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Parity Invariance and Effective Light-Front Hamiltonians M. Burkardt Department of Physics New Mexico State University Las Cruces, NM 88003-0001 U.S.A. (P , which is conjugate to x0 in ET and P , which is 0 + In the light-front form of field theory, boost invariance conjugate to the LF-time x+ in LF quantization) con- is a manifest symmetry. On the downside, parity and ro- tain interactions among the fields and are typically very tational invariance are not manifest, leaving the possibility complicated. Generators which leave the initial surface that approximations or incorrect renormalization might lead invariant are also called kinematic generators, while the 6 toviolations ofthese symmetriesfor physicalobservables. In othersarecalleddynamic generators. Obviouslyit is ad- 9 this paper, it is discussed how one can turn this deficiency vantageous to have as many of the ten generators kine- 9 into an advantage and utilize parity violations (or the ab- 1 sencethereof)inpracticeforconstrainingeffectivelight-front matic as possible. There are seven kinematic generators n Hamiltonians. More precisely, we will identify observables on the LF but only six in ET quantization. The fact that P , the generator of x− translations, is a that are both sensitive to parity violations and easily cal- − J culable numerically in a non-perturbativeframework and we kinematic (obviously it leaves x+ = 0 invariant!) and 7 will use these observables to constrain thefinite part of non- positive has striking consequences for the LF vacuum 1 covariant counter-termsin effectivelight-front Hamiltonians. [2]. For free fields p2 = m2 implies for the LF energy p+ = m2+p~⊥ /2p−. Hencepositiveenergyexcitations 1 havepositivep . Aftertheusualre-interpretationofthe − v I. INTRODUCTION negati(cid:0)ve energy(cid:1)states this implies that p for a single − 9 particle is non-negative [which makes sense, considering 8 2 Light-Front(LF)quantizationisverysimilartocanon- that p− = (p0 p3)/√2]. P− being kinematic means − 1 ical equal time (ET) quantization [1] (here we closely that it is given by the sum of single particle momenta 0 follow Ref. [2]). Both are Hamiltonian formulations of p−. Combined with the non-negativity of p− this im- 6 field theory, where one specifies the fields on a partic- plies that, evenin the presence ofinteractions,the phys- 9 ular initial surface. The evolution of the fields off the icalvacuum(groundstate ofthe theory)differs fromthe / h initialsurfaceisdeterminedbytheLagrangianequations Fock vacuum (no particle excitations) only by so-called p ofmotion. Themaindifferenceisthechoiceoftheinitial zero-modeexcitations,i.e. byexcitationsofmodeswhich - surface, x0 = 0 for ET and x+ = 0 for the LF respec- are independent of the longitudinal LF-space coordinate p e tively. In both frameworks states are expanded in terms x−. Due to this simplified vacuum structure, the LF- h of fields (and their derivatives) on this surface. There- frameworkseemstobetheonlyframework,whereacon- : fore,thesamephysicalstatemayhaveverydifferentwave stituent quark picture in a strongly interacting relativis- v i functions1 in the ET and LF approaches because fields tic field theory has a chance to make sense [3–6]. X at x0 =0 provide a different basis for expanding a state Whenever the generator of a symmetry is dynamical r than fields at x+ = 0. The reason is that the micro- (contains interactions)it is somewhere between very dif- a scopic degrees of freedom — field amplitudes at x0 = 0 ficult and impossible to monitor and maintain that sym- versus field amplitudes at x+ =0 — are in generalquite metry at each step of a calculation — unless of course different from each other in the two formalisms. on can solve the theory exactly. A typical example is From the purely theoretical point of view, various ad- the boost invariance,which, in the context of equal time vantagesofLFquantizationderivefrompropertiesofthe quantization, is generated by a dynamical operator. It tengeneratorsofthePoincarégroup(translationsPµ,ro- is thus not manifestly true that En2 = m2n+p~2, i.e. the tations L~ and boosts K~) [1,2]. Those generators which eigenvalues satisfy the correct dispersion