iDµmψv.

(5.2)It is well known that this matrix element should have only two independent form factors,f+ and f−. The reparameterisation invariance on the states may be used to show that onecan eliminate one of the form factors, and write Eq.

(5.1) in the form⟨v, k′| jµ |v, k⟩= f1vµ + kµ + k′µ2m+ f3 (kµ −k′µ) ,(5.3)where f1 and f3 are functions of v + k/m and v′ + k/m. This is equivalent to the f± formfactor decomposition, and is a trivial application of reparameterisation invariance; there10

are redundant variables in the effective theory which lead to redundant form factors whichcan then be eliminated.Finally, one can easily see that the formulæ of Secs. 2–3 can be applied to externalstates with velocity v, residual momentum k, and spin, by replacing p/m by v + k/m.There is no operator ordering ambiguity because the residual momentum k for externalstates is a number.

The redundant form factors for particles with spin can be eliminatedusing the methods used above for the form factors of spinless particles.AcknowledgementsWe would like to thank A. Falk, E. Jenkins, M. Savage and M. B. Wise for usefuldiscussions.

This work was supported in part by DOE grant #DOE-FG03-90ER40546,and by a NSF Presidential Young Investigator award PHY-8958081.Appendix A. Lorentz BoostsThe Lorentz boostΛ(w, v, θ) = expiJαβwαvβθ,[Jαβ]µν = −i (gαµgβν −gανgβµ) ,(A.1)is a Lorentz boost in the w −v plane with boost parameter θ. To compute Λ(w, v, θ)explicitly, define the matrixN αβ = wαvβ −vαwβ,Λ(w, v, θ) = eθN,(A.2)A straightforward computation by expanding the exponential in a power series givesΛ(w, v, θ)αβ =gαβ +1 −cosh λθλ2(wαwβ + vαvβ) + sinh λθλ(wαvβ −vαwβ)+ (w · v)cosh λθ −1λ2(wαvβ + vαwβ) ,(A.3)whereλ2 = (w · v)2 −1.

(A.4)To obtain the boost matrix Λ(w, v) which rotates v into w, the boost parameter θ musthave the valuesinh λθ = λ,(A.5)11

so thatΛ(w, v)αβ =gαβ −11 + v · w (wαwβ + vαvβ) + (wαvβ −vαwβ)+v · w1 + v · w (wαvβ + vαwβ) . (A.6)The corresponding transformations eΛ(w, v, θ) and eΛ(w, v) in the spinor representationmay be obtained by using Eq.

(A.1), and replacing the Lorentz generators Jαβ by theirvalues in the spinor representation,Jαβwαvβ = −12σαβwαvβ = −i4 [w/ , v/] . (A.7)The exponential is evaluated explicitly using the identity[w/ , v/]2 = 4λ2,(A.8)to giveeΛ(w, v, θ) = coshλθ2+ 12λ [w/ , v/] sinhλθ2.

(A.9)For the transformation that rotates v into w, θ has the value Eq. (A.5), so thateΛ(w, v) =1 + w/ v/p2 (1 + v · w).

(A.10)12

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