CHIRAL DYNAMICS IN NUCLEONS AND NUCLEI

V. Bernard

Centre de Recherche Nucleaire et Université Louis Pasteur de Strasbourg

Physique Théorique, BP 28, F–67037 Strasbourg Cedex 2, France

N. Kaiser

Technische Universität München, Physik Department T30

James-Franck-Straße, D–85747 Garching, Germany

Ulf-G. Meißner

Universität Bonn, Institut für Theoretische Kernphysik,

Nussallee 14–16, D–53115 Bonn, Germany

email: ,

ABSTRACT:

We review the implications of the spontaneous chiral symmetry breaking in QCD

for processes involving one, two or more nucleons.

Commissioned article for Int. J. Mod. Phys. E

CRN 95/3

TK 95 1 February 1995

CONTENTS

1. INTRODUCTION 2 2. CHIRAL SYMMETRY IN QCD 8 2.1.  Elementary introduction to chiral symmetry 8

2.2.  Three–flavor QCD 9

2.3.  Chiral perturbation theory 11

2.4.  Modelling the pion 21 3. THE PION–NUCLEON SYSTEM 26 3.1.  Effective Lagrangian 26

3.2.  Extreme non–relativistic limit 33

3.3.  Renormalization 38

3.4.  Low–energy constants and the role of the 42

3.5.  Aspects of pion–nucleon scattering 46

3.6.  The reaction 50

3.7.  The pion–nucleon vertex 54 4. NUCLEON STRUCTURE FROM ELECTROWEAK PROBES 58 4.1.  Electromagnetic form factors of the nucleon 58

4.2.  Nucleon Compton scattering 63

4.3.  Axial properties of the nucleon 76

4.4.  Threshold pion photo– and electroproduction 79

4.5.  Two–pion production 88

4.6.  Weak pion production 94 5. THE NUCLEON–NUCLEON INTERACTION 102 5.1.  General considerations 102

5.2.  Nucleon–nucleon potential 106

5.3.  More than two nucleons 112

5.4.  Three–body interactions between nucleons, pions and photons 114

5.5.  Exchange currents 118 6. THREE FLAVORS, DENSE MATTER AND ALL THAT 122 6.1.  Flavor SU(3), baryon masses and –terms 122

6.2.  Kaon—nucleon scattering 130

6.3.  The pion in matter 134

6.4.  Miscellaneous omissions 138  APPENDICES 142

I.  INTRODUCTION Effective field theories (EFTs) have become a popular tool in particle and nuclear physics. An effective field theory differs from a conventional renormalizable (”fundamental”) quantum field theory in the following respect. In EFT, one only works at low energies (where ”low” is defined with respect to some scale specified later) and expands the theory in powers of the energy/characteristic scale. In that case, renormalizability at all scales is not an issue and one has to handle strings of non–renormalizable interactions. Therefore, at a given order in the energy expansion, the theory is specified by a finite number of coupling (low–energy) constants (this allows e.g. for an order–by–order renormalization). All observables are parametrized in terms of these few constants and thus there is a host of predictions for many different processes. Obviously, at some high energy this effective theory fails and one has to go over to a better theory (which again might be an EFT of some fundamental theory). The trace of this underlying theory are the particular values of the constants. The EFT presumably studied in most detail is chiral perturbation theory (CHPT). The central topic of this review will be the application of this framework when nucleons (baryons) are present, with particular emphasis on processes with exactly one nucleon in the initial and one nucleon in the final state. Before elaborating on these particular aspects of CHPT, it is useful to make some general comments concerning the applications of EFTs. EFTs come into play when the underlying fundamental theory contains massless (or very light) particles. These induce poles and cuts and conventional Taylor expansions in powers of momenta fail. A typical example is QED where gauge invariance protects the photon from acquiring a mass. One photon exchange involves a propagator , with the invariant four–momentum transfer squared. Such a potential can not be Taylor expanded. A classic example to deal with such effects is the work of Euler and Heisenberg [1.1] who considered the scattering of light by light at very low energies, , with the photon energy and the electron mass. To calculate the scattering amplitude, one does not need full QED but rather integrates out the electron from the theory. This leads to an effective Lagrangian of the form

