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Chiral Symmetry and Low Energy Pion-Nucleon Scattering

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arXiv:nucl-th/9906011v1 4 Jun 1999

Chiral Symmetry and Low Energy Pion-Nucleon Scattering
Sidney A. Coon?)
Physics Department, New Mexico State University, Las Cruces, NM 88003
Received XXX
In these lectures, I examine the e?ect of the meson factory πN data on the current algebra/PCAC program which describes chiral symmetry breaking in this system. After historical remarks on the current algebra/PCAC versus chiral Lagrangians approaches to chiral symmetry, and description of the need for πN amplitudes with virtual (o?-massshell) pions in nuclear force models and other nuclear physics problems, I begin with kinematics and isospin aspects of the invariant amplitudes. A detailed introduction to the hadronic vector and axial-vector currents and the hypothesis of partially conserved axial-vector currents (PCAC) follows. I review and test against contemporary data the PCAC predictions of the Goldberger-Treiman relation, and the Adler consistency condition for a πN amplitude. Then comes a detailed description of the current algebra WardTakahashi identities in the chiral limit and a brief account of the on-shell current algebra Ward-Takahashi identities. The latter identities form the basis of so-called current algebra models of πN scattering. I then test these models against the contemporary empirical πN amplitudes extrapolated into the subthreshold region via dispersion relations. The scale and the t dependence of the “sigma term” is determined by the recent data.

1 Introduction
The implementation of chiral symmetry in hadronic physics began around 1960. Its consequences were examined with two basic approaches. One is based on the concept of a partially conserved axial-vector current (PCAC) coupled with the algebra of vector and axial-vector hadronic currents. This current algebra is expressed as equal time commutation relations (the analogue of angular momentum commutation relations in quantum mechanics). The other (Lagrangian form) is based on chiral Lagrangians with small explicit chiral symmetry breaking terms. A famous example of the latter is the linear sigma model of Gell-Mann and Levy [1] which explicitly exhibits both PCAC and the current algebra. To adapt J. B. S. Haldane’s famous remark about the Deity and beetles, chiral symmetry seems to be inordinately fond of pions. Single pion exchange accounts for about 70% of the binding of light nuclei [4] (and perhaps all nuclei) and pions make up the most prominent non-nucleon degree of freedom in nuclear physics. Thus it should not be surprising that chiral symmetry was applied to nuclear physics as soon as 1967. It is the aim of these lectures, motivated by the nuclear physics problems brie?y mentioned below, to discuss the PCAC-current algebra approach with the aid of contemporary experimental knowledge of the low energy pion nucleon interaction. Relationships between the two approaches to chiral symmetry breaking will be mentioned when useful. This introductory lecture is primarily for motivation and will freely employ unde?ned concepts that will be de?ned and derived in detail in the following lectures.

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One of the earliest and boldest uses, in nuclear physics, of the PCAC form of chiral symmetry was the relationship obtained, by Blin-Stoyle and Tint, between the β-decay pion-exchange operator and a phenomenological two-body (nucleons) pion production operator [2]. With this relation, they attempted to analyze the process p+p → d+π+ using two-body terms obtained from a comparison of β-decay of the tritium nucleus and β-decay of the neutron. Neither the pion production data, the three-body wavefunction, nor the f t values of the two β-decays were known in the 60’s well enough to obtain a quantitative conclusion. Nearly the same technique was used 30 years later to obtain a rather reliable calculation of the process p + p → d + e+ + νe, so important for stellar nucleosynthesis [3]. It may seem reasonable to anyone that the latter process of weak capture of protons by protons might be related to weak β-decay. But it is the introduction of an isovector axial-vector hadronic current to play a role in both strong and weak interactions which lead to the perhaps more startling recognition of a relation between a strong (pion production) and a weak (β-decay) process. We shall see how this comes about later.
Another explicit use of PCAC alone (in the form of Adler’s consistency condition) was in an envisioned re-scattering of a virtual pion from one nucleon of a three-nucleon system. This process establishes a three-nucleon interaction due to two-pion exchange. Brown et al. showed that the three-nucleon force contribution to the binding energy of nuclear matter could be obtained from the isospin symmetric pion-nucleon forward scattering amplitude extrapolated o? the pion mass shell, and was quite small [5, 6]. This analysis knowingly [7] neglected the pion-nucleon sigma term, a measure of chiral symmetry breaking (the sigma term is proportional to the non-conserved axial-vector current). Somewhat later the full panoply of current algebra and PCAC constraints (labelled current algebra/PCAC) was brought to bear on the o?-shell pion-nucleon amplitude. These current algebra/PCAC “soft pion theorems” led to a scenario in which the chiral symmetry breaking sigma term could not be neglected, but instead was quite prominent in the three-body interaction [8]. However, the original insight of Refs. [5, 6, 7, 8], based on PCAC and later on current algebra/PCAC, that pion exchange based three-nucleon interactions are small compared to two-nucleon interactions remains true in the Lagrangian form of chiral symmetric theories. A currently employed three-nucleon interaction according to a chiral Lagrangian is the Brazil three-body force (TBF). The ?rst version of this TBF [9] had a sigma term contribution which did not come from a Lagrangian and in a later version [10] the sigma term contribution was altered to conform to the current algebra/PCAC constraints which had previously guided the Tucson-Melbourne two-pion exchange TBF [11].
A technical trick in Refs. [6, 8], which directly relates pion-nucleon scattering to a TBF contribution in nuclear matter, leads me to my third (and ?nal) illustrative example of chiral symmetry in nuclear physics: pion condensation in nuclear matter. An approximate evaluation of a three-body diagram in an translationally invariant system like nuclear matter can be made by summing and averaging the active nucleon over the Fermi sea. Then one obtains a modi?ed one-pion-exchange-potential between the other two nucleons, which can easily be evaluated in a many-body

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system. That is, the (now) single exchanged pion has an e?ective mass m? which is proportional to the isospin even, forward πN amplitude multiplied by the density (from the summation). A useful way to think about pion condensation is to extend the idea of a virtual pion rescattering from the active nucleon of a three-nucleon cluster to the picture of a pion rescattering again and again from the nucleons of nuclear matter. The criterion for pion condensation can be expressed in terms of m? (or the self energy in the pion propagator) which again is directly related to forward πN amplitudes. Pion condensation in neutron matter was examined with such amplitudes subject to the current algebra/PCAC constraints [12]. Actually, this third example of chiral symmetry had been discussed earlier with the aid of chiral Lagrangians [13]. This last problem has a contemporary reverberation which is somewhat amusing in that, of the three problems so far, it surely is the least constrained by experiment. Yet the relationships between the two forms of chiral symmetry have been clari?ed by a small debate on, of all thing, kaon condensation in dense nuclear matter. This debate was between a group [14] who, ?fteen years after the Tucson group [12], re-examined the current algebra/PCAC program of pion condensation, and practitioners [15] of the contemporary e?ective Lagrangian form of chiral symmetry known as chiral perturbation theory.
These three examples share the idea of a virtual pion rescattering from a nucleon (pion production in N N collisions and two-pion exchange TBFs) or from the many nucleons of nuclear matter (pion condensation). Another example which I will not discuss much is the two-pion exchange part of the N N interaction itself. As a Feynman diagram, this process has pion loops and the other three problems need only tree diagrams. As with the three-nucleon interaction, the ?rst use of chiral symmetry in the two-nucleon interaction was again by Gerry Brown who applied the current algebra/PCAC constraints on the πN amplitudes (and dropped the sigma term) in a series of articles titled“Isn’t it time to calculate the nucleon-nucleon force?” and “Soft pioneering determination of the intermediate range nucleonnucleon interaction” [16]. The chiral symmetry aspect of two-pion exchange NN diagram can (as expected) and has been treated with chiral Lagrangian techniques recently [17, 18]. I now cut o? this introductory and historical survey and turn to the concept of “soft pions”.
Each of these nuclear physics problems can be thought of as dependent on a πN scattering amplitude with at least one of the exchanged pions o? its mass shell: q2 = q02 ? q 2 = m2π. For example, short range correlations between two nucleons of nuclear matter suggests that the virtual pions of a TBF are spacelike with q0 ≈ 0 and q 2 ≤ 10m2π[6], and the calculation of Ref. [12] found that, at the condensation density, qc2 ≤ ?2m2π. The “soft pion theorems” strictly apply to pions with q → 0 which means that every component of the four-vector goes to zero. In particular, since q0 → 0 then q02 = m2π → 0 and a soft pion is a massless pion. In the language of QCD, this means that the quark mass goes to zero and the chiral symmetry of the QCD Lagrangian is restored. The axial-vector current would be conserved if the pion was massless. The mass of the pion is small on the scale of the other hadrons (m2π/m2N ≈ 1/45) so one of the ideas of PCAC is that the non-conservation of the axial vector current is small. Another formulation of PCAC suggests that

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one can make a smooth extrapolation from the exact amplitudes with soft pions to obtain either theoretical amplitudes with on-mass-shell pions (“hard” pions in the old jargon) or the o?-shell amplitudes of the nuclear physics problems. Certainly the soft pion constraints of current algebra/PCAC are within the assumed o?-shell extrapolations used in these problems.
In the 1960’s the current algebra/PCAC approach and the chiral Lagrangian approach to chiral symmetry (and how it is broken in the non-chiral world we do experiments in) developed in parallel and each approach paid close attention to the other. The 1970 lectures by Treiman on current algebra and by Jackiw on ?eld theory provide a useful (and pedagogical!) summary of this development[19]. For example, the linear σ model was an early chiral Lagrangian motivated by the current algebra/PCAC program. This model re?ects the feeling in the 1960’s that the ultimate justi?cation of the results obtained from a chiral Lagrangian rests on the foundation of current algebra. On the other hand, an early puzzle was the current algebra demonstration that the (observed) decay π → γγ should be zero. In his lectures, Jackiw used the linear σ model to demonstrate that the “conventional current algebra” techniques were inadequate. He went on from this demonstration of a violation of the axial-vector Ward identity with the nucleon level linear σ model to introduce a study of anomalies which is documented in Ref. [19]. (The quark level linear σ model, however, does appear to describe the decay π → γγ and 22 other radiative meson decays [20], so the ?nal denouement of this dialogue may be still to come.) In any event, anomalies play no role in the nuclear physics problems of these lectures, and will not be discussed here. A very useful pedagogical paper, speci?cally aimed at the nuclear physicist, commented on the relation between the two approaches to chiral symmetry. In it, David Campbell showed that, in a given chiral model ?eld theory with a speci?c choice of canonical pion ?elds, certain of the theorems expected from current algebra/PCAC (in particular the Adler consistency condition) will not be true [21]. This is one of the excellent papers which I hope the present lectures will prepare the student to appreciate.
In 1979, Weinberg [22] introduced a “most general chiral Lagrangian” constructed from powers of a chiral-covariant derivative of a dimensionless pion ?eld. This Lagrangian was aimed at calculating purely pionic processes with low energy pions. The most general such phenomenological Lagrangian, unlike earlier closed form models, is an in?nite series of such operators of higher and higher dimensionality. He was easily able to show that the lowest order Feynman diagrams constructed from the Lagrangian are tree graphs. These tree graphs reproduce his earlier [23] analysis of low energy ππ scattering obtained from i) the Ward identities of current algebra and ii) PCAC in the form of a smooth extrapolation from soft to physical pions. The importance of the 1979 paper lies in its analysis of the more complicated Feynman diagrams of the in?nite perturbation expansion of the chiral Lagrangian. The phenomenological Lagrangian would produce amplitudes of the form: T ? Eν , where E is the energy. This fact was obtained using dimensional analysis and ν is an integer determined by the structure of the Feynman diagram. The QCD picture of chiral symmetry breaking (for example in a world with only light u and d quark ?elds) imposes a further constraint upon ν: that more complicated diagrams

