Effective Field Theory and χpt 0 0 Barry R. Holstein 0 2 n Department of Physics and Astronomy a J University of Massachusetts 7 Amherst, MA 01003 2 1 February 1, 2008 v 1 8 2 Abstract 1 0 0 A brief introduction to the subject of chiral perturbation theory 0 (χpt) is given, including a discussion of effective field theory and ap- / h plication to the upcoming Bates virtual Compton scattering measure- p ment. - p e h : v i X r a 0 1 Introduction We have gathered to celebrate the fact that Bates has been delivering beam successfully for twenty five years and to review some of the things which have been learned and which are still to be studied. One thing that has changed theoretically during this period is that we now have a new paradigm for anal- ysis of low energy processes such as studied at Bates. I was a student in the 1960’s and at that time our goal was to attempt to find a renormalizable field theory which describes all particle interactions with the same sort of success as quantum electrodynamics (QED). In 1967 we went part of the way with development of the Weinberg-Salam theory, which incorporated the weak interaction as a sibling to the electromagnetic. Because the inter- action was weak it could be treated via the same perturbative techniques as coulditselectromagnetic kinandwhathasresultedisanextremely successful description of all weak and electromagnetic processes. Forthestronginteractionsarenormalizablepicturehasalsobeendeveloped— quantum chromodynamics or QCD. The theory is, of course, deceptively simple on the surface. Indeed the form of the Lagrangian1 1 = q¯(iD m)q tr G Gµν . (3) LQCD 6 − − 2 µν is elegant, and the theory is renormalizable. So why are we not satisfied? While at the very largest energies, asymptotic freedom allows the use of per- turbative techniques, for those who are interested in making contact with low energy experimental findings there exist at least three fundamental dif- ficulties: i) QCD is written in terms of the ”wrong” degrees of freedom—quarks 1 Here the covariantderivative is λa iD =i∂ gAa , (1) µ µ− µ 2 where λa (with a=1,...,8) are the SU(3) Gell-Mann matrices, operating in color space, and the color-field tensor is defined by G =∂ A ∂ A g[A ,A ], (2) µν µ ν ν µ µ ν − − 1 andgluons—whilelowenergyexperiments areperformedwithhadronic bound states; ii) the theory is non-linear due to gluon self interactions; iii) the theory is one of strong coupling—g2/4π 1—so that perturbative ∼ methods are not practical. Nevertheless, therehasbeen agreat dealofrecent progress inmaking contact betweentheoryandexperiment usingthetechniqueof”effectivefieldtheory”, which exploits the chiral symmetry of the QCD interaction. In order to understandhowthisisaccomplished, weshallfirstreviewthisideaofeffective field theory in the simple context of quantum mechanics. Then we show how these ideas can be married via chiral perturbation theory and indicate applications at Bates. 2 Effective Field Theory The power of effective field theory is associated with the feature that there exist many situations in physics involving two scales, one heavy and one light. Then, provided one is working at energies small compared to the heavy scale, it is possible to fully describe the interactions in terms of an “effective” picture, which is written only in terms of the light degrees of freedom, but which fully includes the influence of the heavy mass scale through virtual effects. A number of very nice review articles on effective field theory can be found in ref. [1]. Before proceeding to QCD, however, it is useful to study this idea in the simpler context of ordinary quantum mechanics, in order to get familiar with the concept. Specifically, we examine the question of why the sky is blue, whose answer can be found in an analysis of the scattering of photons from the sun by atoms in the atmosphere—Compton scattering[2]. First we examine the problem using traditional quantum mechanics and consider elastic (Rayleigh) scattering from, for simplicity, single-electron (hydrogen) atoms. The appropriate Hamiltonian is then (p~ eA~)2 H = − +eφ (4) 2m 2 (a) (b) (c) Figure 1: Feynman diagrams for nonrelativistic photonl-atom scattering. and the leading— (e2)—amplitude for Compton scattering is found from O calculating the diagrams shown in Figure 1, yielding the familiar Kramers- Heisenberg form e2/m 1 ǫˆ∗ < 0 p~e−iq~f·~r n > ǫˆ < n p~eiq~i·~r 0 > Amp = ˆǫ ǫˆ∗ + f· | | i· | | −√2ωi2ωf " i · f m n ωi +E0 En X − ǫˆ < 0 p~eiq~i·~r n > ǫˆ∗ < n p~e−iq~f·~r 0 > + i· | | f· | | (5) E0 ωf En !