 Leff=12(→E2−→B2)+e4360π2m4e[(→E2−→B2)2+7(→E⋅→B)2]+…

which is nothing but a derivative expansion since and contain derivatives of the gauge potential. Stated differently, since the photon energy is small, the electromagnetic fields are slowly varying. From eq.(1.1) one reads off that corrections to the leading term are suppressed by powers of . Straightforward calculation leads to the cross section . This can, of course, also be done using full QED [1.2], but the EFT caculation is much simpler. The results of [1.2] nicely agree with the ones making use of the EFT for . A similar situation arises in QCD which is a non–abelian gauge theory of colored quarks and gluons,

 LQCD=−14g2GaμνGμν,a+¯qiγμDμq−¯qMq=L0QCD+LIQCD

with the quark mass matrix. For the full theory, there is a conserved charge for every quark flavor separately since the quark masses are all different. However, for the first three flavors ( it is legitimate to set the quark masses to zero since they are small on a typical hadronic scale like e.g. the –meson mass. The absolute values of the running quark masses at 1 GeV are MeV, MeV, MeV, i.e. , and [1.3]. If one sets the quark masses to zero, the left– and right–handed quarks defined by

 qL=12(1−γ5)q,qR=12(1+γ5)q

do not interact with each other and the whole theory admits an symmetry. This is further reduced by the axial anomaly, so that the actual symmetry group of three flavor massless QCD is

 G=SU(3)L×SU(3)R×U(1)L+R

The U(1) symmetry related to baryon number conservation will not be discussed in any further detail. The conserved charges which come along with the chiral symmetry generate the corresponding Lie algebra. In the sixties and seventies, manipulations of the commutation relations between the conserved vector (L+R) and axial–vector (L-R) charges were called ”PCAC relations” or ”current algebra calculations” and lead to a host of theorems and predictions [1.4]. These rather tedious manipulations have nowadays been replaced by EFT methods, in particular by CHPT (as will be discussed later on). Let us come back to QCD. One quickly realizes that the ground state does not have the full symmetry , eq.(1.4). If that were the case, every known hadron would have a partner of the same mass but with opposite parity. Clearly, this is in contradiction with the observed particle spectrum. Further arguments that the chiral symmetry is not realized in the Wigner–Weyl mode are given in section 2. The physical ground state must therefore be asymmetric under the chiral [1.5]. In fact, the chiral symmetry is spontaneously broken down (hidden) to the vectorial subgroup of isospin and hypercharge, generated by the vector currents,

 H=SU(3)L+R×U(1)L+R

As mandated by Goldstone’s theorem [1.6], the spectrum of massless QCD must therefore contain massless bosons with quantum numbers (pseudoscalars) since the axial charges do not annihilate the vacuum. Reality is a bit more complex. The quark masses are not exactly zero which gives rise to an explicit chiral symmetry breaking (as indicated by the term in eq.(1.2)). This is in agreement with the observed particle spectrum – there are no massless strongly interacting particles. However, the eight lightest hadrons are indeed pseudoscalar mesons. These are the pions (), the kaons (

REFERENCES 1.1H. Euler, Ann. Phys. (Leipzig) 26 (1936) 398;

H. Euler and W. Heisenberg, Z. Phys. 98 (1936) 714. 1.2R. Karplus and M. Neuman, Phys. Rev. 83 (1951) 776. 1.3J. Gasser and H. Leutwyler, Phys. Reports C87 (1982) 77. 1.4S. L. Adler and R. F. Dashen, ”Current Algebras and applications to particle physics”, Benjamin, New York, 1968. 1.5Y. Nambu and G. Jona–Lasinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246. 1.6J. Goldstone, Nuovo Cim. 19 (1961) 154;

J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965. 1.7H. Pagels, Phys. Reports 16 (1975) 219. 1.8R. Dashen and M. Weinstein, Phys. Rev. 183 (1969) 1261. 1.9S. Weinberg, Phys. Rev. 166 (1968) 1568. 1.10S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239;

C. G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247. 1.11S. Weinberg, Physica 96A (1979) 327. 1.12J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158 (1984) 142. 1.13J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465. 1.14H. Leutwyler, Ann. Phys. (N.Y.) 235 (1994) 165. 1.15J. Gasser, M.E. Sainio and A. varc, Nucl. Phys. B307 (1988) 779. 1.16E. Jenkins and A.V. Manohar, Phys. Lett. 255 (1991) 558. 1.17S. Weinberg, Phys. Lett. 251 (1990) 288.