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necessarily have larger values of ν. Thus, provided that E is smaller than some intrinsic energy scale, Λ, the perturbation series is a decreasing series in E/Λ. The derivative structure of the Lagrangian guarantees that amplitudes from loops and other products of higher order perturbation theory produce only larger values of ν. The Lagrangian cannot be renormalized because this is an e?ective ?eld theory where all possible terms consistent with the symmetries assumed must be included. The non-renormalizability means that more and more unknown constants appear at higher (arranged in powers of ν) and higher orders of perturbation theory but their e?ect is suppressed by factors of E/Λ. Since it is a phenomenological Lagrangian the unknown constants must be determined by experiment, and one hopes that meaningful results can be obtained at a low enough energy such that the number of terms needed remains tractable. That is the disadvantage of this approach. An advantage is the systematic nature of the scheme with respect to the breaking of chiral symmetry. I quote from the seminal paper: “the soft π and soft K results of current algebra, which would be precise theorems in the limit of exact chiral symmetry, become somewhat fuzzy, depending for their interpretation on a good deal of unsystematic guesswork about the smoothness of extrapolations o? the mass shell. . . . phenomenological Lagrangians can serve as the basis of an approach to chiral symmetry breaking, which has at least the virtue of being entirely systematic” [22]
The introduction of the nucleon into this scheme (now called chiral perturbation theory or ChPT) led to a major industry in particle physics and to a reversal of the old idea that a symmetry imposed on an e?ective Lagrangian can only be legitimized by an underlying theory such as current algebra [24]. The new e?ective ?eld theory program does not attempt to ?nd really fundamental laws of nature, but does exploit systematically the symmetries encoded in the phenomenological Lagrangian. The belief in the power of this program leads to astounding remarks in chiral perturbation theory papers. Consider:
“Although the purpose of this comment is not to discuss the experimental situation, it may be one of nature’s follies that experiments seem to favour the original LEG [Low Energy Guesses of pion photoproduction from the nucleon] over the correct LET [Low Energy Theorems from ChPT]. One plausible explanation for the seeming failure of the LET is the very slow convergence of the expansion in mπ.” [25]
Or:
“We can compare the situation with that of the decay η → π0γγ where the one loop ChPT prediction is approximately 170 times smaller than the experimental result. The O(p6) contributions [the next order in the expansion] then bring the ChPT result into satisfactory accord with experiment.” [26]
On a more cautious note:
“The one-loop calculation [of γγ → π0π0] in ChPT disagrees with the data even near threshold.” . . . “In conclusion, a self-consistent, quantitative description of γγ → π0π0 and η → π0γγ data at O(p6) is still

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problematic. A good description of the γγ → π0π0 cross section has been achieved whereas a satisfactory, quantitative prediction of the decay width seems to be beyond the reach of an ordinary calculation at O(p6) [such a calculation involves tree-level, one- and two-loop Feynman diagrams].” [27]
Finally, on elastic πN scattering, the subject of these lectures:
“the chiral expansion converges to the experimental values, but the convergence seems to be rather slow, in a sense that contributions to di?erent orders are comparable. This fact seems to show that despite of the relative success in describing elastic πN scattering at threshold, the third order is de?nitely not the whole story. A complete one-loop calculation, which will include the fourth order of the chiral expansion, is probably needed for su?ciently reliable description of this process” [28]
Although nature seems to have pulled up its socks since the ?rst comment was made (more recent measurements of pion photoproduction seem to favor ChPT results near threshold), one is still left with a not fully satis?ed feeling by these comments.
More recently Weinberg applied this procedure to systems with more than one nucleon [29] so that e?ective ?eld methods could be extended to nuclear forces and nuclei [30]. This program is being continued vigorously by van Kolck and others, thereby generating another minor industry in nuclear physics a decade or so after ChPT hit particle physics.
In the following lectures, I will review the current algebra and PCAC program and its applications to the three nuclear physics problems of this introduction. These problems have also been attacked by the e?ective ?eld theory program. The former approach to chiral symmetry can be closely tied to the experimental program in pion-nucleon scattering and the latter approach takes some of its undetermined constants from pion-nucleon scattering. Before proceeding, I recommend the following review articles on this ?eld. Pion-nucleon scattering is treated, more extensively than I will, in Field Theory, Chiral Symmetry, and Pion-Nucleus Interactions by D. K. Campbell [31]. The mathematical aspects of global symmetries in Lagrangian forms of ?eld theory is discussed cogently in lectures at an earlier Indian-Summer School: Elements of Chiral Symmetry by M. Kirchbach [32, 33]. A very useful account (which I shall freely borrow from) of the original approach to chiral symmetry is Current algebra, PCAC, and the quark model by M. D. Scadron [34]. The nuclear physics aspects of e?ective ?eld theories are well described in E?ective Field Theory of Nuclear Forces by U. van Kolck [35] and Dimensional Power Counting in Nuclei by J. L. Friar [36]. The titles of these review articles should suggest to the student where to go for further studies.

2 Kinematics
We begin with the scattering amplitude for πj (q) + N (p) → πi(q′) + N (p′) where p, q, p′, q′ are nucleon and pion initial and ?nal momenta. We ignore for the time

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being spin and isospin aspects of the problem (i and j are pion (Cartesian) isospin indices). For elastic scattering q + p = q′ + p′ and the scalar product of these
four-vectors is ab ≡ a0b0 ? a · b. De?ne the “s-channel” Mandelstam invariants

q′, i

p′

t

s ≡ (p + q)2 = (p′ + q′)2 t ≡ (q ? q′)2 = (p′ ? p)2 u ≡ (p ? q′)2 = (p′ ? q)2 .

q, j

p

The invasriant s in this s-channel corresponds to the square of the total energy

for the process. Since four-momentum conservation is but one constraint upon

the four momenta, there are three independent combinations of these momenta

(and energies), but only two independent combinations of Lorentz scalar products.

So s, t, and u are not independent and it can quickly be shown with the aid of

q + p = q′ + p′ that

s + t + u = p2 + p′2 + q2 + q′2 .

If the nucleons are on-mass-shell (p2 = p20 ? p 2 = m2) and the pions are on-massshell (q2 = q02 ? q 2 = ?2), this relation becomes s + t + u = 2m2 + 2?2.
These Lorentz invariants s, t, and u can be visualized in di?erent coordinate
systems. For example, in the s-channel center of mass frame the incoming (on-
mass-shell) momenta are pion q = (q0, qcm) and nucleon p = (E, ?qcm), the ?nal (on-mass-shell) momenta are q′ = (q0, qcm′) and p′ = (E, ?qcm′), where |qcm| = |qcm′| and the three-vector qcm is simply rotated by the angle θcm.

q = (q0, qcm)

p = (E, ?qcm)

q′ = (q0, qcm′) θcm
p′ = (E, ?qcm′)

In this frame one can evaluate

1

1

s ≡ (p + q)2 = m2 + ?2 + 2[(m2 + qc2m) 2 (?2 + qc2m) 2 + qc2m]

t ≡ (q ? q′)2 = ?2qc2m(1 ? cos θcm)

u



(p ? q′)2

=

m2

+

?2

?

2[(m2

+

qc2m)

1 2

(?2

+

qc2m)

1 2

+ qc2m cos θcm]

.

Note that s ≥ (m + ?)2 and t ≤ 0 for physical πN scattering where qc2m ≥ 0 and ?1 ≤ cos θcm ≤ 1. The cm energy of the on-mass-shell nucleon is E = (s ? m2 ?

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?2)/2√s. Partial wave phase shifts are naturally expressed in terms of qc2m and its associated Mandelstam variable s.
Now consider the s-channel laboratory frame in which the target nucleon is at rest and the kinetic energy of the incoming pion is de?ned by Tπ ≡ ω ? ? =
k2 + ?2 ? ?, where ω is the lab energy of the incoming pion:

q = (ω, k)

p = (m, 0)

q′ = (ω′, k′) θlab
p′ = (E, p)

In this laboratory frame the Mandelstam invariants s and u take the form
s ≡ (p + q)2 = m2 + ?2 + 2mω = (m + ?)2 + 2mTπ u ≡ (p ? q′)2 = m2 + ?2 ? 2mω′ .

In either frame, it is clear that the threshold for physical πN scattering is sth = (m + ?)2 from qc2m = 0 or Tπ = 0, tth = 0 from qc2m = 0, and uth = (m ? ?)2 from qc2m = 0 or ω′ = ?.
Now introduce the variable

ν

=

s?u 4m

,

which has the threshold value νth = ? in the s-channel. For on-shell nucleons and pions ν = ω + t/(4m) so that in the forward direction, t = 0, the variable ν represents the lab energy of the incoming pion. Pion-nucleon scattering amplitudes are often given in terms of the pair of variables (ν, t) rather than (s, t) which would be appropriate for a partial wave representation, for example. A reason for this is that the variable ν has a de?nite symmetry under crossing, a concept to which we now turn. Crossing is the interchange of a particle with its antiparticle with opposite four-momentum. I can “cross” the pions by adding nothing to the s-channel relation πj(q) + N (p) → πi(q′) + N (p′) as follows:

πj (q)

+N (p) → πi(q′)

+N (p′)

πi(q′) + π?i(?q′)

πj(q) + π?j (?q)

Since I have changed nothing this s-channel process is equivalent to π?i(?q′) + N (p) → π?j(?q) + N (p′):

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?q, j

p′

Chiral Symmetry . . .