relation if and only if one starts from the correct renormalized Hamil- leave the initial surface invariant (P~ and L~ for ET and tonian, with the right counter-terms (for the regulators P ,P~ ,L andK~ forLF)are“simple”inthesensethat − ⊥ 3 employed). Now suppose, one does not know the Hamil- they have very simple representations in terms of the tonian precisely but has some idea how it may look like: fields (typically just sums of single particle operators). for example one knows the operators that appear in the The other generators,which include the “Hamiltonians” Hamiltonian but not their coefficients. In such a situa- 1By“wavefunction”wemeanherethecollectionofallFock space amplitudes. 1 tion it should, at least in principle,2 be possible to use ply to rotationalinvariance, i.e. at least in principle this relativisticcovarianceasarenormalizationconditionthat paper could also have been written about rotational in- canbeusedtopindownsomeoftherenormalizationcon- variance [16]! However, in practical LF calculations, ro- stants in the Hamiltonian3. tational invariance is usually broken much more badly InHamiltonianLFcalculationsonefacesaverysimilar than parity invariance: for example, in the transverse problem: in practical non-perturbative calculations one lattice formulation of LF field theory [17–21], the classi- often leaves out degrees of freedom, such as zero-modes cal actionis still invariantunder parity (which maps the and other high-energy degrees of freedom [3]. Because 1+1 dimensional sheets onto themselves), but not under ofsuch(inpracticalnon-perturbativecalculationsnearly general rotations which mix the continuous longitudinal unavoidable!) approximations it is in general not guar- direction and the discrete transverse direction. Thus for anteed that one recovers non-manifest symmetries (on agiventransverselattice(withfixedtransversespacing), the LF: parity and rotationalinvariance) in the end. On if one does a perfect numerical job and if one does the the contrary, without appropriate4 counter-terms in the renormalizationright,one shouldobtain a perfectly par- Hamiltonian one is almost guaranteed to violate these ityinvarianttheory,whereas,underthesameconditions, symmetries. rotationalinvarianceshouldonly be recoveredif onefur- In this work, an attempt will be made to turn these thermore takes the limit of zero lattice spacing and infi- problemsintoanadvantage. Moreprecisely,wewilliden- nite lattice volume. tify physical observables which are easily accessible in a Thepaperisorganizedasfollows. InSectionIIandIII, practicalnon-perturbativecalculationandwhicharesen- someobservablesthataresensitivetoviolationsofparity sitive to violations of covariance. are identified and we will discuss their usefulness in the Parity transformations take x+ x−, i.e. LF-time contextofpracticalnon-perturbativeLF-calculations. In ←→ and LF-space are interchanged. Within the LF formal- Section IV, we will illustrate the formalism by studying ism, this is a very complicated transformation: in the the these observables in the context of a concrete exam- above language, parity transformations are obtained by ple: 1+1 dimensional Yukawa theory. In Section V, we dynamical operatorsbecause the initial surface (x+ =0) will summarize the findings and discuss potential appli- isnotinvariantunderparity. Fromthepracticalpoint-of- cations of the formalism to QCD. view, this has the following consequences. First, given a LF Hamiltonian, most approximations to that Hamilto- nianarelikelytobreakparityinvariance. Secondly,even II. THE DIFFICULTY IN FINDING SENSITIVE if one does a “perfect numerical job”, parity invariance AND SENSIBLE OBSERVABLES may still be broken because it may have been broken already at the level of regularization and renormaliza- There are of course infinitely many relations between tion: most regulators that are practically useful within matrix elements one can write down that are potentially the LF-formalism break parity invariance. This also in- sensitivetoparityviolations. However,mostofthemare cludeseffectsthatarisewhenzero-modesareomitted(or not useful here because of a number of practical consid- eliminated) as dynamical degrees of freedom [7–15]. erations: The main limitations arise since The counter-terms introduced in the renormalization procedure are thus not only supposed to cancel the in- A) certain relations arising from parity invariance are finities but also to restore parity invariance (in the limit “protected” by some manifest symmetry, or as the cutoff goes away). In general, restoring parity re- B) the matrix elements appearing in those relations quires an additional finite renormalization! This issue are incalculable in praxis. will be the main subject in the rest of this paper. Many of the general statements made so far also ap- These points can be illustrated by considering a few examples. 2In practice, this example has a serious flaw: Most non- A. Protected Relations perturbativenumericaltechniquesproject most efficientlyon the ground state with zero momentum. Energies of excited Charge conjugation is a manifest symmetry in the LF statesandstateswithnonzeromomentumaretypicallymuch formalism. Therefore, certain matrix elements that are more difficult to evaluate. inprinciplesensitivetoparitycouldbe“protected”byC- 3Thisisofcourseonlypossibleaslongastheenergyscaleof parity. For example,if n is aneigenstateofparity then the approximations involved is much larger than the kinetic | i its vacuum to meson scalar and pseudoscalar couplings energy associated with themomentum p. ( 0ψψ n and 0ψiγ ψ n respectively) cannot both be 4Appropriate means here not only the correct infinite part h | | i h | 5 | i nonzeroforthe samestate n . However,the samestate- ofthecounter-term,whichcanoftenbeobtainedonthebasis | i ment is true for eigenstates of C-parity — irrespective of perturbative arguments, but also the correct finite part of whether n is an eigenstate of parity. Thus, no matter thecounter-term. | i 2 howbadlyparityisviolated,aslongasmanifestC-parity invariance in the Hamiltonian LF formalism they are all is maintained, either 0ψψ n or 0ψiγ ψ n (or both) totally useless! 5 h | | i h | | i will always vanish and one cannot exploit these vacuum The flaw in all these examples is that at least one of to meson matrix elements to investigate whether or not the Lorentz components of the currents involved in such parity is violated. relationscontainsabadcurrent: intheLFformalismone The situation changes when one introduces different usuallydecomposesthe fermionfieldinto dynamicaland quark flavors and considers states that have net flavor non-dynamical components and hence are not eigenstates of C-parity, such as su5. However, even in a multi-flavor theory, these vacuum to ψ =ψ(+)+ψ(−), (2.3) meson matrix elements are not very useful in practice for investigations of parity invariance because of some where ψ(±) ≡ 21γ∓γ±ψ. The point is that the (LF-) nontrivialoperatorrenormalizationissues, which we will time derivative of ψ(−) does not enter the Lagrangian discuss below in the context of j−. and thus ψ(−) satisfies a constraint equation and is usually eliminated by solving this constraint equation.7 An explicit example will be given in section IV. Since the B. Incalculable Matrix Elements solution to these constraint equations are typically nonlinear expressions (in term of the dynamical degrees of Otherselectionrulesandrelationscanbederivedfrom freedom),anyoperatorcontainingψ(−)naturallyendsup the Lorentz transformation properties of vector currents being highly nonlinear when expressed in terms of ψ(+) inaparityinvarianttheory. Forexample,forthevacuum and the other (dynamical) fields involved in the interac- to meson matrix element of a vector/pseudovector one tions. For example, in a gauge theory (A+ = 0 gauge) obtains or Yukawa theory, ψ(−) contains a product of ψ(+) and 3+1 dimensions: the boson field. These nonlinear terms generally lead to nastydivergencesin composite operatorsinvolvingψ , (−) 0ψγ ψ n,s,p =s c which is the reason why composite operators involving µ µ n h | | i ψ are called bad currents.8 However, the motivation 0ψγ γ ψ n,s,p =p c (2.1) (−) µ 5 µ n h | | i for this terminology is not only the occurrence of diver- 1+1 dimensions:6 gences—afterallwehavebecomeusedtodivergencesin quantum field theory and we have learned how to renor- 0ψγµψ n,p =εµνpνcn malize the infinities by adding counter-terms — but the h | | i 0ψγ γ ψ n,p =p c , (2.2) fact that the finite part of the counter-term remains a µ 5 µ n h | | i priory ambiguous in this procedure. The bottom line is where p , s are the momentum and spin vector respec- the following: the goal of this paper is to find ways to µ µ tively. Notethatthere isnospinin1+1dimensions. ε usespace-timesymmetriestoconstrainthefinitepartsof µν is the antisymmetric tensor in 1+1 dimensions. the non-covariant counter-terms in the LF-Hamiltonian. Naively one may think that these relations are use- Matrix elements of bad currents contain unknown finite ful to detect violations of parity invariance. For exam- renormalizations themselves. In a sense, by consider- ple,one can calculate both c(+) 0ψγ+γ ψ n,p /p+ ing matrix elements of bad currents we have increased n 5 and c(−) 0ψγ−γ ψ n,p /p− in≡dehpe|ndently a|ndithen not only the number of equations (renormalization con- n 5 ≡ h | | i ditions) but also the number of “unknowns” and the net compare the results: in a parity invariant theory the re- result is questionable. 9 sults for c extracted from the two Lorentz components n After these sobering insights about formal limitations of the current should be the same. There are many such in studying parity (and also rotationalinvariance)viola- relationsthatonecanderiveformatrixelementsandcou- tionswithintheLFframework,letusnowturntopracti- pling constantscalculatedfromdifferentLorentzcompo- nents. For the purpose of detecting violations of parity 7Thisprocedureresemblesverymuchtheelimination ofthe 5Ofcourse,foraflavorsymmetrictheoryG-paritytakesthe ‘Coulomb component’ of thephoton field in Coulomb gauge. 8Sometimesonereferstooperatorwhicharebilinearinψ , role of C-parity in this case, but we will assume mu =ms in (−) this example. 6 such as j− =ψ(†−)ψ(−), as very bad operators. 6Even if one is only interested in 3+1 dimensional theories, 9Certain bad currents also enter the Hamiltonian and thus itisusefultoconsiderthe1+1dimensionalrelations because studyingtheirmatrix elementsdoes not increase thenumber the 3+1 dimensional relations assume rotational invariance of unknowns because these operators have to be renormal- and are thuslikely tobebroken. Furthermorethetransverse ized anyway in order to construct the Hamiltonian. How- latticeformulationof3+1dimensionalfieldtheoriesexplicitly ever, we will not exploit this fact here any further. See Ref. utilizes 1+1 dimensional degrees of freedom to approximate [22]forsomeexampleswherebadcurrentsacquireadditional the3+1 dimensional continuum. renormalizations. 3 cal limitations: the most powerful numerical techniques whose solutions typically come in pairs x (except for 1/2 for non-perturbative LF-calculations, the Lanzcos algo- M = M q2). Physically these two solutions cor- n m rithm[23]andMonteCarlotechniques[24]areonlygood respond to t±he two cases where the momentum is trans- for groundandlow lying states(for givengoodquantum ferred to the ipnitial state by “hitting it” from the left or numbers). Excited states and scattering states are very from the right. Of course, in a parity invariant theory difficulttohandleandanalyze. Thisimpliesthatwecan- one should (up to a kinematic factor) obtain the same not(forpracticalreasons)useselectionrulesinthedecay formfactorfromthesetwovaluesofx. However,inaLF of resonances to study parity violations either! calculationthis is in generalnotmanifestly true and one can use the equality of the form factor F as extracted mn from x and x as a condition to test parity invariance. III. PARITY SENSITIVE OBSERVABLES THAT 1 2 Note that this test only works for m = n because for ARE SENSIBLE 6 m = n invariance under PT (which is usually manifest in the LF formalism), combined with longitudinal boost Thecombinationofthesetworestrictions,exclusionof invariance, guarantees equality of the two form factors bad currents and ground state (for given good quantum extracted from x and x . 