II. CHIRAL SYMMETRY OF THE STRONG INTERACTIONS

In this section, we first discuss chiral symmetry on an elementary level. We extend these considerations to three–flavor QCD and the formulation of its effective low–energy field theory in terms of the Goldstone bosons related to the spontaneous chiral symmetry breaking. We also outline briefly how the structure of the pion can be modeled in a four–quark interaction cut–off theory of the Nambu–Jona-Lasinio type. II.1. ELEMENTARY INTRODUCTION TO CHIRAL SYMMETRY Before discussing full QCD, let us give a few very introductory remarks about chiral symmetry. The reader familiar with this concept is invited to skip this section. To be specific, consider a free and massless spin–1/2 (Dirac) field,

 L[Ψ]=i¯Ψγμ∂μΨ

The state of a free relativistic fermion (of arbitrary mass) is completely characterized by its energy , its momentum and its helicity

 Ψ=12(1−γ5)Ψ+12(1+γ5)Ψ=PLΨ+PRΨ=ΨL+ΨR

Obviously, the operators are projectors,

 P2L=PL,P2R=PR,PL⋅PR=0,PL+PR=1

with the property**For a massive fermion, the are still projectors but do not yield exactly the helicity.

 12^hΨL,R=±12ΨL,R

This shows that the states are helicity eigenstates. In terms of these fields, the Lagrangian (2.1) takes the form

 L[ΨL,ΨR]=i¯ΨLγμ∂μΨL+i¯ΨRγμ∂μΨR

One notices that the left– and right–handed fermion modes do not communicate. Stated differently, one can apply separate transformations which leave the Lagrangian invariant,

 ΨL→eiϵLΨL,ΨR→eiϵRΨR

leading to conserved left– and right–handed currents,

 JIμ=¯ΨIγμΨI,I=L,R

with

 ∂μJμ,I=0,I=L,R

Equivalently, one can construct conserved vector and axial–vector currents,

 Vμ=¯ΨγμΨ,∂μVμ=0Aμ=¯Ψγμγ5Ψ,∂μAμ=0

since . To reiterate, chiral symmetry means that for massless fermions chirality is a constant of motion. A fermion mass term explicitely breaks this symmetry since it mixes the left– and right–handed components,

 ¯ΨMΨ=¯ΨLMΨR+¯ΨRMΨL

To make chiral symmetry a viable concept for massive fermions, the corresponding eigenvalues of the mass matrix have to be small compared to a typical energy scale of the system under consideration. As an example, we will now consider the case of three–flavor Quantumchromodynamics (QCD).

II.2. THREE–FLAVOR QCD The standard model of the strong, electromagnetic and weak interactions involves three generations of fermion doublets, alas six different quark flavors. From these six quark types, three are labelled ’light’ () and the other three ’heavy’ (). Here light and heavy refers to a typical hadronic scale GeV. In fact, and whereas typical values of the light quark masses at a renormalization point of 1 GeV are [2.1]

 md=5±2MeV,md=9±3MeV,ms=175±55MeV

Note that there exist some controversy about these values, for reviews with detailed references see e.g. [2.2,2.3,2.4]. In the three–flavor sector, the QCD Lagrangian takes the form

 LQCD=−12g2GaμνGμν,a+¯qiγμ(∂μ−iGμ)q−¯qMq−Θ16π2Gaμν~Gμν,a

with , the gluon field, the corresponding field strength tensor and its dual. The last term in (2.11) is related to the strong CP–problem. In what follows, we will set . The quark mass matrix can be chosen to be diagonal,

 M=diag(mu,md,ms)

In (2.11), we have not made explicit the generators related to the local transformations. From the chiral symmetry point of view we rewrite (2.11) as

 LQCD=L0QCD−¯qMq

and is invariant under the global transformations of the group

 G=SU(3)L×SU(3)R×U(1)V×U(1)A

Projecting onto left– and right–handed quark fields, , these transform under the chiral group as

 qL→eiTaαaLqL,qR→eiTaαaRqR,a=1,…,8

with the generators (a=1,,8) given in terms of the Gell-Mann SU(3) matrices via with . In what follows, we will not be concerned with the vectorial symmetry related to the baryon current and the anomalous current. It is believed that the axial is broken by instanton effects [2.5]. To the global symmetry of one associates conserved currents,

 Vaμ=¯qγμTaq,∂μVμ,a=0Aaμ=¯qγμγ5Taq,∂μAμ,a=0

with the corresponding conserved charges

Of course, in the presence of quark mass terms, this symmetry is explicitely broken.