?q′, i p

u ? channel which is called the u-channel because u = (p ? q′)2 is now the sum of the incoming momenta and in this channel u is the square of the total energy. Because the antiparticle of a pion is still a pion (π+ = π? and π0 = π0) both the s-channel and the u-channel describe πN scattering.
Carrying on with “crossing” one can convince oneself that
s-channel In this channel s is the total energy squared and for physical scattering s ≥ sth = (m + ?)2 and t ≤ 0 (and u ≤ 0).
u-channel In this channel u is the total energy squared and for physical scattering u ≥ uth = (m + ?)2 and t ≤ 0 (and s ≤ 0).
t-channel In this channel the incoming particles are a pion and an anti-pion and the outgoing particles are N and N? . For physical scattering the total energy squared must be larger than the rest mass of the heaviest particles, so that t ≥ tth = (m + m)2, s ≤ 0, and u ≤ 0.
The scattering amplitude T (s, t, u) is a function of the three (not all independent) variables. The physical regions of the variables of the three channels are disjunct. We have determined the threshold values “by inspection”. It is slightly more complicated to work out the boundaries of the physical regions in the Mandelstam plane.They are given by the zeros of the Kibble function[37]
Φ = t[su ? (m2 ? ?2)] .
The physical regions correspond to the regions where Φ ≥ 0. This criterion essentially characterizes the need for the scattering angle to satisfy ?1 ≤ cos θcm ≤ 1. Clearly t = 0 or cos θcm = 1 is a boundary of the physical region no matter what the values of s and u are. The other zero of Φ then shows the dependence of a lower limit to t for s-channel πN scattering (for example) which depends upon the values of s ≥ sth = (m + ?)2 and u ≤ 0. Elastic scattering depends upon two independent variables: some sort of energy and some sort of scattering angle. As mentioned before, if you wanted to end up with a partial wave representation the natural variables are the pair (s, t) because t has a simple interpretation in terms of qc2m and θcm, for example. In the following discussion, the pair (ν, t) is more natural because ν → ?ν under the interchange of s and u: s ? ? u. Then the physical regions of the Mandelstam plane are bounded by a hyperbola in the (ν, t) plane

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and the straight line t = 0 (See Fig 1). The boundaries of the physical regions for π + N → π + N and for π + π → N + N? form branches of the same hyperbola. Note that the asymptotes of the boundary hyperbola are the lines s = 0 and u = 0.
t-channel
NN !
t = 4m2

u=0

150

u

s

100

t= 2 s=0

50

0 uN-ch!anneNl

t=4 2

=

=

sN-ch!anneNl

-50 -4

-2

0

2

4

=

Figure 1: Mandelstam diagram of pion-nucleon scattering

Let us turn from the relativistic invariants in T (ν, t) to isospin considerations in

the three (nucleon)

channels. The incoming and 1 (pion). The total

particles in the s isospin Is is then

and u either

channels have

1 2

or

3 2

.

In

the

isospin

1 2

t-channel

(ππ? → N N? ) It = 0, 1 so there are also two amplitudes in isospin: T (+) with It = 0

and T (?) which has It = 1. The t-channel isospin is especially convenient because

the pions obey Bose symmetry when crossed: ie It0 → It0 and It1 → ?It1 when s ?? u. To make contact with the s-channel amplitudes (and the charge states) we

need the s ?? t crossing relations:

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Chiral Symmetry . . .

T (+)

T

(

1 2

)

= =

1 3

(T

(

1 2

)

+

2T

(

3 2

))

T (+) + 2T (?)

T (?)

T

(

3 2

)

= =

1 3

(T

(

1 2

)

?

T

(

3 2

))

T (+) ? T (?) .

The pion ?eld operators transform as components of a vector in isospin space with Cartesian components de?ned as [38]

√ π+ = +(π1 + iπ2)/ 2

π0 π?

= =

π3



+(π1 ? iπ2)/ 2 ,

and

T (π+p → π+p) =

T

(

3 2

)

= T (+) ? T (?)

T (π?p → π?p) T (π?p → π0n)

= =

√3132T(T( 23()32

+

2 3

)?

T T

( (

1 2
1 2

) ))

= =

T (+) + T (?) √
? 2T (?) ,

for example.

Now we are in a position to examine the isospin structure of the T-matrix elements which describe the scattering πj(q) + N (p) → πi(q′) + N (p′). They are

de?ned as:

q′p′|S ? 1|qp ≡ +i(2π)4δ4(p′ + q′ ? p ? q)T ij(ν, t; p2 = m2, p′2 = m2, q2, q′2) , (1)

where we have displayed the (assumed) on-mass-shell nucleons and left the fourmomentum of the pions as a variable. The isospin structure of T ij is perhaps most
easily visualized from the Feynman diagram with an intermediate nucleon pole state and the isospin “scalar” vertex N? τ N · π:

T ijτ iτ j

=

T

(+)

1 2



i

τ

j

+

τjτi)

+

T

(?)

1 2



i

τ

j

?

τjτi)

= T (+)δij + T (?)i?ijkτ k .

(2)

The second equality is easily proved from the properties of the SU (2) τ -matrices: {τ i, τ j } = 2δij and [τ i, τ j] = 2i?ijkτ k. With this representation, it is clear that T (+) (T (?)) must be even (odd) under the interchange of the pions i ? j and s ? u. We also note that πiπj δij = π · π = π2 could be realized by the t-channel exchange of an isoscalar scalar ππ resonance–the putative sigma meson. In a similar manner, the t-channel odd exchange, πiπji?ijkτ k = iπ × π · τ could be realized by the isovector ρ meson. We will discuss these models in a later lecture.
Finally, we let the Dirac spinors u(p) carry the spin of the nucleons and write the Lorentz invariant T = +u?(p′){M (ν, t)}u(p) where M could be made up of scalars, vectors, and higher order tensors constructed of the vectors p, p′, q, q′ and the gamma matrices 1, γ?, γ?γν, γ5, γ?γ5. That is, one could form
M = A + B?γ? + C?ν [γ?, γν ] + D?γ?γ5 + Eγ5

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but conservation of parity eliminates D? and E as candidates. With the aid of the Dirac equation for free (on-mass-shell) nucleons,

(p?γ? ? m)u(p) = 0 = u?(p′)(p′?γ? ? m) ,

all the combinations one can write down for C?ν reduce to A + B?γ? where A is a scalar and B is a four-vector formed of those available: p, p′, q, q′. B? cannot be p or p′ because the Dirac equation would make T ? mu?(p′)u(p) already included in A. So B? must be linear in q and q′, but B? cannot be (q ? q′)? = (p′ ? p)? for
the same reason. We conclude that

T

±

=

u?(p′){A±(ν,

t)

+

1 2

(q/′

+

q/)B±(ν,

t)}u(p)

where

/q



q?γ?

and

the

factor

1 2

is

inserted

to

make

the

expressions

for

s

and

u-channel nucleon poles in B simple. With the aid of ν = (s ? u)/4m = (q′ + q) ·

(p′ + p)/4m and the free particle Dirac equation, one can rewrite this as

T

±

=

u?(p′){A±(ν,

t)

+

νB±(ν,

t)

?

1 4m

[q/,

/q′]B±(ν,

t))}u(p)

.

(3)

De?ne the combination A + νB = F , which is called D in Ho¨hlers book [39] and in
much of the literature. It can be shown that this combination of invariant amplitudes corresponds to the non-relativistic forward (p = p′) scattering of a nucleon from a pion in which the spin of the nucleon remains unchanged (non-spin ?ip); for example, see Ref. [31], pp 612. For this reason the invariant amplitude F is sometimes called the “forward amplitude” but obviously we can study the combination A + νB for any value of ν and t.
Expressions of chiral symmetry in the form of soft pion theorems and their on-mass-shell analogues are most naturally expressed as conditions on the four amplitudes F ±(ν, t) and B±(ν, t), rather than the set A±(ν, t) and B±(ν, t). As
we shall see in the following, the predictions of chiral symmetry breaking are all in the subthreshold crescent of the Mandelstam representation (Figure 2). This crescent is below the s-channel threshold sth = (m + ?)2 for πN → πN , below the u-channel threshold uth = (m + ?)2 for π?N → π?N , and below the t-channel threshold tth = (m + m)2 for π?π → N? N . Therefore the invariant amplitudes in this subthreshold crescent are real functions of the real variables ν and t.

3 Current Algebra and PCAC
The formalism in this lecture follows very closely the exposition of Scadron in his review article [34] and textbook [40]. It is included here to make the lectures somewhat self-contained and to enable me to compare the current algebra predictions for pion-nucleon scattering with the current experimental results, a comparison which has not been emphasized in most contemporary discussions of the meson factory data.
We begin with the basic ideas of the current algebra-Partially Conserved Axialvector Current (PCAC) implementation of chiral symmetry in hadronic physics.

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Czech. J. Phys. 48 (1998)

Chiral Symmetry . . .

t= 2

4

u = m2

s = m2

3

2

1

0 -1

0

1

=

Figure 2: The subthreshold crescent of the Mandelstam diagram

Recall that in non-relativistic quantum mechanics the charge operator Q(t) obeys the Heisenberg equation of motion:

dQ(t) dt

=

?Q(t) ?t

?

i[Q(t), H(t)]

.

(4)

Then, if Q is explicitly independent of time, conservation of charge is equivalent to Q commuting with the Hamiltonian. In relativistic quantum mechanics we can de?ne current densities and Hamilton densities as

Q(t) ≡ d3xJ0(t, x)

(5)

H(t) ≡ d3xH(t, x) .

(6)

The equation of continuity for charge

dQ(t) dt

=

d3x

?J0(t, x) ?t

+ ? · J(t, x) ≡

d3x? J (x)

(7)

allows one to rewrite (4) in the local density form

i?J(x) = [Q, H(x)]

(8)

if Q does not depend explicitly on time.