1 2 numbers) properties only, severely constrains the pos- WhenonewantstotestwhethersomeLFHamiltonian sibilities for studying parity violations on the LF in a gives rise to a parity invariant theory, one can use the non-perturbative framework,which is probably why this following procedure: subject has so far not been studied in more detail. Fortunately, despite these limitations, there are a few One diagonalizes the Hamiltonian and determines • observablesleftwhicharearenotexcludedfromthestart: the meson wave functions Our first example is the inelastic electro-magnetic form Then one calculates the inelastic transition form factor. Current conservation, i.e. qµjµ = 0 and parity • factor F using Eq.(3.2) as a function of the lon- invariance imply that it should be possible to write the mn gitudinal momentum transfer x for two arbitrary matrix elements of the current operator in the form meson states m and n10. (pµ+p′µ)F (q2) same parity hm,p′|jµ(0)|n,pi=( qµF (qm2n) opposite parity, • Ttrhaennsfeornqe2aalslosocaaslcaulfautnescttihone ionfvxaruiasnintgmEoqm.(e3n.3tu)m mn Finally, one parametrically (parameter x) plots (3.1) • F versus q2. If F does not turn out to be mn mn whereq =p p′. Forthe“good”component,thisimplies a unique function of q2 then parity is violated − Below the practicality of this procedure will be demon- (p++p′+)F (q2) same parity mn m,p′ j+(0)n,p = strated in a concrete example. However, before doing h | | i ( q+F (q2) opposite parity. this,we shouldmentionapossible caveat: Inorderto be mn abletoevaluatethematrixelementappearinginEq.(3.2) (3.2) one needs two ingredients: the states in some basis and the current operator in the same basis. For the “good” It is implied that the states have been normalized co- current appearing in Eq.(3.2) we will always assume the variantly. Otherwise one has to multiply by appropriate canonical form in this paper. There are several rea- normalization factors. After factoring out the kinematic sons for doing this. First, it is the most simple form. coefficient [(p++p′+) for transitions between states of Second, since no Tamm-Dancoff approximation will be equal parity and q+ for parity changing transitions] the employed in this paper and since we extrapolate to the form factor should depend on q2 only, but no longer on limit where there is no cutoff in this paper, there is no q+ =p+ p′+ andq− =M2/2p+ M2/2p′+ separately. reason to believe that the good current should be mod- − n − m Of course if parity is violated then any linear combina- ified from its canonical form.11 Third, our numerical tion of (p++p′+) and q+ (with two independent form results explicitly demonstrate that this leads to a self- factors)mayoccuronther.h.s. anditisnolongerpossi- consistent procedure. Nevertheless, especially when em- bletoobtainaresultthatdependsonq2onlybyfactoring ploying Tamm-Dancoff truncations, or other parity vio- outa kinematiccoefficient. Denoting x=q+/p+, energy conservation implies q2 M2 M2 = + m , (3.3) 10In practice one preferably calculates the form factor be- n x 1 x tween the two lightest mesons since for those the numerical − i.e. for fixed q2 one obtains a quadratic equation in x convergence is fastest). 11In this limit, zero-mode corrections are expected only for x2M2+x M2 M2 q2 +q2 =0, (3.4) bad currents! n m− n− (cid:0) (cid:1) 4 lating(un-extrapolated)cutoffs,onemustconsidermod- of view of renormalizationit is thus natural to make the ifying even the good current operator from its canonical following ansatz for the renormalized Hamiltonian den- form. A detailed discussion of this would go beyond the sity12 intended scope of this paper, but it should be emphasized that parity conditions might also be helpful in a = m2φ2 M2ψ† i ψ + (4.7) self-consistent determination of the current operator in Hren 2 − √2 (+)∂− (+) Hn.o. those cases. c i i c i 3 ψ† φ+φ ψ 4 ψ† φ φψ . − √2 (+)(cid:20)∂− ∂−(cid:21) (+)− √2 (+) ∂− (+) IV. A NON-PERTURBATIVE EXAMPLE The canonical Hamiltonian is obtained by taking [25] The most simple example, where the issue of parity c = M2c (canonical Hamiltonian). (4.8) 3 4 invariance and LF quantization can be studied, is the p Yukawa model in 1+1 dimensions In perturbation theory with can, infinities in the lon- H gitudinal momentum integral occur only at the one-loop 1 m2 level (for both fermion and boson self-energies) and are =ψ¯(i ∂ M gφ)ψ+ ∂ φ∂µφ φ2, (4.1) L 6 − − 2 µ − 2 calculable [25]. The corresponding counter term (whose infinite part is unique) is denoted by the “normal order- whereφis somescalarfield. The LFquantizationofthis ingterm” . Thefinitepartof hastheoperator n.o. n.o. model has been studied in Refs. [25–29]. With ψ(±) as structure oHf kinetic terms. Since sHuch operators are al- defined above, the spinor part of the above Lagrangian ready explicitly included in the above ansatz [Eq.