One might now ask the question whether this chiral symmetry is also manifest in the ground state or the particle spectrum of QCD? In fact, there are numerous indications that this is not the case. The realization of the chiral symmetry in the Wigner mode (i.e. all generators defined in (2.17) annihilate the vacuum) would lead to degenerate hadron doublets of opposite parity in plain contradiction to the observed spectrum. Furthermore, in the Wigner phase the vector–vector and axial-vector–axial-vector correlators in the ground state would be equal,

 <0|Vaμ(x)Vbν(y)|0>=<0|Aaμ(x)Abν(y)|0>.

These correlators can be extracted from decay data, with even (odd) containing the information about the VV (AA) correlation function. As shown in refs.[2.6], the VV correlator strongly peaks around GeV whereas the AA correlator has a broad maximum around GeV. From that and the approximate flavour SU(3) symmetry of the hadron spectrum we conclude that the chiral symmetry is spontaneously broken down to its vectorial subgroup,

 SU(3)L×SU(3)R→SU(3)V

with the appearance of massless pseudoscalar mesons, the Goldstone bosons [2.7]. These are the analog to the spin waves in a ferromagnet which underwent spontaneous magnetization (thus breaking the rotational symmetry of the magnet Hamiltonian). In nature, however, these Goldstone bosons are not exactly massless but acquire a small mass due to the explicit symmetry breaking from the quark masses, , where is a generic symbol for the three pions, the four kaons and the eta. From the sytematics of the hadron spectrum, we can immediately conclude that since the pions do not contain any strange quarks. These Goldstone bosons are in fact the lightest observed hadrons and they saturate the pertinent Ward identities of the strong interactions at low energies. To calculate QCD Green functions in the non–perturbative regime, one therefore makes use of an effective field theory (EFT) with the pseudoscalar mesons as the relevant degrees of freedom. The essential feature which makes this EFT amenable to a systematic perturbative expansion is the fact that the interaction between the Goldstone bosons at low energies is weak. To be more precise, consider the elastic scattering process (for massless pions) [2.8]

 T(π+π0→π+π0)=tF2π

with the invariant four–momentum transfer squared. Indeed, as approaches zero, the Goldstone boson interaction vanishes. This fact is at the heart of the systematic low energy expansion in terms of small momenta and quark masses - chiral perturbation theory (CHPT) - as discussed in some detail in the next section. For a more detailed account see e.g. the monograph [2.9], the original papers by Gasser and Leutwyler [2.10] or the review [2.3]. II.3.  CHIRAL PERTURBATION THEORY In this section, we briefly review how to construct the effective chiral Lagrangian of the strong interactions at next–to–leading order, following closely the work of Gasser and Leutwyler [2.10]. It is most economical to use the external field technique since it avoids any complication related to the non–linear transformation properties of the pions. The basic objects to consider are currents and densities with external fields coupled to them [2.11] in accordance with the symmetry requirements. The associated Green functions automatically obey the pertinent Ward identities and higher derivative terms can be constructed systematically. The S–matrix elements for processes involving physical mesons follow then via standard LSZ reduction. To be specific, consider the vacuum–to–vacuum transition amplitude in the presence of external fields

 eiZ[v,a,s,p]=<0out|0in>v,a,s,p

based on the QCD Lagrangian

 L=L0QCD+¯q(γμvμ(x)+γ5γμaμ(x))q−¯q(s(x)−ip(x))q

The external vector , axial–vector (, pseudoscalar and scalar fields are hermitean matrices in flavor space. The quark mass matrix (2.12) is contained in the scalar field . The Green functions of massless QCD are obtained by expanding the generating functional around . For the real world, one has to expand around . The Lagrangian is invariant even under local chiral transformations if the quark and external fields transform as follows:

 q′R=Rq;q′L=Lqv′μ+a′μ=R(vμ+aμ)R†+iR∂μR†v′μ−a′μ=L(vμ−aμ)L†+iL∂μL†s′+ip′=R(s+ip)L†

with elements of (in general, these are elements of , but we already account for the axial anomaly to be discussed later). The path integral representation of reads:

 eiZ[v,a,s,p]=∫[DGμ][Dq][D¯q]e∫id4xL(q,¯q,Gμν;v,a,s,p)