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Sidney A. Coon

Commutators such as this one form the underlying dynamics in current algebra. We now know that at the hadronic level QCD is spontaneously broken down into a vector SU (2) algebra and an axial-vector SU (2) algebra. Current algebra is based on the SU (2) equal-time commutation relations of isotopic vector charges

[Qi, Qj] = i?ijkQk

(9)

which was extended in the 1960’s by Gell-Man’s suggestion of adding axial charge (Qi5) commutators

[Qi, Qj5] = i?ijkQk5 ,

[Qi5, Qj5] = i?ijkQk

(10)

to complete the chiral algebra. Models of H for strong, electromagnetic, and weak
transitions as products of currents then predict observable hadron current diver-
gences according to (8) with the aid of the charge algebra and its current algebra
generalizations. We defer discussion of the current algebra per se until after this
introductory material is discussed. The SU (2) notation is the same as before with states |πi , and (to be de?ned)
vector currents Ji and axial-vector currents Ai, where i = 1, 2, 3. De?ne isotopic charges from current densities as Qi = d3xJ0i(t, x). The SU (2) hadron states transform irreducibly as Qi|P j = if ijk|P k , where f ijk = ?ijk. In the generalization to SU (3) the anti-symmetric structure constant f ijk is related to the Gell-Man λi matrices, i = 1, · · · 8 (for a tabulation, see Ref. [34]). Now consider the SU (2) and SU (3) structure of the electromagnetic current

J?γ

= J?S + J?V

=

√1 3

J?8

+

J?3

,

(11)

where J?V

= J?3

is the isovector current and J?S

=

√1 3

J?8

corresponds to 2J?Y

the

hypercharge current. The corresponding charges are

Q = d3xJ0γ (x),

1 2

Y

=

d3 xJ0S (x),

I3 = d3xJ0V (x) . (12)

The equation of continuity (7), coupled with the fact that the electromagnetic

charge Q is conserved in the strong interaction, implies ?Jγ(x) = 0. The SU (3)

structure of the photon is consistent with the separate conservation of isospin and

hypercharge in the strong interactions and suggests the Gell-Mann-Nishijima rela-

tion

Q

=

1 2

Y

+

I3.

3.1 Conserved SU (2) Vector Currents
We want to treat J?S and J?V,i (where J?V ≡ J?V,i) as conserved hadronic currents for the strong interactions, ?JV,i = 0 and ?JS = 0. To illustrate this, de?ne the isovector part of the SU (2) strong vector current by its nucleon matrix elements:

Np′ |J?i (x)|Np

=

N?p′

τi 2

[F1V

(q2)γ?

+

F2V

(q2)iσ?ν qν /2m]Npeiq·x

,

(13)

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Czech. J. Phys. 48 (1998)

Chiral Symmetry . . .

where (q = p′ ? p). F1V (q2) is the nucleon isovector charge form factor and F2V (q2)

the nucleon isovector magnetic moment form factor. The isoscalar and isovector

decomposition is de?ned as F1S,2(q2) = F1p,2(q2) + F1n,2(q2) and F1V,2(q2) = F1p,2(q2) ?

F1n,2(q2). This de?nition is made because we identify the i = 3 component of J?V

plus J?S (with a similar de?nition) as the electromagnetic current for the proton

or the neutron. For the complete electromagnetic current, J?γ = J?S + J?V , charge

Fis1Vc(o0n)se=rveFd1p(a0n)d+FF1V1n(0(0))==11. ,Inwhteerrme sF1opf(0i)so=spi1n

this and

conservation law becomes F1n(0) = 0. With the aid

of the free Dirac equation one can show u?p′q?γ?up = 0 and q?σ?ν qν = 0, thus

demonstrating that the divergence of the isovector current (13) (and the analogue

isoscalar current) is indeed zero.

In a similar manner, we can extend the electromagnetic charged pion current to

the isovector-vector hadron current:

πpi ′ |J?V,j(x)|πpk = ?ijkFπ (q2)(p′ + p)?eiq·x ,

(14)

where the charge form factor of the pion is normalized to Fπ(0) = 1. In our isospin convention J?γ = J?V,3, and I note that J?S does not couple to pions; this would violate G-parity. The current of (14) is conserved for p′ 2 = p2 up to a term proportional to (p′ ? p)? which disappears in π|?JV,j|π = 0. But the general SU (2)
vector current is conserved as an operator

?Ji(x) = 0 i = 1, 2, 3

(15)

such that the nucleon and pion matrix elements of (15) vanish, consistent with the vanishing divergences of (13) and (14) for on-shell equal-mass hadrons.
One can continue to demonstrate the vanishing divergence of other matrix elements of the hadronic vector current. For example, the existence of the vector mesons ρ and ω suggests a direct ρ-γ and ω-γ transition. We write the ρ-to-vacuum matrix elements of the hadronic isovector vector current as

0|J?V,i(x)|ρj (q)

=

m2ρ gρ

??

(q)δij

e?iq·x

,

(16)

and the hadronic isoscalar vector current as

0|J?S (x)|ω(q)

=

m2ω gω

??(q)e?iq·x

.

(17)

These currents are conserved as well because ?JV,S ∝ q · ?(q) = 0 for on-shell spin-1 polarization vectors ??(q).

3.2 SU (2) Axial-vector Current Ai?

We introduce this current with the simplest matrix element (and the analogue of the ρ-to-vacuum matrix element of the vector current) π-to-vacuum:

0|Ai?(x)|πj (q) = ifπq?δij e?iq·x ,

(18)

Czech. J. Phys. 48 (1998)

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Sidney A. Coon

where fπ ≈ 93 MeV is called the pion decay constant and its value is measured in the weak decay π+ → ?+ν?. The divergence of (18) is

0|?Ai(0)|πj = δij fπ?2

(19)

for i, j = 1, 2, 3 and an on-shell pion q2 = ?2 ≡ m2π. From this exercise we learn that axial-currents are not conserved, even if SU (2) is an exact symmetry. But the
pion mass is small relative to all other hadrons: ?2/m2 ≈ 1/45. In 1960 Nambu suggested that 0|?Ai(0)|πj ≈ 0 and even ?Ai ≈ 0 in an operator sense [41]. Next we de?ne the nucleon matrix elements of Ai?:

Np′ |Ai?(x)|Np

=

N?p′

τi 2

[gA

(q2

)(t)iγ?

γ5

+

hA(q2)iq?γ5]Npeiq·x

,

(20)

where q = (p′ ? p) as usual and γ?5 = γ5 as in Refs.[34, 40]. Finally we present a diagrammatic representation of these matrix elements:

p′

A

π

>

q

0|A|π(q)

A >

Np′ |A?(x)|Np

p

Figure 3: Matrix elements of the axial-vector current which will be useful in the discussion of PCAC and later on of current algebra.

3.3 PCAC

We now review three ways of looking at the partial conservation of the axial
vector current (PCAC) and establish a soft-pion theorem which will be used and
tested against data in the following. The ?rst (Nambu) statement of PCAC is simply that ?Ai ≈ 0 in an operator sense. We now consider the general emission of a very low energy pion A → B + πi so that the emitted pion is soft (m2π ≈ 0). Replace the pion by an axial-vector and then remove the pion pole in this diagrammatic way:
This diagrammatic equation relates the axial-vector M -function M? and the pion pole contribution Mπ as

M?i = (?i)(?ifπq?)

i q2 ? m2π + i?

Mπi (q)

+

M

i ?

.

(21)

To establish this (second) S-matrix form of PCAC, let i) m2π ≈ 0 in the pion propagator, ii) take the divergence of both sides of (21) and iii) use the Nambu version in the form of q?M?i ≈ 0 to arrive at

ifπMπi (q)

=

q?M

i ?

(22)

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Czech. J. Phys. 48 (1998)

Chiral Symmetry . . .

B

B

B

A
<

=

A
<

q

+

A
<

A

A

A

Figure 4: Pion-pole dominance of axial-vector current matrix elements.

Relation (22) can also be derived ( [34],pp 221) for m2π = 0 with the aid of the ?eld theoretic statement ?A = fπm2πφiπ(x) where 0|φiπ|πj = δij and φiπ(x) is some pseudoscalar ?eld operator with the quantum numbers of the pion. It can be

shown [34] that equation (22) holds for either m2π → 0 or q2 → 0, provided that the pion pole is ?rst removed from q?M?i .

The third and most useful form of PCAC (for our study of πN scattering) is

obtained

from

the

soft-pion

limit

(q



0)

of

(22)

rewritten

as

Mπi (q)

=

?i fπ

q?

M

i ?

.

The

right-hand

side

of

this

relation

has

contribution

M

i ?

?

O(1)

which

vanish

as

q? → 0. However, the O(1/q) poles from “tagging on” the axial-vector to external

nucleon lines will not vanish, giving the soft pion theorem:

Mπi (q)

?q→→0

=

?i fπ

q?

M

i ?

(poles)

+

O(q)

,

(23)

and the soft pion version of PCAC: after removal of the pion poles and O(1/q) poles
from the axial-vector amplitude the (truly) background amplitude is a smoothly varying function of q2 such that

q?M i?(non ? pole) ≈ 0 ,

(24)

and

Mπi (q2) ≈ Mπi (0) .

(25)

This soft pion version of PCAC (25) is now a statement about pion amplitudes and can be used as such. It is a sharp statement, comparable to other characterizations of PCAC, such as “What is special is that the pion mass is small, compared to the characteristic masses of strong interaction physics; thus extrapolation over a distance of m2π introduces only small errors” [42], pp 43.

3.4 The Goldberger-Treiman Relation
The Goldberger-Treiman (GT) relation between strong and weak interaction parameters was displayed already in 1958 [43] and explained by Nambu a short time later [41]. Here we show that the GT relation can be regarded as a single soft pion prediction of PCAC and pion pole dominance of axial-vector, hadronic transitions. First let us notice that the divergence of (20) coupled with the (Nambu)

Czech. J. Phys. 48 (1998)

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17

Sidney A. Coon

PCAC statement that Np′ |?Ai?(x)|Np ≈ 0 implies that the axial form factors obey

2mgA(q2) + q2hA(q2) ≈ 0 .

(26)

where we have used γ?q?γ5 → 2mγ5, when sandwiched between the spinors of on-
mass-shell nucleons . To go farther, we dominate the axial-vector matrix element
with the pion pole exactly as in Fig. 4, but this time we have an e?ective πN N coupling HπNN = gπNN N? τ · πγ5N which gives an explicit form to the pion pole Mπi (q) of (21). Carrying this out we ?nd

Np′ |Ai?(x)|Np



gπNN u?p′ τ

iγ5up

q2

?

i m2π

+

i? (?i)(?ifπq?)

.

(27)

Neglecting m2π and comparing with (20) shows that it is the form factor hA(q2)iq?γ5

which has the pion pole:

hA(q2)



?

2fπ gπN N q2

.

(28)

Now let q2 → 0 to suppress the non-pion-pole terms, and the ≈ in (28) becomes an equality, turning (26) into the exact relation

2mgA(q2 = 0) ? 2fπgπNN (q2 = 0) = 0 ,

(29)

which takes the familiar Goldberger-Treiman form

mgA(0) = fπgπNN ,

our ?rst soft pion prediction. To test the GT relation empirically in the chirally broken real world, convert it
to a Goldberger-Treiman discrepancy

?

=

1

?

mN gA(0) fπ g

.

(30)

The experimental values are [44]

mN

=

1 2

(mp

+

mn)

=

938.91897 ±

0.00028

MeV,

and [45]

fπ = 92.6 ± 0.2 MeV,

We use the current best value of gA(t = 0) as determined by two consortia at the Institute for Nuclear Theory [3, 46]. They ?nd, by averaging modern results for the neutron lifetime and decay asymmetries,

gA(0) = 1.2654 ± 0.0042.

The least well known, and somewhat controversial, strong interaction parameter is

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Czech. J. Phys. 48 (1998)

Chiral Symmetry . . .

gπNN (q2 = m2π) ≈ 13.12 [47] gπNN (q2 = m2π) ≈ 13.02 [48]
down about 2% from the pre-meson factory value of gπNN ≈ 13.40 [49, 50]. The GT discrepancy then becomes

? ≈ 0.023 [47]

(31)

? ≈ 0.015 [48]

(32)

or a discrepancy of only 2%! This numerical fact is an, better than usual, example of the soft pion form of PCAC; Mπi (q2) ≈ Mπi (0).
The Goldberger-Treiman relation is exact in the chiral limit m2π → 0 (?A = 0). In our derivation we neglected m2π and then took the limit q2 → 0. Both limits are necessary to make the relationship exact. This distinction becomes important as one attempts to use the q2 → 0 limit to guide the low q2 variation of the πN N vertex
function for an o?-mass shell pion in models of the N N and N N N forces [49].
The value of ? indicates a 2% decrease in the coupling from the on-shell coupling q2 = m2π to q2 = 0. One should parameterize the πN N “form factor” to have this “GT slope” which re?ects chiral symmetry breaking. The usual πN N form factor
of the Tucson-Melbourne N N N force [11] has been parameterized to have about a 3% GT slope. That is, if FπNN (q2) = (Λ2 ? m2π)/(Λ2 ? q2) then Λ ≈ 800 MeV.

3.5 The Adler Consistency Condition

Another soft pion result which is independent of current algebra follows from an

Adler-Dothan

version

of

the

soft

pion

theorem.

We

start

with

Mπi (q)

=

?i fπ

q?

M

i ?

for the general hadronic amplitude A → B + πi, and examine the origin of the

O(q?1) nucleon poles which survive in the limit q → 0. Then only the axial vector

(designated by ? “tagging” onto external nucleon lines of the general hadronic

ingoing line A and outgoing line B in the diagram below) will generate nucleon

propagators O(q?1). The ? represents a gA(q2)-type coupling of the axial vector, since hA(q2) is already included in Mπi (q2) from the pion pole in hA(q2), see (28).

B?

B

+

A

A?

Now we apply the GT relation mgA(0) = fπgπNN to identify the nucleon pole

parts

of

the

axial

background

(i.e.,pion-pole

removed)

amplitude

q?M

i ?

with

the

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19

Sidney A. Coon

pseudoscalar πN interaction, so that the diagrammatic representation of the background becomes:

B

B

+

+ (M0γ5τ i + γ5τ iM0)

A

A

The hadronic amplitude A → B labeled M0 contains nucleons but has had, as

we have seen, the soft pion removed and the axial vector removed. Now let q? → 0

in

Mπi (q)

=

?i fπ

q?M

i ?

,

to

suppress

all

further

background

parts

in

M

i ?

of

O(q0).

Only the nucleon poles O(q?1) with pseudoscalar pion-nucleon coupling are left

and we have the soft pion theorem proved by Adler and Dothan [51]:

Mπi (q) ?q→→0

MpisNpoles(q)

+

M

i π

(q



0)

(33)

where

M

i π

(q



0)

=

gπN N 2m

(M0γ5τ i

+ γ5τ iM0)

.

(34)

This version of the soft pion theorem, valid for either an incoming or an outgoing

soft pion, is the analog of the soft photon theorem of Low [52]. It allows us to turn

the Adler zero [53], Mπi (q) ?q→→0

0

provided

that

M

i ?

in

(22)

has

no

poles,

into

the

Adler PCAC consistency condition for πN scattering.

To begin this demonstration, let us display explicitly the s-channel and u-channel

nucleon poles in πj (q) + N (p) → πi(q′) + N (p′):

q0; i

p0

q; j p

q0; i
=
q; j

p0

q0; i

+

p

q; j

q0; i

p0

p0
+

p q; j p

Figure 5: Nucleon pole terms in N scattering

Both nucleon poles are present and are added together by the Feynman rules

because the crossed pions are bosons. Now let the ?nal pion become soft (q′ →

0) and the other three particle be on-mass-shell. As we have separated out the

pseudoscalar nucleon poles, we can apply (34) with M0 = ?gπNN τ j γ5 from N → N + π. Then

M

ij π

(q



0)

=

?

gπN N 2m

(gπNN τ j γ5γ5τ i

+

γ5τ igπNN

τ j γ5)

(35)

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Czech. J. Phys. 48 (1998)

Chiral Symmetry . . .

=

+

gπ2N N m

δij

(36)

where we have used {τ i, τ j} = 2δij and remind the reader that γ52 = ?1 in this (Schweber) convention.
Now we restate the Adler consistency condition (36) as a condition on the invariant amplitude F + = A+ + νB+ (since the isospin condition is t-channel even). The kinematic variables for q2 = m2π = ?2 and q′ → 0 are t = (q2 ? q′2) = ?2 and ν = 0 because s = u = m2. Then the Adler consistency condition becomes

F +(ν

=

0, t

=

?2; q2

=

?2, q′2

=

0)

=

A+(ν

=

0, t

=

?2; q2

=

?2, q′2

=

0)

=

g2 m

,

(37)

and we note the pseudoscalar nucleon poles do not contribute to A(±) but only to

B(±). Speci?cally,

Ap(±s ) = 0

Bp+s

=

g2 m

νB2

ν ? ν2

Bp?s

=

g2 m

νB νB2 ? ν2

(38)

To make contact with πN data analyses obtained from dispersion relations, it
is natural to evaluate pseudoscalar nucleon poles, not as ?eld theory Feynman diagrams but in the sense of dispersion theory so that the residue in ν2 of FP+ is evaluated at the value of ν at the s-channel nucleon pole, 2m(νB ? ν) = (m2 ? s) or νB = ?q · q′/2m:

FP+

=

g2 m

νB2 νB2 ? ν2

FP?

=

g2 m

ννB νB2 ? ν2

,

(39)

(see Ref. [40], pp 340-343). The di?erence between the two prescriptions lies only

in F +:

FP+(ν, t)

=

Fp+s(ν, t)

+

g2 m

.

(40)

Now restate the Adler consistency condition in the form of a condition on the background πN amplitude de?ned as

F (±) = FP(±) + F?(±) ,

(41)

so that

F?+

=

F?p+s

?

g2 m

.

(42)

In the single soft (Adler) limit (37) becomes

FP+ → 0 ,

F?+



g2 m

?

g2 m

=

0.

(43)

The knowledgeable reader may have noticed that for F + the dispersion-theoretic nucleon pole with pseudoscalar coupling is the same as the ?eld theoretic nucleon

Czech. J. Phys. 48 (1998)

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21

Sidney A. Coon
pole of F + with pseudovector coupling, so one can think, if one wishes, think of the background F?+ as the full amplitude minus the pseudovector poles. It is often said that the Adler PCAC consistency condition of chiral symmetry forces the use of pseudovector coupling, but it is obvious from the above that this soft pion theorem makes no such demand. In the future, the phrase “nucleon poles” refer to dispersion theory poles with pseudoscalar coupling.
Invoking PCAC in the form of (25), we can expect that putting the ?nal pion back on-mass-shell (and holding ?xed t and ν) should not change the Adler consistency condition much:

F?+(0, ?2) ≡ F?+(ν = 0, t = ?2; q2 = ?2, q′2 = ?2) ≈ 0 .

(44)

This “Adler Low Energy Theorem” (LET) point is in the subthreshold crescent region of the Mandelstam plane (see Fig. 2) and the value of F?+ can be reliably determined from πN scattering data with the aid of dispersion relations. It is F?+ ≈ ?0.03??1 [54] or F?+ ≈ ?0.08??1 [55], extrapolations obtained from the most recent phase shift analysis called SM98 [56]. As the amplitude, unlike the Goldberger-Treiman discrepancy, has dimensions we must compare this result to the overall scale ?1.3??1 ≤ F?+(ν, t) ≤ 6??1 within the subthreshold crescent. Then we see that this PCAC low energy theorem (44) is also rather impressively con?rmed by the data [57]. Indeed one can go a step further and notice that this background amplitude has a zero in the subthreshold crescent which, beginning at the Adler LET, passes very near the threshold point (ν = ?, t = 0) [55]. The nucleon pole contribution is quite small (≈ ?0.13??1) at threshold, so the overall unbarred isoscalar scattering length a0 ≈ F +(?, 0)/4π ≈ 0.01??1 ≈ 0 is a threshold consequence of the PCAC Adler consistency condition and has nothing to do with current algebra.

3.6 Current Algebra and πN Scattering

The current algebra representation of low-energy πN scattering not only utilises on-mass-shell (q2 = q′2 = ?2) axial Ward-Takahashi identities (analogous to the conditions gauge invariance imposes on photon-target scattering, but incorporating
a current algebra commutation relation) but also make a speci?c prediction for the amplitude πj(q) + N (p) → πi(q′) + N (p′) when both pions are soft [23]. This latter prediction can be compared to the data by invoking PCAC in the form of (25) to
bring each pion back to the mass-shell. But ?rst we must establish this current algebra representation [53, 34].