(4.7)], (4.1) reads [26] it is not necessary to discuss the finite part of here n.o. H any further. ψ¯(i ∂ M gφ) Lψ ≡ 6 − − The Lagrangian as well as the canonical Hamiltonian =√2 ψ† i∂ ψ +ψ† i∂ ψ (4.5) contain only 3 free parameters:M,m,g. Therefore (+) + (+) (−) − (−) the most general situation for the Yukawa model should h i (M +gφ) ψ† γ0ψ +ψ† γ0ψ (4.2) thusbedescribedbyfixingonly3parametersaswell. On − (+) (−) (−) (+) the other hand, it is known [16,15] that the above rela- h i and ψ satisfies the constraint equation tion (4.8) is not valid after renormalization,i.e. it seems (−) that all 4 parameters in Eq.(4.7) get renormalized inde- i pendently. However, the 4 parameters in Eq.(4.7) are ψ = (M +gφ)γ0ψ . (4.3) (−) −√2∂ (+) not really independent! The point is that arbitrary val- − uesfor the parametersm2, M2, c , c donot correspond 2 4 Upon inserting the solution to this constraint equation to the Yukawa model but rather something else. Only (4.3) into (4.2) one finds for a 3-dimensional subspace of the 4-dimensional pa- ψ L rameter space spanned by m2, M2, c , c does Eq.(4.7) 2 4 M2 i actually describe a version of the Yukawa model. At =√2ψ† i∂ ψ + ψ† ψ (4.4) Lψ (+) + (+) √2 (+)∂− (+) the tree level, this 3-dimensional subspace is character- ized by the canonical relation (4.8). Beyond the tree Mg i i g2 i + ψ† φ+φ ψ + ψ† φ φψ , level, the relation between the 4 parameters, for which √2 (+)(cid:20)∂− ∂−(cid:21) (+) √2 (+) ∂− (+) Eq.(4.7) describes a Yukawa model, is in general more complex. Hencethecrucialquestionis: howcanonefind which contains only dynamical degrees of freedom and that relation? One possibility (at least in principle) is can be quantized straightforwardly. One thus obtains thatone makescalculations bothusing equaltime quan- the canonical LF Hamiltonian tization as well as LF quantization. One then calculates 4 physical quantities in both schemes and fine-tunes the P− = dx− , (4.5) can Hcan 4 parameters in the LF calculation until the 4 physical Z quantities have been reproduced. Even though there is where nothing fundamentally wrong with this procedure, it is very unattractive since it requires one to go back to an m2 M2 i = φ2 ψ† ψ (4.6) equal time quantized theory in order to define the LF Hcan 2 − √2 (+)∂− (+) theory. Mg i i g2 i ψ† φ+φ ψ ψ† φ φψ . − √2 (+)(cid:20)∂− ∂−(cid:21) (+)− √2 (+) ∂− (+) ThecanonicalHamiltonian(4.5)contains4terms: atwo 12In fact, in perturbation theory, it is both necessary but point function for both fermions and bosons, a three also sufficient to generalize the Hamiltonian as in Eq.(4.7) pointfunctionandafourpointfunction. Fromthepoint [16]. 5 A much more satisfying approach is based on the ob- K = 24, 32, and40. Typical results are shown in Figs. servation that, in general, a “wrong” combination of the 1 and 2. Fig. 1 corresponds to intermediate coupling 4 parameters m2, M2, c2, c4 leads to a parity violating [ Mphys 2 = Mphys 2 = 4 in units of the coupling theory! 13 One can exploit this fact by means of the F F following renormalization procedure: first one picks (or c(cid:16)onstant(cid:17)],while(cid:16)Fig.2[(cid:17)Mphys 2 = Mphys 2 =2]isan F F fixes, using some at this point unspecified renormaliza- example for strong cou(cid:16)pling. (cid:17) (cid:16) (cid:17) tioncondition)threeoftheabovefourparameters. Then onefinetunesthefourthparameteruntilsomeparitysen- sitiveobservableindicatesno violation. As aconsistency condition one can check more than one parity sensitive K=24 K=32 K=40 observable. 