It allows one to make contact to the effective meson theory. Since we are interested in processes were the momenta are small (the low energy sector of the theory), we can expand the Green functions in powers of the external momenta. This amounts to an expansion in derivatives of the external fields. This low energy expansion is not a simple Taylor expansion since the Goldstone bosons generate poles at (in the chiral limit) or (for finite quark masses). The expansion involves two small parameters, the external momenta and the quark masses (or the Goldstone masses ). One expands in powers of these with the ratio fixed. The effective meson Lagrangian to carry out this procedure follows from the representation of the generating functional

 eiZ[v,a,s,p]=∫[DU]e∫id4xLeff(U;v,a,s,p)

where the matrix collects the pseudoscalar Goldstone fields. The low energy expansion is now obtained from a perturbative expansion of the meson EFT,

 Leff=L2+L4+…

where the subscript (, 4, ) denotes the low energy dimension (number of derivatives and/or quark mass terms). Let us now discuss the various terms in this expansion. The leading term (called ) in the expansion (2.26) can easily be written down in terms of the mesons which are described by a unitary matrix in flavor space,

 U†U=1,detU=1

The matrix transforms linearly under chiral symmetry, . The lowest order Lagrangian consistent with Lorentz invariance, chiral symmetry, parity, G–parity and charge conjugation reads [2.10]

 L2=14F2{Tr[∇μU†∇μU+χ†U+χU†]}

The covariant derivative transforms linearly under chiral and contains the couplings to the external vector and axial fields,

 ∇μU=∂μU−i(vμ+aμ)U+iU(vμ−aμ)

The field embodies the scalar and pseudoscalar externals,

 χ=2B(s+ip)

There are two constants appearing in eqs.(2.28,2.30). The scale is related to the axial vector currents, and thus can be identified with the pion decay constant in the chiral limit, , by direct comparison with the matrix–element .**Strictly speaking the axial-axial correlator in the vacuum has a pion pole term with its residuum given by . The constant , which appears in the field , is related to the explicit chiral symmetry breaking. Consider the symmetry breaking part of the Lagrangian and expand it in powers of the pion fields (with , so that )

 LSB2=12F2BTr[M(U+U†)]=(mu+md)B[F2−π22+π424F2+O(π6)]+…

where the ellipsis denotes the contributions for the kaons and the eta. The first term on the right hand side of eq.(2.31) is obviously related to the vacuum energy, while the second and third are meson mass and interaction terms, respectively. Since it follows from (2.31) that

 <0|¯uu|0>=<0|¯dd|0>=<0|¯ss|0>=−F2B{1+O(M)}

This shows that the constant is related to the vev’s of the scalar quark densities , the order parameter of the spontaneous chiral symmetry breaking. The relation (2.32) is only correct modulo higher order corrections in the quark masses as indicated by the term . One can furthermore read off the pseudoscalar mass terms from (2.31). In the case of isospin symmetry (, one finds

with denoting the leading term in the quark mass expansion of the pseudoscalar meson masses. For the , the Gell–Mann–Okubo relation is exact, . In the case of isospin breaking, which leads to mixing, these mass formulae are somewhat more complicated (see e.g. ref.[2.10]). Eq.(2.33) exhibits nicely the Goldstone character of the pions – when the quark masses are set to zero, the pseudoscalars are massless and is an exact symmetry. For small symmetry breaking, the mass of the pions is proportional to the square root of the symmetry breaking parameter, i.e. the quark masses. From eqs.(2.31) and (2.33) one can eliminate the constant and gets the celebrated Gell–Mann–Oakes–Renner [2.12] relations

 F2πM2π=−2^m<0|¯uu|0>+O(M2)F2KM2K=−(^m+ms)<0|¯uu|0>+O(M2)F2ηM2η=−23(^m+ms)<0|¯uu|0>+O(M2)

where we have used (, , ), i.e. the differences in the physical decay constants appear in the terms of order in eq.(2.34). From this discussion we realize that to leading order the strong interactions are characterized by two scales, namely and . Numerically, using the QCD sum rule value MeV) one has and . The large value of the ratio has triggered some investigations of alternative scenarios concerning the mode of quark condensation [2.13].