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Czech. J. Phys. 48 (1998)

Chiral Symmetry . . .

Ai?(q′)

p′

Ajν (q)

p

Figure 6: Compton-like two-current scattering diagram.
Begin with SU (2) axial currents “scattering” o? target nucleons and write the covariant amplitude (to be sandwiched between on-shell nucleon spinors) as

M?ijν = i d4xeiq′·xT [Ai?(x), Aj?(0)]θ(x0) ,

(45)

where ? = q ? q′ = p′ ? p and the momentum transfer is t = ?2. Contract (45) with q′ (i.e. take a divergence in coordinate space), integrate the right hand side by parts, and drop the surface term at in?nity. Using the identity ??T (A?(x) . . .) =
T (?A . . .) + δ(x0)A0 . . ., we ?nd

q′?M?ijν = i d4xeiq′·xT [??Ai?(x), Aj?(0)]θ(x0) ? i?ijkΓkν (?) ,

(46)

where we have used the Equal Time Commutation relationship (47) to bring in the three-point vertex function Γkν(?) which depends only on the momentum transfer ? = (p′ ? p).

p′

Jk >

Γkν (?)

p

Figure 7: The three-point vertex M function for the isovector-vector current.

Before going on, let us pause to examine the extension to currents of the charge algebra of (9) and (10):

[Qi, Jνj (x)]ET C = if ijkJνk(x) [Qi, Aj (x)]ET C = if ijkAkν (x) [Qi5, J j (x)]ET C = if ijkAkν (x) [Qi5, Aj (x)]ET C = if ijkJνk(x) .

(47)

Czech. J. Phys. 48 (1998)

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23

Sidney A. Coon

The axial charge is, of course, de?ned as Qi5 = d3xAi0(t, x), by analogy to Qi = d3xJ0i(t, x), and the SU (3) structure constants f ijk reduce to ?ijk for i = 1, 2, 3
of SU (2) pion-nucleon scattering. Then one can recover the charge algebra (9) and
(10) from the current algebra relations (47) by setting ν = 0 and integrating over all
space. Notice that if the currents in (45) were the conserved isovector-vector current Jnju(k) and Jmi u(k′) then (46) would become simply k′?M?ijν = ?i?ijkΓkν (?). The latter is the isovector-vector Ward-Takahashi identity for virtual isovector photons
which replaces the gauge invariance equation for real photons. Now we go on, by contracting (46) by qν and converting the x dependence of the
currents from ?Ai to Ajν so that we can integrate by parts once again. The result is the “double” Ward-Takahashi identity

q′?M?ijν qν =

(48)

d4xe?iq·xT [??Ai?(0), Aj?(x)]θ(x0) ? i?ijkΓkν (?)qν + [i??Ai?(0), Qj5]ET C

We can symmetrise the current algebra term i?ijkΓkν(?)qν = i?ijkΓkν (?)Qν by

utilising

Γkν (?)?ν

=

0

and

the

de?nition

Q

=

1 2

(q

+

q′

).

This

term

can

be

identi?ed

with the measured electromagnetic form factors of nucleons [58]:

Γkν (?)Qν

=

τ 2

F1V

(t)γν Qν

?

1 4m

F2V

(t)[γν qν′ , γν qν ]

,

(49)

where we have put back in the suppressed nucleon spinors and used the de?ning equation (13). The second commutator term on the RHS of (49), reinstating the nucleon spinors, is the pion-nucleon “sigma” term

Np′ |[i??Ai?(0), Qj5]ET C |Np = δij σN (t)N?p′ Np ,

(50)

which like the current algebra term can be only a function of t. The amount of

the t dependence of the sigma term cannot be determined by theory, but like

the t dependence of the the current algebra term is obtained by comparing with

measurement. The sigma term is isospin symmetric, as can can be established by

reversing the order of momentum contractions of M?ijν . The sigma term is a measure of chiral symmetry breaking since it is proportional to the non-conserved current

?A = 0.

Having discussed the two t dependent terms on the RHS of (49), we now relate

the LHS to pion-nucleon scattering by dominating the LHS by pion poles according

to Figure 8.

The

S-matrix

version

of

PCAC

ifπMπi (q)

=

q?

M

i ?

holds

no

matter

if

we

let

?2 → 0 or q2 → 0, or keep both non-zero. So we can take the double divergence of

Fig. 8 as in (49), ?rst setting ?A = 0, which gets rid of the integral on the RHS of

(49), and equate the result to the RHS of (49), giving

? fπ2Mπijπ + q′?M? ?ijν qν

=

?i?ij

k

τk 2

F1V

(t)γν Qν

?

1 4m

F2V

(t)[γν qν′ , γνqν ]

+ δijσN (t) .

(51)

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Czech. J. Phys. 48 (1998)

Ai (q )0 p0

Chiral Symmetry . . .

=

+

+

+

Aj (q) p
Figure 8: Pion-pole dominance of A N ! A N.

The minus sign of Mππ is the reverse of the sign associated with the third (the one we really want) diagram on the RHS of the ?gure because of the ?rst two diagrams.
This relationship between πN scattering and a double divergence has been obtained from the covariant amplitude (45) with the aid of pion-pole dominance and the S-Matrix version of PCAC (22). In order to reproduce two noteworthy current algebra/PCAC theorems, the Weinberg double soft-pion theorem [23] and the Adler-Weisberger double soft-pion relation [59] we now use the soft pion theorem (25). The latter can be applied only if all poles are removed from both amplitudes: Mπijπ and the non-pion pole axial-Compton amplitude M? ?ijν. The nucleon poles are removed from Mπijπ as in Fig. 5 and in an analogous ?gure for nucleon poles in M? ?ijν, with the important distinction that the nucleon-axial coupling is not pseudoscalar gπNN γ5 ≡ gγ5 but instead is de?ned by (20). This distinction

?

MπNπ

+ fπ?2q′?M? ?Nν qν

=

δij

g2 m

+

i?ijk

τ

k

g2ν 2m2

(52)

introduces the Adler contact term back into the isospin-even πN amplitude. With the removal of the nucleon poles the resulting q′?M? ?′iνjqν vanishes as q → 0 , q′ → 0. The upshot is the generic double soft-pion result:

Mππ(q, q′) ≈ MπpπsN (q, q′) + M? ππ(q → 0 , q′ → 0) ,

(53)

which has the isospin decomposition (2)

M? π+π(q



0

, q′



0)

=

g2 m

?

σN (0) fπ2

,

(54)

derived by Weinberg [23], and

ν?1M? π?π(q



0

, q′



0)

=

1 fπ2

?

g2 2m2

=

1 fπ2 (1

? gA2 )

,

(55)

the Adler-Weisberger relation [59]. To obtain the crossing symmetric relation (55),

we divide F1V (t)γν Qν of (51) by ν = (p + p′) · (q + q′)/4m, before taking the

q



0, q′



0

limit,

to

get

1 2

F1V

(t)



1 2

because

F1V (0)

=

1.

We

then

use

the

GT

relation to obtain the far RHS of (55).

Czech. J. Phys. 48 (1998)

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Sidney A. Coon

3.7 On-pion-mass-shell current algebra Ward-Takahashi identities
We have derived the Ward identities in the chiral limit such that q′?M? ?′iνjqν vanishes as q → 0, q′ → 0, where M? ?′iνj is the background axial-Compton amplitude with pion and nucleon poles removed. A derivation similar to the above, but keeping both pions on-mass-shell at all stages, yields on-shell Ward identities which impose current algebra constraints on pion-nucleon scattering [60]. These identities are most conveniently expressed by writing the (not necessarily zero) double divergence in the same manner as the M function for πN scattering (3)

Mπ(±π )

=

F

±(ν,

t)

?

1 4m

[q/,

/q′]B±(ν,

t)

q′?M? ?′±ν qν

=

C ± (ν,

t)

?

1 4m

[q/,

/q′]D±(ν,

t)

(56)

Now de?ne the background πN amplitude as M? ≡ M ? MP where MP is the dispersion-theoretic pole of (39), see Figs. (2) and (5), for pseudoscalar πN N coupling. The on-pion-mass-shell Ward-Takahashi identities take the form

F?+(ν, t)

=

σN (t) fπ2

+

C+(ν, t)

(57)

ν?1F??(ν, t)

=

F1V (t) 2fπ2

?

g2 2m2

+

ν?1C?(ν, t)

(58)

ν?1B?+(ν, t) = ν?1D+(ν, t)

(59)

B??(ν, t)

=

F1V

(t) + F2V 2fπ2

(t)

?

g2 2m2

+

D?(ν, t)

,

(60)

where we notice that removal of the dispersion theory pole removes the contact term g2/m from (57). The on-shell analogue (58) of the Adler-Weisberger double soft pion point (55) goes to (55) because C?(ν, t) is de?ned as a double divergence in coordinate space which vanishes as (q → 0, q′ → 0). Comparing with the double-
soft pion Weinberg limit (54), which can be written as

F? + (q



0,

q′



0)

=

?

σN (0) fπ2

,

(61)

we note that the sign change is due to an analytic power series expansion in q and q′ (scaled to a typical hadron mass such as m) which obeys the Adler zero
(43) [61]. In fact, Brown, Pardee, and Peccei [60] suggest the sigma-term structure σN (t)[(q2 + q′2)/?2 ? 1] which manifests the sign change and the Adler zero. They
also con?rm the on-shell Cheng-Dashen (CD) low energy theorem of 1971 [61] :

F?+(0, 2?2)

=

σN (2?2) fπ2

+

O(?4)

.

(62)

With the on-shell Ward identities we are now in a position to test the current algebra representation with the data of πN scattering extrapolated to the subthreshold crescent of Fig. 2. These tests can establish the magnitude of the sigma

26

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Czech. J. Phys. 48 (1998)

Chiral Symmetry . . .
term in (57), suggest (with the aid of the PCAC hypothesis) the t dependence of the sigma term, and con?rm or deny the t dependence of the current algebra terms (from (47)) in (58) and (60). We turn to these tests in the next section.