# − − where 0 > represents the hydrogen ground state having binding energy E . 0 | Here the leading component is the familiar ω-independent Thomson am- plitude and would appear naively to lead to an energy-independent cross- section. However, this is not the case. Indeed, by expanding in ω and using a few quantum mechanical identities one can show that, provided that the energy of the photon is much smaller than a typical excitation energy—as is the case for optical photons, the cross section can be written as dσ ω2 = λ2ω4 ǫˆ∗ ǫˆ 2 1+ (6) dΩ | f · i| O (∆E)2!! where 2 z 2 n0 λ = α | | (7) em E E n 0 X − is the atomic electric polarizability, α = e2/4π is the fine structure con- em stant, and∆E mα2 is atypical hydrogen excitationenergy. We notethat ∼ em α λ a2 αem a3 is of order the atomic volume, as will be exploited em ∼ 0 × ∆E ∼ 0 below, and that the cross section itself has the characteristic ω4 dependence 3 whichleadstothebluenessofthesky—bluelightscattersmuchmorestrongly than red[3]. Now while the above derivation is certainly correct, it requires somewhat detailed and lengthy quantum mechanical manipulations which obscure the relatively simple physics involved. One can avoid these problems by the use of effective field theory methods. The key point is that of scale. Since the incident photons have wavelengths λ 5000A much larger than the ∼ 1A atomic size, then at leading order the photon is insensitive to the ∼ presence of the atom, since the latter is electrically neutral. If χ represents the wavefunction of the atom then the effective leading order Hamiltonian is simply ~p2 H(0) = χ∗ +eφ χ (8) eff 2m ! and there is no interaction with the field. In higher orders, there can ex- ist such atom-field interactions and this is where the effective Hamiltonian comes in to play. In order to construct the effective interaction, we demand certain general principles—this Hamiltonian must satisfy fundamental sym- metry requirements. In particular H must be gauge invariant, must be a eff scalar under rotations, and must be even under both parity and time reversal transformations. Also, since we are dealing with Compton scattering, H eff should be quadratic in the vector potential. Actually, from the requirement of gauge invariance, it is clear that the effective interaction can utilize A~ only via the electric and magnetic fields, rather than the vector potential itself— ∂ E~ = ~ φ A~, B~ = ~ A~ (9) −∇ − ∂t ∇× since these are invariant under a gauge transformation ∂ ~ ~ ~ φ φ+ Λ, A A Λ (10) → ∂t → −∇ while the vector and/or scalar potentials are not. The lowest order inter- action then can involve only the rotational invariants E~2,B~2 and E~ B~. · However, under spatial inversion—~r ~r—electric and magnetic fields be- → − have oppositely—E~ E~ while B~ B~—so that parity invariance rules out any dependence→on−E~ B~. Lik→ewise under time reversal invariance · 4 E~ E~, B~ B~ so such a term is also T-odd. The simplest such effective → → − Hamiltonian must then have the form 1 1 H(1) = χ∗χ[ c E~2 c B~2] (11) eff −2 E − 2 B (Terms involving time or spatial derivatives are much smaller.) We know from electrodynamics that 1(E~2 + B~2) represents the field energy per unit 2 volume, so by dimensional arguments, in order to represent an energy in Eq. 11, c ,c must have dimensions of volume. Also, since the photon has E B such a long wavelength, there is no penetration of the atom, so only classical scattering is allowed. The relevant scale must then be atomic size so that we can write c = k a3, c = k a3 (12) E E 0 B B 0 where we anticipate k ,k (1). Finally, since for photons with polariza- E B ∼ O tion ǫˆ and four-momentum q we identify A~(x) = ǫˆexp( iq x), then from µ − · Eq. 9, E~ ω, B~ ~k = ω and | | ∼ | | ∼ | | dσ < f H i > 2 ω4a6 (13) dΩ ∝ | | eff| | ∼ 0 as found in the previous section via detailed calculation. This is a nice example of the power of simple effective field theory arguments. 