One can now calculate tree diagrams using the effective Lagrangian and derive with ease all so–called current algebra predictions ( theorems). Current algebra is, as should have become evident by now, only the first term in a systematic expansion. Working out tree graphs using can not be the whole story – tree diagrams are always real and thus unitarity is violated. One has to include higher order corrections to cure this. To do this in a consistent fashion, one needs a counting scheme. The leading term in the expansion of (2.26) was denoted because it has dimension (chiral power) two. It contains two derivatives or one power of the quark mass matrix. If one assumes the matrix to be order one, , a consistent power counting scheme for local terms containing , , , , , , goes as follows. Denote by a generic small momentum (for an exact definition of ’small’, see below). Derivatives count as order and so do the external fields which occur linearly in the covariant derivative . For the scalar and pseudoscalar fields, it is most convenient to book them as order . This can be traced back to the fact that the scalar field contains the quark mass matrix, thus . With these rules, all terms appearing in (2.28) are of order , thus the notation (notice that a term of order one is a constant since and can therefore be disregarded. Odd powers of clash with parity requirements). To summarize, the building blocks of all terms containing derivatives and/or quark masses have the following dimension:

 ∂μU(x),vμ(x),aμ(x)=O(q)s(x),p(x),FL,Rμν(x)=O(q2)

where we have introduced the field strengths for later use. They are defined via

 FIμν=∂μFIν−∂νFIμ−i[FIμ,FIν],I=L,RFRμν=vμ+aμ;FLμν=vμ−aμ

As already mentioned, unitarity calls for pion loop graphs. Weinberg [2.14] made the important observation that diagrams with (, , ) meson loops are suppressed by powers of with respect to the leading term. His rather elegant argument goes as follows. Consider the S–matrix for a reaction involving external pions

 S=δ(p1+p2+…+pNe)M

with the transition amplitude. Now depends on the total momentum flowing through the amplitude, on the pertinent coupling constants and the renormalization scale (the loop diagrams are in general divergent and need to be regularized**It is advantegoeus to use dimensional regularization since it that case one avoids the appearance of power–law divergences.),

 M=M(q,g,μ)=qDf(q/μ,g)

with the total scaling dimension of given by [2.14]

 D=2+∑dNd(d−2)+2NL.

Here, is the number of pion loops and the number of vertices with derivatives (or quark mass insertions). The dominant graphs at low energy carry the smallest value of . The leading terms with scale like at tree level , like at one loop level () and so on. Higher derivative terms with scale as at tree level, as at one–loop order etc. This power suppression of loop diagrams is at the heart of the expansion in EFTs like e.g. chiral perturbation theory (CHPT).

Up to now, we have been rather casual with the meaning of the word ”small”. By small momentum or small quark mass we mean this with respect to some typical hadronic scale, also called the scale of chiral symmetry breaking (denoted by ). Georgi and Manohar [2.15] have argued that a consistent chiral expansion is possible if . Their argument is based on the observation that under a change of the renormalization scale of order one typical loop contributions (say to the scattering amplitude) will correspond to changes in the effective couplings of the order . Setting and cutting the logarithmically divergent loop integrals at this scale, quantum corrections are of the same order of magnitude as changes in the renormalized interaction terms. The factor is generic for one–loop integrals (in 3+1 dimensions). Another type of argument is related to the non–Goldstone spectrum. Consider scattering in the channel. There, at MeV, one hits the –resonance. This is a natural barrier to the derivative expansion of the Goldstone mesons and therefore serves as a cut off. The appearance of the signals the regime of the non–Goldstone particles and describes new physics. It is therefore appropriate to choose , which is not terribly different from the previous estimate. In summary, small external momenta and small quark masses means

We have now assembeld all tools to discuss the generating functional at next–to–leading order, i.e. at . It consists of three different contributions: (1) The anomaly functional is of order (it contains four derivatives). We denote the corresponding functional by . The explicit construction was given by Wess and Zumino [2.16] and can also be found in ref.[2.10]. A geometric interpretation is provided by Witten [2.17]. (2) The most general effective Lagrangian of order which is gauge invariant. It leads to the action . (3) One loop graphs associated with the lowest order term, . These also scale as terms of order .