4 Tests of Current Algebra and Soft Pion Theorems

In this lecture we use contemporary analyses of on-mass-shell πN scattering to i) test the structure of current algebra in this context, and to ii) examine the validity of the PCAC hypothesis (already validated for the Adler LET (44)) that the exact double soft-pion theorems of Weinberg (54) and of Adler and Weisberger (55) should be evident in the πN data. The former tests of current algebra were initiated by the data analysis of the Karlsruhe group [62] which use ?xed-t dispersion relations to extrapolate from the(ir) s-channel experimental phase shifts into the subthreshold region around (ν = 0, t = 0) (see Fig. 1). In fact, the experimental information in this region, once the nucleon pole contributions (see Fig. 2) have been removed, can be expressed in terms of the expansion coe?cients of the four background πN invariant amplitudes about this point. The values of these Ho¨hler expansion coe?cients obtained from data taken in the 1970’s, before the meson factories were built, are summarized in the encyclopedic Ref. [39]. The two current algebra models [63, 64] of πN amplitudes, which have been adapted to the construction of two-pion exchange three-nucleon forces [8-11], were calibrated against these pre-meson factory Ho¨hler coe?cients.
Recently two of the twenty-eight subthreshold subthreshold coe?cients have been re-evaluated [54] via ?xed-t dispersion relations from the latest partial wave analysis [56] of meson factory data. We will use the more comprehensive determination [55] of the amplitudes F?+(ν, t) and ν?1F??(ν, t) in the subthreshold crescent to fully carry out the tests i) and ii) of chiral symmetry. This determination also is from the VPI phase-shift analysis SM98 [56], but the analysis used interior dispersion relations (IDR), pioneered by Hite, et al. [65], and advocated by Ho¨hler for the purpose of testing chiral symmetry [66]. Interior dispersion relations are evaluated along hyperbolas in the Mandelstam plane which correspond to a ?xed angle in the s-channel laboratory frame. For t < 0 the path of ?xed lab angle lies entirely within the s-channel physical region and passes through the s-channel threshold point. (The IDR paths are similar to the lines of ?xed center of mass three-momentum q2 and x = cos θcm of Fig. 9, from Ref. [67] and illustrating a construction of invariant amplitudes from a simple summation of partial-wave amplitudes continued into the subthreshold crescent). With the IDR paths, one can reliably extrapolate the invariant amplitudes to any point in the subthreshold crescent and in particular evaluate the amplitudes along the vertical axis (ν = 0, 0 ≤ t ≤ 4?2) to test the on-shell analogues of the soft pion theorems.

Czech. J. Phys. 48 (1998)

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27

Sidney A. Coon

Fig. 9. A portion of the (all four particles on-mass-shell) Mandelstam (ν, t) plane, which includes the s-channel physical region and the subthreshold crescent.

But ?rst we must attempt to calibrate the IDR invariant amplitudes from SM98 phase shifts against independent measurements. The IDR value at the s-channel threshold point (ν = ?, t = 0) the scattering lengths

a(+)

=

4π(1

1 +

?/m)

F

+(?,

0)



?0.005

??1

a(?)

=

4π(1

1 +

?/m) F

?(?,

0)



+0.087

??1

in good agreement with the preliminary values a(+) = .0016 ± .0013 ??1 and a(?) = 0.0868 ± .0014 ??1 from the 1s level shifts and widths in pionic hydrogen and deuterium [68]. In addition, at the pseudothreshold point ν = 0, t = 4?2 (a
focus of the boundary hyperbola; see Fig. 1) the I = 0 ππ scattering length, a00, can be evaluated with the method of Ref. [69]. The IDR value a00 ≈ 0.20??1 again agrees well with the recent determination of a00 = 0.204 ± 0.014 ± 0.008 ??1 from the totally independent reaction πN → ππN [70]. That the IDR give reasonable values of a00 and a(+) at opposite points on the boundary of the subthreshold
crescent adds con?dence to the values in the central region.

28

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Czech. J. Phys. 48 (1998)

Chiral Symmetry . . .

Our ?rst test of a combined current algebra-PCAC prediction (55) uses the IDR determination of the isospin odd amplitude ν?1F??(ν, t) at the point (ν = 0, t = 0). The exact current algebra Adler-Weisberger result (55) is

ν?1F??(q → 0, q′ → 0) = ν?1F??(0, 0; 0, 0) =

1 fπ2

(1

?

gA2 )

= ?0.62 ??2 ,

(63)

where we have used the values of section 3.4. The empirical IDR ν?1F??(0, 0) ≈

?0.44 ??2, indicating that if ν and t are kept ?xed, the PCAC extrapolation from

the chiral symmetric Adler-Weisberger limit to the chirally broken real world is

minimal. Perhaps not as small as the 2% single soft pion Goldberger-Treiman result

of section 3.4 nor the (≈ 5%) single soft pion Adler consistency condition result

of section 3.5, but still the PCAC hypothesis appears to work well in this more

stringent test. Since the πN amplitude can be written with the choice of variables

ν, t; q2, q′2 or ν, q · q′; q2, q′2 or indeed some other combination, the magnitude of

the PCAC corrections will depend on which pair (ν, t) or (ν, νB = q · q′/2m) is held ?xed during the extrapolation from q → 0 to q2 = ?2 [71]. From our experience

with (44) in Section 3.5, we argue that holding ?xed (ν, t) (see Fig. 10) is the correct

way to apply PCAC. Given these empirical values of the on-shell amplitude F?+(ν, t) and ν?1F??(ν, t),

one can now visualize in Fig. 10 the proposed tests of the (isospin-even) PCAC

low energy theorems(LET) labeled Adler LET (44)) and Weinberg LET, the latter

the on-shell analogue of (54). Fig. 10 depicts the projection onto the hyperspace

ν = 0 of the coordinates (ν, t, q2, q′2) needed to describe a fully (pion) o?-shell

amplitude. The extrapolations shown are from the soft pion points A and A’ where F?+(q → 0) = F?+(q′ → 0) = 0 and from the double soft Weinberg point F?+(q →

0, q′



0)

=

F?+(0, 0; 0, 0) =

? σN (0)
fπ2

to

the

on-pion-mass-shell

line,

holding

?xed

ν

and t. The scale of the various points on the ?gure and of the isospin-even tests is

given by the Cheng-Dashen LET (62) and the (expected by PCAC) “anti-Cheng-

Dashen” value at the Weinberg point (54).

The IDR amplitude takes the following values on the on-mass-shell line:

F?+(0, t = 2?2) F?+(0, t = ?2)

= =

σN (2?2) fπ2
0

+ +

O(?4) O(?2)

≈ +1.35 ??1 ≈ ?0.08 ??1

F?+(0, t = 0)

=

? σN (0)
fπ2

+ O(?2) ≈ ?1.34 ??1

(64)

Let us make a careful distinction between the expected corrections indicated in (64). The O(?4) corrections of the top line are the result of a rigorous on-shell derivation of a Ward identity [60, 61]. The putative O(?2) corrections of the lower two lines are what one might expect from the already discussed corrections to the Goldberger-Treiman relation, the Adler zero, and the Adler-Weisberger relation as one goes from the chiral symmetric world to the world of πN scattering. It is those latter presumed PCAC corrections which we are trying to test. If one extends the observed pattern of small PCAC corrections to the bottom line of (64), one can interpret F?+(0, t = 0) ≈ ?F?+(0, t = 2?2) as indicating that the t

Czech. J. Phys. 48 (1998)

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29

Sidney A. Coon

dependence of σ(t) (50), as determined from the πN scattering data, is quite small

indeed.

The

alternative

picture

of

σN (2?2) fπ2

?

σN (0) fπ2



0.25

??1

[73]

would

demand

a quite large PCAC correction along the lower plane of Fig. 10 to get back to the

empirical amplitude; much larger than any other PCAC correction evaluated in the

πN system or elsewhere (Ref. [34]). Furthermore, we will see in Fig. 11 that the amplitude F?+(0, t) is nearly linear in t in the interval between t = 0 and t = 2?2 (with increasing curvature at t approaches the ππ → N N? pseudo-threshold at 4?2).

We will return to this issue after investigating the uniqueness of the IDR amplitudes

from which these conclusions are drawn.

t
On-shell Line

t=2 2

t= 2

PCAC
A0
t=0 W

2
q02

PCAC

Cheng-Dashen LET
A
PCAC

Adler LET PCAC

2
q2

Weinberg LET

Fig. 10. The geometry of the o?-mass-shell πN amplitude F?+(ν, t, q2, q′2) for ν = 0.

The value of the IDR amplitude F?+(ν = 0, t = 2?2) ≈ 1.35 ??1 at the ChengDashen point leads to a value of the sigma term from (62) of
σN (2?2) = F?+(0, 2?2)fπ2 ≈ 83 MeV

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Czech. J. Phys. 48 (1998)

Chiral Symmetry . . .

Fig. 11. The values of F?+(0, t) at the on-mass-shell line of Fig.10. The solid line and the ?lled circles are from the pre-meson factory data analysed in Ref. [39]. The double line corresponds to the IDR analyses of meson factory phaseshifts SM98 (Ref. [55]) and the short dashed line is constructed with the subthreshold coe?cients determined from SM98 with forward dispersion relations (Ref. [54]). The star includes the “curvature corrections”
to estimate the latter amplitude at the Cheng-Dashen point.
This value is at the high end of a range of 40 MeV to 80 MeV presented in 1997 at the MENU97 conference and reviewed there by Wagner [72]. An independent application of forward dispersion relations to the SM98 partial wave analysis (itself heavily in?uenced by various sets of dispersion relation constraints [54]) gives
σN (2?2) = F?+(0, 2?2)fπ2 ≈ (f1+ + 2?2f2+)fπ2 ≈ 77 MeV
where we use the notation of Appendix A of Ref. [11] for the Ho¨hler subthreshold coe?cients. To this the authors of Ref. [54] estimate in various ways a “curvature correction” to arrive at a ?nal value of 82 to 92 MeV for σN (2?2). They also obtain at the Weinberg LET F?+(0, t = 0) ≡ f1+ = ?1.30 ??1 which agrees well with the IDR value. We have already noted in section 3.5 that the Adler zero is closely emulated both by the IDR value F?+(0, t = ?2) ≈ ?0.08 ??1 and by the forward dispersion relation value F?+(0, t = ?2) = f1+ + ?2f2+ = (?1.30 + 1.27) ??1 = ?0.03 ??1. These two di?erent dispersion relation analyses of the same set of phase shifts agree well with each other on the subthreshold (but on-shell) line (ν = 0, 0 ≤ t ≤ 2?2) important for tests of PCAC (See Fig. 11). Both analyses

Czech. J. Phys. 48 (1998)

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Sidney A. Coon

include the statements that if the authors replace the SM98 phase shifts by the old Karlsruhe phase shifts (from the pre-meson factory data) they reproduce the Karlsruhe dispersion relation results. It would seem that this value of the sigma term follows from the SM98 phase shifts and is not an artifact of a particular type of dispersion theory analysis. Be warned, however, that extrapolations to the unphysical Cheng-Dashen point with potential model amplitudes ?t to (perhaps) di?erent data sets give sigma terms at the low end of the MENU97 range and the reader is encouraged to continue monitoring the situation, especially the CNIexperiment with CHAOS at TRIUMF [68].
We have established the scale set by the size of the sigma term, the near linearity and change of sign of the empirical amplitude F?+(0, t) in the range 0 ≤ t ≤ 2?2, and the validity of the PCAC hypothesis for the o?-shell amplitudes F?+(0, t; q2, q′2) and F??(0, t; q2, q′2). It remains to discuss the t dependence of the ?rst terms in the on-shell current algebra Ward identities of Eqs. (57), (58), and (60). The current algebra terms (49) are simply given by the measured electromagnetic form factors. Given that the intrinsic t dependence of σN (t) is quite small, one can set

F?+(ν, t) =

σN (2?2) fπ2

[1

+

β(

t ?2

?