1.5 IntheapplicationtotheYukawamodelweusedavari- ationofthisgenericprocedure. First,thefourpointcou- plingc isassignedthevalue2π. 14 Thisonlydetermines F 4 theoverallmassscale. Acompletelyequivalentapproach wouldhavebeennottofixc atallbuttomeasurealldi- 4 1.0 mensionfulparametersandphysicalquantitiesinunitsof λ c4/2π (which carries the dimension mass squared). m2=6. m2=6. m2=6. ≡ V V V Then arbitrary values for the physical masses of the lightest fermion as well as the lightest scalar meson (strictly speaking, the C = 1 meson, which is supposed K=24 K=32 K=40 to be scalar!) were selected. Including the condition 1.5 c = 2π, this implies 3 conditions and we can now start 4 the actual fine tuning procedure which allows to fix all four bare parameters: For given values of Mphys and F F Mphys an arbitrary value for the vertex mass (i.e. for S c ) was selected. Then the bare masses of both fermion 3 1.0 and boson were fine tuned so that the physical masses 2 2 2 of both fermion and boson equal Mphys and Mphys. Af- mV=5. mV=5. mV=5. F S ter this had been achieved, the inelastic transition form factorbetweenthe twolightestmesonswascalculatedas K=24 K=32 K=40 described above. This procedure was repeated for differ- 1.5 ent values of c until the inelastic form factor satisfied 3 the parity condition discussed in section III. In the numerical procedure, DLCQ [25], with anti- F periodic boundary conditions for the fermions and periodic boundary conditions for the bosons, was employed. No cutoff beyond the DLCQ cutoff was used, i.e. parti- 1.0 cle number was allowed to reach arbitrarily large val- 2 2 2 m =4. m =4. m =4. ues — the only limit was set by the DLCQ parame- V V V ter K, which measures the total momentum of the ini- 1.0 2.0 3.0 1.0 2.0 3.0 1.0 2.0 3.0 tial meson in the transition matrix element [Eq.(3.2)] in units of 2π/L, where L is the box length in DLCQ. The 2 2 2 Q Q Q DLCQ-cutoff itself violates parity. It is therefore necessary, to verify that the results have numerically con- verged in K. The matrix elements were calculated for FIG. 1. Inelastic transition form factor (3.2) between the two lightest meson states of the Yukawa model, calculated for various vertex masses mv and for various DLCQ parameters K. The physical masses for the fermion and 13Onecaneasilyconvinceoneselfaboutthisfactforexample the scalar meson have been renormalized to the values by calculating some physical observables at tree level, but mpFhys 2 = mpFhys 2 = 4. All masses and momenta are with a combination of parameters that does not satisfy the in units of √λ= c /2π. (cid:0) (cid:1) (cid:0) (cid:1)4 canonical relation [Eq.(4.8)]. 14Note,Yukawa1+1 issuperrenormalizable andthereisonly Inthecalculatiopns,equalphysicalmassesforthefermion afiniterenormalizationofthefourpointcouplingwhichweig- and the scalar meson were chosen because if the fermion norehereforsimplicity. Alternativelywecouldhaveimposed andmesonhavesimilarmassesthereisonlyonescaleand a condition on a physical observablehere. the numerical convergence (in K) is faster. In principle, 6 there is nothing wrong with repeating this procedure for tion formfactor F (3.2) is not a unique function mn unequal masses. ofq2 (evenforK large),thusclearlydemonstrating Since momenta assume only discrete values in DLCQ, a violation of parity in the general case. the form factor could only be evaluated at a discrete set For specific combinations of physical masses (or ofpoints. Forexample,withaninitialmomentumofK = • bare kinetic masses) and vertex masses, the tran- 40 for the “pseudoscalar”meson, the finalmomentum of sition form factor F (3.2) is a unique function the “scalar” meson was taken to be K = 38,36,34,.. . mn of q2. For given values for the physical masses, a In the plots, the form factors, evaluated between states unique vertex mass, which renders the form fac- with these momenta, were connected by a smooth curve tor parity invariant, was found: In the case of to guide the eye. 2 2 mphys = mphys = 4, the correct value F F f(cid:16)or the(cid:17)vertex(cid:16)mass is(cid:17)about m2 5, while for V ≈ K=24 K=32 K=40 2 2 2.0 mphys = mphys = 2 the correct value is F F F (cid:16)near m2V(cid:17)≈2.8(cid:16). (cid:17) It should be emphasized that the transition form 1.5 • factor is a function and not just one number, i.e. the mere fact that F is parity invariantover the mn wholerangeofq2 consideredprovidesaconsistency 1.0 2 2 2 checkoftheproceduredescribedinthiswork. This m =3.0 m =3.0 m =3.0 V V V fact, plus the uniqueness mentioned above, give a strongindicationthatwehavereallyfoundthecor- K=24 K=32 K=40 rect renormalizedHamiltonianand that the proce- 2.0 dure outlined