Let us first discuss the anomaly functional . It subsumes all interactions which break the intrinsic parity and is responsible e.g. for the decay . It also generates interactions between five or more Goldstone bosons [2.17]. In what follows, we will not consider this sector in great detail (for a review, see ref.[2.18]).

What is now the most general Lagrangian at order ? The building blocks and their dimensions were already discussed – we can have terms with four derivatives or with two derivatives and one quark mass or with two quark masses (and, correspondingly, the other external fields). In , the only invariant tensors are and , so one is left with (imposing also P, G and gauge invariance) [2.10]

 L4=10∑i=1LiPi+2∑j=1Hj~Pj

with

 P1=Tr(∇μU†∇μU)2,P2=Tr(∇μU†∇νU)Tr(∇μU†∇νU)P3=Tr(∇μU†∇μU∇νU†∇νU),P4=Tr(∇μU†∇μU)Tr(χ†U+χU†)P5=Tr(∇μU†∇μU)(χ†U+χU†),P6=[Tr(χ†U+χU†)]2P7=[Tr(χ†U−χU†)]2,P8=Tr(χ†Uχ†U+χU†χU†)P9=−iTr(FRμν∇μU∇νU†)−iTr(FLμν∇μU†∇νU)P10=Tr(U†FRμνUFL,μν)~P1=Tr(FRμνFR,μν+FLμνFL,μν),~P2=Tr(χ†χ)

For the two flavor case, not all of these terms are independent. The pertinent effective Lagrangian is discussed in ref.[2.10]. The first ten terms of (2.40) are of physical relevance for the sector, the last two are only necessary for the consistent renormalization procedure discussed below. These terms proportional to , do not contain the Goldstone fields and are therefore not directly measurable at low energies. The constants appearing in (2.40) are the so–called low–energy constants. They are not fixed by the symmetry and have the generic structure

 Li=Lri+Linfi

These constants serve to renormalize the infinities of the pion loops and the remaining finite pieces have to be fixed phenomenologically or to be estimated by some model (see below). It should be noted that a few of the low–energy constants are in fact finite. At next–to–leading order, the strong interactions dynamics is therefore determined in terms of twelve parameters – , , , (remember that we have disregarded the singlet vector and axial currents). In the absence of external fields, only the first three terms in (2.40) have to be retained.

Finally, we have to consider the loops generated by the lowest order effective Lagrangian. These are of dimension (one loop approximation) as mandated by Weinberg’s scaling rule. To evaluate these loop graphs one considers the neighbourhood of the solution to the classical equations of motion. In terms of the generating functional, this reads

 eiZ=ei∫d4x[L2(¯U)+L4(¯U)]∫[DU]ei∫d4x[L2(U)−L2(¯U)]

The bar indicates that the Lagrangian is evaluated at the classical solution. According to the chiral counting, in the second factor of (2.43) only the term is kept. This leads to

 Z=∫d4x(¯L2+¯L4)+i2lndetD

The operator is singular at short distances. The ultraviolet divergences contained in can be determined via the heat kernel expansion. Using dimensional regularization, the UV divergences in four dimensions take the form

 −1(4π)21d−4Sp(12^σ2+112^Γμν^Γμν)

The explicit form of the operators and can be found in ref.[2.10]. Using their explicit expressions, the poles in can be absorbed by the following renormalization of the constants:

 Li=Lri+ΓiL,i=1,…,10Hj=Hrj+~ΓjL,j=1,2

with

 L=116π2λd−4{1d−4−12[ln(4π)+Γ′(1)+1]}Γ1=332,Γ2=316,Γ3=0,Γ4=18,Γ5=38,Γ6=11144,Γ7=0,Γ8=548,Γ9=14,Γ10=−14,~Γ1=−18,~Γ2=524,

and is the scale of dimensional regularization. The contribution is finite at when expressed in terms of the renormalized coupling constants and . The next step consists in the expansion of the differential operator in powers of the external fields. This gives the explicit contributions of the one–loop graphs to a given Green function. The full machinery is spelled out in Gasser and Leutwyler [2.10]. In general, one groups the loop contributions into tadpole and unitarity corrections. While the tadpoles contain one vertex and one loop, the unitarity corrections contain one loop and two vertices. The tadpole contributions renormalize the couplings of the effective Lagrangian. Both of these loop contributions also depend on the scale of dimensional regularization. In contrast, physical observables are –independent. For actual calculations, however, it is sometimes convenient to choose a particular value of , say, or . To one–loop order, the generating functional therefore takes the form