3 Application to QCD: Chiral Perturbation Theory Now let’s apply these ideas to the case of QCD. In this case the invariance we wish to exploit is “chiral symmetry.” The idea of ”chirality” is defined by the operators 1 1 1 1 Γ = (1 γ ) = ∓ (14) L,R 2 ± 5 2 1 1 ! ∓ which project “left-” and “right-handed” components of the Dirac wavefunc- tion via ψ = Γ ψ ψ = Γ ψ with ψ = ψ +ψ (15) L L R R L R 5 IntermsofthesechiralitystatesthequarkcomponentoftheQCDLagrangian can be written as q¯(iD m)q = q¯ iDq +q¯ iDq q¯ mq q¯ mq (16) L L R R L R R L 6 − 6 6 − − The reason that these chirality states are called left- and right-handed is that in the limit m 0 they coincide with quark helicity projection operators. → With this background, we note that QCD, in the mathematical limit as m 0 has the structure → m=0 q¯ iDq +q¯ iDq (17) QCD L L R R L −→ 6 6 and is invariant under independent global left- and right-handed rotations q exp(i λ α )q , q exp(i λ β )q (18) L j j L R j j R → → j j X X This invariance is called SU(3) SU(3) or chiral SU(3) SU(3). Con- L R × tinuing to neglect the light quark masses, we see that in a chiral symmetric N world one would expect to have sixteen—eight left-handed and eight right- handed—conserved Noether currents 1 1 q¯ γ λ q , q¯ γ λ q (19) L µ i L R µ i R 2 2 Equivalently, bytaking thesumanddifference we wouldhaveeight conserved vector and eight conserved axial vector currents 1 1 Vi = q¯γ λ q, Ai = q¯γ γ λ q (20) µ µ2 i µ µ 52 i In the vector case, this is just a simple generalization of isospin (SU(2)) invariance to the case of SU(3). There exist eight (32 1) time-independent − charges F = d3xVi(~x,t) (21) i 0 Z and there exist various supermultiplets of particles having identical spin- parity and (approximately) the same mass in the configurations—singlet, octet, decuplet, etc. demanded by SU(3)-invariance. If chiral symmetry were realized in the conventional fashion one would expect there also to exist corresponding nearly degenerate same spin but 6 opposite parity states generated by the action of the time-independent axial charges F5 = d3xAi(~x,t) on these states. However, it is known that the ax- i 0 ial symmetry is broken spontaneously, whereby Goldstone’s theorem requires R theexistence ofeight massless pseudoscalar bosons, which couplederivatively to the rest of the universe[4]. Of course, in the real world such massless 0− states do not exist, because in the real world exact chiral invariance is bro- ken by the small quark mass terms which we have neglected up to this point. Thus what we have are eight very light (but not massless) pseudo-Goldstone bosons which make up the pseudoscalar octet. Since such states are lighter than their other hadronic counterparts, we have a situation wherein effective field theory can be applied—provided one is working at energy-momenta small compared to the 1 GeV scale which is typical of hadrons, one can ∼ describe the interactions of the pseudoscalar mesons using an effective La- grangian. Actually this has been known since the 1960’s, where a good deal of work was done with a lowest order effective chiral Lagrangian[5] F2 m2 = πTr(∂ U∂µU†)+ πF2Tr(U +U†). (22) L2 4 µ 4 π where thesubscript 2indicates thatwe areworking attwo-derivative order or one power of chiral symmetry breaking—i.e. m2. Here U exp( λ φ /F ), π ≡ i i π where F = 92.4 is the pion decay constant. This Lagrangian is unique—if π P ~ we expand to lowest order in φ i i 2 Tr∂ U∂µU† = Tr ~τ ∂ φ~ − ~τ ∂µφ~ = ∂ φ~ ∂µφ~ µ µ µ F · × F · F2 · π π π 1 2 Tr(U +U†) = Tr(2 ~τ φ~~τ φ~) = const. φ~ φ~ (23) − F2 · · − F2 · π π we reproduce the free pion Lagrangian, as required, At the SU(3) level, including an appropriately generalized chiral symme- try breaking term, there is even predictive power—one has F2 1 8 πTr∂ U∂µU† = ∂ φ ∂µφ + (24) µ µ j j 4 2 ··· j=1 X F2 1 3 πTr2B m(U +U†) = const. (m +m )B φ2 4 0 − 2 u d 0 j j=1 X 7 1 7 1 (m +m +2m )B φ2 (m +m +4m )B φ2 + − 4 u d s 0 j − 6 u d s 0 8 ··· j=4 X (25) where B is a constant and m is the quark mass matrix. We can then identify 0 the meson masses as m2 = 2mˆB π 0 m2 = (mˆ +m )B K s 0 2 m2 = (mˆ +2m )B , (26) η 3 s 0 where mˆ = 1(m +m ) is the mean light quark mass. This system of three 2 u d equations is overdetermined, and we find by simple algebra 3m2 +m2 4m2 = 0 . (27) η π − K whichistheGell-Mann-Okubomassrelationandiswell-satisfiedexperimentally[6]. Expanding to fourth order in the fields we also reproduce the well-known and experimentally successful Weinberg ππ scattering lengths[7] 7m2 m2 m2 a0 = π , a2 = π , a1 = π (28) 0 32πF2 0 −16πF2 1 24πF2 π π π However, when one attempts to go beyond tree level in order to unitarize the results, divergences arise and that is where the field stopped at the end of the 1960’s. The solution, as pointed out ten years later by Weinberg[8] and carried out by Gasser and Leutwyler[9], is to absorb these divergences in phenomenological constants, just as done in QED. A new wrinkle in this case is that the theory is nonrenormalizabile in that the forms of the divergences are different from the terms that one started with. That means that the form of the counterterms that are used to absorb these divergences must also be different, and Gasser and Leutwyler wrote down the most general counterterm Lagrangian that one can have at one loop, which involves four- derivative interactions 10 2 = L = L tr(D UDµU†) +L tr(D UD U†) tr(DµUDνU†) 4 i i 1 µ 2 µ ν L O · i=1 (cid:20) (cid:21) X 8 + L tr(D UDµU†D UDνU†)+L tr(D UDµU†)tr(χU† +Uχ†) 3 µ ν 4 µ 2 + L tr D UDµU† χU† +Uχ† +L tr χU† +Uχ† 5 µ 6 (cid:16) (cid:16) (cid:17)(cid:17) (cid:20) (cid:16) (cid:17)(cid:21) 2 + L tr χ†U Uχ† +L tr χU†χU† +Uχ†Uχ† 7 8 − (cid:20) (cid:16) (cid:17)(cid:21) (cid:16) (cid:17) + iL tr FLDµUDνU† +FRDµU†DνU +L tr FLUFRµνU† 9 µν µν 10 µν (cid:16) (cid:17) (cid:16) (cid:17)(29) where the covariant derivative is defined via D U = ∂ U + A ,U +[V ,U] (30) µ µ µ µ { } the constants L ,i = 1,2,...10 are arbitrary (not determined from chiral i symmetry alone) and FL,FR are external field strength tensors defined via µν µν FL,R = ∂ FL,R ∂ FL,R i[FL,R,FL,R], FL,R = V A . (31) µν µ ν − ν µ − µ ν µ µ ± µ Now just as in the case of QED the bare parameters L which appear in this i Lagrangian are not physical quantities. Instead the experimentally relevant (renormalized) valuesofthese parametersareobtainedbyappending tothese bare values the divergent one-loop contributions— γ 2 Lr = L i − ln(4π)+γ 1 (32) i i − 32π2 ǫ − − (cid:20) (cid:21) By comparing predictions with experiment, Gasser and Leutwyler were able to determine empirical values for each of these ten parameters. Typical results are shown in Table 1, together with the way in which they were determined. The important question to ask at this point is why stop at order four derivatives? Clearly if two-loop amplitudes from or one-loop 2 L corrections from are calculated, divergences will arise which are of six- 4 L derivative character. Why not include these? The answer is that the chiral procedure represents an expansion in energy-momentum. Corrections to the lowest order (tree level) predictions from one-loopcorrections from or tree 2 L level contributions from are (E2/Λ2) where Λ 4πF 1 GeV is the L4 O χ χ ∼ π ∼ chiral scale[10]. Thus chiral perturbation theory is a low energy procedure. It is only to the extent that the energy is small compared to the chiral scale thatitmakessensetotruncatetheexpansionattheone-loop(four-derivative) 9