in this paper is practical. F As a side remark,it should be explained here how the baremasseswerefixedinpractice. Therearetwoslightly 1.5 different procedures one can imagine. In the first procedure one first adds a momentum dependent kinetic term that takes care of the one loop divergences in the self- 1.0 2 2 2 energies. Then one adds a finite (momentum indepen- mV=2.8 mV=2.8 mV=2.8 dent) bare masses for boson and fermion which are fine tuneduntilthephysicalmassestakethevaluesdesiredin the large K limit. In this work, a slightly different pro- K=24 K=32 K=40 2.0 cedure was used: momentum dependent bare masses for both boson and fermion were introduced in such a way F (byfine-tuning)thatthe physicalmassesareK independent. This can be easily done in a successive procedure. 1.5 In the large K limit, the momentum dependence of the kinetic term thus obtained reproduces the momentum dependence as derived from the one loop counter term 1.0 2 2 2 and therefore both procedures agree with each other in m =2.5 m =2.5 m =2.5 V V V the large K limit. However, it was found that, typically, when the physical masses of the lightest particles 0.4 0.8 1.2 0.4 0.8 1.2 0.4 0.8 1.2 are exactly K independent (and not only in the limit 2 2 2 K ), other physical observables converge faster to Q Q Q the→ir K∞ values. →∞ FIG.2. Same as Fig.1 but for mphys 2 = mphys 2 =2. V. SUMMARY F F (cid:0) (cid:1) (cid:0) (cid:1) From the fact that the form factors for K = 32 and We have investigated several classes of observables, K = 40 hardly differ, one can conclude that the results which are potentially sensitive to parity violations, and have convergednumerically. The examples in Figs.1 and found that most of them are not suitable for “typical” 2 show several facts: LF calculations. What makes them “unsuitable” is that For arbitrary combinations of physical masses (or theyinvolvescatteringstates(whicharenotoriouslydiffi- • barekinetic masses)andvertex masses,the transi- cult for most non-perturbative numerical algorithms) or 7 they involve matrix elements of “bad” currents (which such as Tamm-Dancoff truncations of longitudinal mo- are often ill defined in the LF formalism). Other observ- mentumcutoffsorenergycutoffsmustbe takenlarge(or ables, which seem a priory sensitive to parity violations small)enoughsothattheynolongeraffectthestatesun- of the formalism are often “protected” by manifest LF der consideration. This is hard, but not impossible! For symmetries, such as C-parity. example,onasmalltransverselattice,onecangetrather We found one observable which is both sensitive to closetothelongitudinalcontinuumlimitinpracticalcal- parity violations and easily accessible in a standard LF culations. However,inexampleswereonecannotremove calculation: inelasticmatrixelementsofthegoodcompo- thosecutoffs,onemustcarefullyrenormalizethecurrents nentofthecurrentoperator. Vectorcurrentconservation before calculating the necessary matrix elements. demands Eventhough we focused on inelastic transition matrix elements of the good component of the vector current, q− m,p′ j+(0)n,p +q+ m,p′ j−(0)n,p =0, (5.1) there are more “useful” parity relations beyond the in- h | | i h | | i elastic vector transition matrix element. For example, while parity invariance implies thereexistalsosomeuseful“parity-relations”amongvir- m,p′ j−(0)n,p = m,p¯′ j+(0)n,p¯, (5.2) tual Compton amplitudes which may also be useful in h | | i ∓h | | i the contextofQCD.However,a detailedinvestigationof where p¯± p∓ = M2/2p±, p¯′± p′∓ = M2/2p′± are such observables will be postponed to some forthcoming ≡ n ≡ m paper. the parity transformed momenta of the initial and final Inanycase,parityrelationssuchastheonesderivedin state and the sign in Eq.(5.2) depends on whether the this papershouldbe usefulinthe searchforP− ,since two states m and n have the same or opposite intrinsic QCD they imply a strongconsistency checkfor the states that parity. Eqs.(5.1)and (5.2) individually are useless, since arise as solutions from diagonalizing P− . they involve a bad current matrix element. However, QCD notice that the bad current matrix element in Eq. (5.1) and in Eq. (5.2) is the same. The trick is to use Eq. 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