 Z=Z2+Z4+Zone−loop+Zanom

and what remains to be done is to determine the values of the renormalized constants, , , . These are in principle calculable from QCD, they depend on and the heavy quark masses

 Lri=Lri(ΛQCD;mc,mb,mt)

In practice, such a calculation is not feasible. One therefore resorts to phenomenology and determines the from data. However, some of these constants are not easily extracted from empirical information. Therefore, one uses constraints from the large world. Using this and experimental information from scattering, , the electromagnetic radius of the pion and so on, one ends up with the values for the given in table 1 (large arguments are used to estimate and ). For comparison, we also give the values at . More accurate data will allow to further pin down this quantity. In the case of , one can define scale–independent couplings (). These are discussed in ref.[2.10]. Can one now understand the values of the from some underlying principles? Already in their 1984 paper, Gasser and Leutwyler [2.10] made the following observation. They considered an effective theory of mesons coupled to the pseudoscalars. Eliminating the heavy field by use of the equations of motion in the region of momenta much smaller than the mass, one ends up with terms of order . The values of the corresponding constants are given in terms of and the –meson coupling strengths to photons and pions. This leads to a fair description of the SU(2) constants. This method has been generalized by Ecker et al. [2.19] and by Donoghue et al. [2.20]. They consider the lowest order effective theory of Goldstone bosons coupled to resonance fields (R). These resonances are of vector (V), axial–vector (A), scalar (S) and non–Goldstone pseudoscalar (P) type. For the latter category, only the is of practical importance. The form of the pertinent couplings is dictated by chiral symmetry**For the vectors and axials, this naturally leads to the tensor–field formulation. in terms of a few coupling constants which can be determined from data (from meson–meson and meson–photon decays). At low momenta, one integrates out the resonance fields. Since their couplings to the Goldstone bosons are of order , resonance exchange produces terms of order and higher. Symbolically, this reads

 ∫[dR]exp(i∫d4x~Leff[U,R])=exp(i∫d4xLeff[U])

So to leading order , one only sees the momentum–independent part of the resonance propagators,

 1M2R−t=1M2R(1+tM2R−t)

and thus the can be expanded in terms of the resonance coupling constants and their masses. This leads to

 Lri(λ)=∑R=V,A,S,PLResi+^Li(λ)

with a remainder. For this scenario to make sense, one has to choose somewhere in the resonance region so that one can neglect the remainder. A preferred choice is (as shown in Ref.[2.19], any value of between 500 MeV and 1 GeV does the job). In table 1, we show the corresponding values for all constants estimated from resonance exchange. It is apparent that the resonances almost completely saturate the , with no need for additional contributions. This method of estimating is sometimes called QCD duality or the QCD version of VMD. In fact, it is rather natural that the higher lying hadronic states leave their imprints in the sector of the light pseudoscalars – as already stated, the typical resonance mass is the scale of new physics not described by the Goldstone bosons.

 i Lri(Mη) Lri(Mρ) LRESi 1 0.9±0.5 0.7±0.5 0.6 2 1.6±0.4 1.2±0.4 1.2 3∗ −3.6±1.3 −3.6±1.3 −3.0 4 0.0±0.5 −0.3±0.5 0.0 5 2.2±0.5 1.4±0.5 1.4 6 0.0±0.3 −0.2±0.3 0.0 7∗ −0.4±0.15 −0.4±0.15 −0.3 8 1.1±0.3 0.9±0.3 0.9 9 7.4±0.2 6.9±0.2 6.9 10 −5.7±0.3 −5.2±0.3 −6.0

Table 1: Low–energy constants for . The first two columns give the phenomenologically determined values at and . The , , 8) are from ref.[2.10], the from ref.[2.21] and the
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