2)]

+

C+(ν, t),

(65)

where the background amplitude C+(ν, t) of (56) is modeled by the overwhelm-
ingly dominant ?(1232) isobar. The two approaches to this background amplitude
have used dispersion theory for the (over 20) invariant amplitudes of the axialvector nucleon amplitude M?ijν [63] or a ?-propagator ?eld theory model [64]. Both models of the background amplitude give quite similar results for C+(0, t) in the low t regime [57]. The dispersion theoretic C+(ν, t; q2q′2) ≈ c1ν2 + c2q · q′ + O(q4) contains an unknown subtraction constant in the g?ν term of the axial-nucleon amplitude M? ?+ν, which is moved into the unknown β of (refeq:pnamp) and ultimately determined by the data.
This particular t dependence of the multiplier of σN is suggested by a low energy expansion similar to the Weinberg amplitude for low-energy ππ scattering. This
amplitude in the linear approximation satisfying all current algebra/PCAC and quark model ((?3, 3) + (3, ?3)) constraints is [23]

Tππ

=

1 fπ2

[(s

?

?2



abδcd

+

(t ? ?2)δacδbd

+ (u

? ?2)δadδbc]

+ O(?4),

(66)

along with the quark model ππ σ term σππ = ?2. Generalizations of (66) to SU (3) pseudoscalar meson-meson scattering were worked out by Osborn [74] and by Li and Pagels [75]. In particular, for πP → πP scattering, the o?-shell low-energy generalization of (66) in the linear approximation includes a (t-channel) isospineven part

TPt?Peven

=

σP P fπ2

[(1

?

βP )( q2

+ q′2 ?2

?

1)

+

βP

(

t ?2

?

1)]

+

O(?4)

(67)

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Czech. J. Phys. 48 (1998)

Chiral Symmetry . . .

for

βπ, βK , βη8

=

1,

1 2

,

0

and σππ, σKK , ση8η8 =

1,

1 2

,

1 3

?2.

(68)

In fact the t dependent structure of (65) follows from (67) (with constant sigma terms) for scattering of on-shell pions from a meson target. Moreover, this linear
(in t) structure of both (65) and (67) manifests the Adler and Weinberg soft pion theorems.
For πN → πN scattering, however, the fact that the nucleon four-momentum cannot become soft means that βP in (65) cannot be a priori predicted as it is in (68) for meson targets. Instead β in (65) for on-mass-shell πN → πN scattering is ?tted to the the IDR curve in Fig. 11 to ?nd β ≈ 0.45, quite near to βK = 1/2 in (68), perhaps re?ecting the same isospin structure of K and N. The above current algebra/PCAC analysis in (65) and in (66-68) does not mean that the σ term occurring in four-point function πN scattering has an intrinsic t dependence. Rather, the linear t dependent factor in (65) is a PCAC realization of the unknown subtraction constant βq′ · qσN , with the β determined by the Adler and Weinberg LETs.
An alternative ansatz for the t dependence of the sigma term stems from the SU(2) linear σ model (LσM) with N, π, σ as elementary ?elds [1]. (For a recent review of the resurgence of interest in the σ meson, see the references in [77].) Using a pseudoscalar rather than pseudovector πNN coupling means that the t-channel σ pole has a background F?+amplitude [76] proportional to (m2σ ? ?2)(m2σ ? t)?1 ? 1. This structure automatically complies with the 3 low-energy theorems ; e. g. it vanishes at t=?2 as does the Adler zero. Thus the isospin-even background πN amplitude can be expressed in the LσM at ν = 0 as [76]

F?L+σM (0, t)

=

gπ2N N m

m2σ ? ?2 m2σ ? t

?

1

(69)

To obtain a quantitative ?t to the empirical amplitude this must be supplemented by the ? contribution. In the dispersion relation model the amplitude becomes

F?+(0, t)

=

gπ2N N m

m2σ ? ?2 m2σ ? t

?

1

+ β′q · q′ + C+(ν, t)

(70)

where again the parameter β′ shifts the unknown subtraction constant in the ? contribution to the PCAC realization β′q′ · qσN . A ?t which is within the two lines of the double line of Fig. 11 for 0 ≤ t ≤ 2?2 can be made with β′ = 1.44??3 and
mσ = 4.68? ≈ 653 MeV [78], the latter a quite reasonable value when compared with the current σ meson phenomenology [77].
With these determinations of the t dependence of the sigma term in (57), we
can ?nally ?nish the tests suggested in the last sentence of Section 3. That is,
how well do the current algebra models describe the πN data extrapolated to the subthreshold crescent of Fig. 2? Away from the ν = 0 line in the amplitude F?+, these tests are given by expanding the isobar modeled C±(ν, t) and D±(ν, t) and

Czech. J. Phys. 48 (1998)

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comparing the theoretical expansion coe?cients with the empirical Ho¨hler coe?cients. This is an old story and the general trends are summarized in Appendix A of Ref. [11]. The 28 subthreshold coe?cients are matched very well indeed by the current algebra amplitudes (57-60), once the t dependence of the sigma term is ?xed empirically (as above). The t dependence of the two current algebra terms in (58) and (60) is indeed given quite well by the isovector vector current of (13). The preliminary determinations of the subthreshold Ho¨hler coe?cients from the meson factory data [79] does not change the qualitative picture given in [11, 63]. Only the scale of F?+(ν, t), set by the size of the sigma term, has changed with the advent of increasingly more precise πN data.
The t dependence of the sigma term was suggested by PCAC o?-shell constraints (67) or by the linear sigma model (69). The success of the on-shell tests and of the examples of the PCAC hypothesis suggests a reliable PCAC o?-pion-mass shell extrapolation for the not-so far extrapolations of the πN amplitude discussed in the Introduction. However, the shorter range parts of the two-pion exchange threebody force will be quite di?erent for the t dependence of the sigma term from (67) or (69). The shorter range parts which follow from (67) have been discussed and compared with those of chiral perturbation theory in Ref. [80] which used the techniques of Ref. [15]. The implications of (69) for three-body forces, threshold pion production, and pion condensation remain to be worked out.
I would like to thank the organizers of the Praha Indian-Summer School for inviting me to this beautiful city. I am indebted to Michael D. Scadron for teaching me about current algebra and PCAC and for reading the early parts of this manuscript. I thank William B. Kaufmann for many discussions of the IDR analysis of the meson factory πN data.

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0 1

1 0

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[38] This convention of Ref. [34] chooses a non-unique phase factor. There are other ?as(sπig1n+meiπn2ts)/i√n 2thaenlditeπr?at=ure+; (fπor1 e?xaiπm2p)/le√t2h.e “Condon-Shortley” convention is π+ =

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[44] Particle Data Group, C. Caso et al. Euro. Phys. J C3, (1998) 1.

[45] B. R. Holstein, Phys. Lett. B244, (1990) 83, ?nds fπ = 92.4 ± 0.2 MeV including radiative corrections. With the most recent rates and masses of the Particle Data Group [44], this value becomes fπ = 92.6 ± 0.2 MeV.

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[47] M. M. Pavan and R. A. Arndt, πN Newsletter, No. 13 (1997) ed. D. Drechsel et al., p. 165.

[48] J. J. de Swart, M. C. M. Rentmeester, and R. G. E. Timmermans, πN Newsletter, No. 13, (1997) ed. D. Drechsel et al., p. 96.

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[50] An analysis of contemporary np charge exchange scattering supports the older higher value of gπNN (q2 = m2π) ≈ 13.51. See B. Loiseau, T. E. O. Ericson, J. Rahm, J.
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Chiral Symmetry . . .

[54] R. A. Arndt, R. L. Workman, I. I. Strakovsky, and M. M. Pavan, nucl-th/9807087; see also Ref.[47].
[55] W. B. Kaufmann, G. E. Hite, and R. J. Jacob, πN Newsletter, No. 13, (1997) ed. D. Drechsel al., p. 16; W. B. Kaufmann and G. E. Hite, submitted to Phys. Rev. D.
[56] VPI phase-shift analysis SM98, courtesy of R. A. Arndt. Internet access is through http://clsaid.phys.vt.edu?CAPS/.
[57] These values are signi?cantly closer to zero than those obtained from subthreshold analyses of pre-meson factory data. A convenient compilation of the early amplitudes can be found in Table 1 of S. A. Coon and M. D. Scadron, J. Phys. G: Nucl. Phys. 18 (1992) 1923.
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[72] G. H. Wagner, Proc. of MENU97, Vancouver, July 28-Aug 1, 1997 in πN Newsletter 13, (1997) ed. D. Drechsel al., pp 385-392. The original contributions concerning the sigma term are also in this issue.
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[76] H. Schnitzer, Phys. Rev. D5(1972)1482; H. F. Jones and M. D. Scadron, Phys. Rev. D11(1975)174.
[77] J. L. Lucio, M. Napsuciale, and M. Ruiz-Altaba, “The Linear Sigma Model at Work: Successful Postdiction for Pion Scattering”, hep-ph/9903420. This paper cites 16 recent data analyses which point to a wide scalar resonance in the vicinity of 600 MeV, and 14 recent theoretical papers on the linear sigma model, both at the nucleon level and at the quark level. More recent references can be found in the talks at the Workshop on Hadron Spectroscopy, March 8-13, 1999 Frascati, Italy which will appear in the Frascati Physics Series.
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