A graphical method for multi-soft-pion processes

ANNALS

OF

PHYSICS:

42, 467475

A Graphical

(1967)

Method HENRY

for

Multi-Soft-Pion

Processes

D. I. ABARBANEL* AND

SHMUEL Palmer Physical

Laboratory,

NUSSINOV~

Princelon

Gniversily,

Princeion,

l\;ew

Jersey

‘4 set of graphical rules is established for writing down all the cont,ributions to t,he hadron process cy---f @+ TZ soft pions dictat,ed by the algebra of current commutators of Cell-Mann and PCAC. Explicit results are given for n = 3. An example of n = 4 is treated in the form of soft ?T--?Tscattering, where the small s-wave scattering lengths found by Weinberg are rederived. I. INTRODUCTION Since the impressive work of Adler and Weisberger (I ) successfully combined the algebra of currents proposed by Gell-blame (2) and the hypothesis of a

partially conserved axial vector current (PCAC) to yield quantitative predictions about processesinvolving ‘%oft pions”, we have learned much about such processesin the form of low energy theorems on the amplitudes involved (3). The application of these low energy theorems has in the past confined itself, in the main, to reactions involving only one or two soft pions. It is not unreasonable to expect,, however, t(hat in bhe future one will discuss multipion processes via these methods. A start in this direction has been t&en by S. Weinberg and his students (4) who have undertaken a detailed st(udy of t(he production process aN --f 27rN. In t.his paper we would like to describe a graphical t,echnique for immediately writing down the co&butions to t,he process a --) p + 7~pions which are dickatcd by the algebra of currents and the use of PCAC. ID our description we shall include the exhibition of all explicit, pion poles corresponding to t,he n soft pions. This allows one to let the II pions be on the mass shell and lets one proceed to approximations on the final ident,ity we shall write

down

by imagining

an expansion

in pion

four-momenta

t,o he made.

Thus

the philosophy in applying our results to physics is that advocaatedby Weinberg, especially, and others (5). * National Science Folmdat ion Postdoctoral Fellow. t Work supported by the U. 8. Air Force Office of Research, Air Research aud L)evelopment Command, under rontract r\F J9(038)-1535. 467

l’lw prohhn of writing 3ow1~ a11 the mntrihutions givw by PCAC and the ~rrctil algebras t#o the amplit,ude for O(---f /3 + 11pions is , after the relevant equal time commutation relations have been given, a combinatorial one, as is the rcmoral of the piou poles. After giving the solution to t.hese problems in the next section, we will discuss a physical application in the form of B-K scattering. This process has been studied in detail by Weinberg (6) and T
THIS

GRAPHICAL

RIETHOI~

In deriving low-energy t,heorems for soft, pions me begins with the nlat#rix clement of an appropriate number of nxinl vec%or currents het#wcen arbitrary hadron states O(and 6. By partial int8cgrations and the usr of the following identity on !Z’ products: (~/~.c~~)T(.~,(.cM

(rljK(.r,).

. .C(.C,~)

= T(a,J,(:c)lilc.~~)K(~~j.

+ 6(X” - .Cl,~)Z’( LJ,,(E), A(XI)]B(.C2).

. .Cix,))

+ 6(x,, - .Ld?‘(A(.r,~JJoix),

. ~C(.c,,))

B(zz)].

+ . . . + 8c.h - .~~~)nA(.hjB(4~~

4m9,

. .C(Z,)) (1) m4J),

one turns the original mat,rix clement into a sum of ot,hcr mat,rix elements between the st,ates (Y and p involving divergences of axial currents and equal-time commutators (ETC) of the axial cm-r&s with whatever objects are contained in the T product. I’CAC is used t,o relate t#he axial vector divergences to the soft pion fields and to provide a “smooth” off-shell extrapolation for those fields. In the following we shall assume t#hat no ot#her oper&ors besides axial currents appear in the T product, and we shall perform partial integrat’ions on all the axial vect,or indices. The extension of our scheme to those cases where other operators appear is st,raightforward once t*he ETC of the axial current, wit,h t,hese objects is given (7). The ETC’s we will encounter are (2) 6(X” - yu)[Ao”(x),

A;(y)]

6(cco - yo)[Ao”(2T), T,“(y)] q.co - yo)[A,“(x), 6(.c” - yo)[Aa”(s),

Db(y)l u(y)]

= iGzbcKy1.)84(.c - y),

(2)

= it,*,il,‘(n)d”(.x

(3)

- y),

= Saba(z)S”(.c - y),

(4)

= D”(.c)S4(n:

(5)

- y).

In these ETC, i2p”(e) is the axial vector current participating in AI = 1, = 0 weak decays; V,“(X), t’he vector current; and D”(z) = dpArra(zc). The isospin indices a, b, and c run from 1 to 3. In (4) we encounter an object called V(X). The fact that its coefficient is &b means we have assumed it to lie in a chiral AY

GRAPHICAL

METHOD

FOR

MULTI-SOFT-PION

PROCESSES

469

quadruplet with D”(X) (5), (6). III the a-model of Gell-JIann and Levy (8) it is precisely the field for the u-meson. The ETC (5) follows from (a), (3), (4), and t,he Jacobi identity. Finally we should note t,hat we have ignored possible “Schwinger” terms. Following the cxhibit’ion of the various t#ermn encountered in the partial integrat’ions we will t,hen subtract out t’he explicit pion poles corresponding to the axial currents turning into soft pions via their divergence. This allows us to pass to the mass shell of t’hcse pious and make the required expansion in pion momenta. Our starting point is, therefore,

We shall represent’ f,his as a “scattering amplitude” for o( ----f p + u axial c:urrent,s as shown in I’ig. 1. When we muhiply t,his by ipI,,, . * . ipnp,L , bring in the momenta t,o become derivatives cm t,he T-product) and perform the multiple derivatives using (1) repeatedly, we encounter

which is represented by Fig. 3, and a whole host of fourier transformed 7’products which include operators arising from ETC’s. We represent each of t,hese T-products by a diagram. The lines entering the central blob of the diagram represent the operators appearing in the T product. Each line occurs at the end of a chain indicating t,he multiple commutat,or which led to the corresponding operator. The chains are constructed out, of the following “commut,ation” vertices (see Fig. 3) : (i) Fig. 3a which represents an axial current divergence. (ii) Fig. 3b which represents (2) and (a). If a is the index of the axial current entering t#he chain, then a factor &be is associated with the vertex, where c is t,he index of the current closest to the “blob”. If the chain begins with t,his vert,ex, then multiply it by fi(~,~ - pb)p, where p is the vect,or index common to the chain. (iii) Fig. 3~: which represents (-1) and carries a coefficients &,h with it. (iv) Fig. 3d which corresponds tjo (5). We can see from (2)-( 5) that we will encounter only pure vector chains in

470

ABARBASEL

a

T

AKD

M aI /’

represent,ation

3c.

API a2 APL

an A!+

FIG. 1. The diagrammatic representation cllrrents, and (Y and 0 are hadron states.

FIG. 2. The diagrammatic axial vector cllrrents.

NUSSINOV

of Eq. (6). The wiggly lines are axial vector

of Eq. (7). The dotted lines are divergences of

3d.

3 abed. The fundamental vertices entering the chains in the reduction They represent the ETC’s, Eqs. (2)-(5). The unbroken lines are a’s, FIG.

of Eq. (6).

which A and V alternate or pure scalar chains in which commutation vertices with A’s entering the chains produce alternating D’s and 2s. We shall call a chain where Fig. 3a enters the central blob directly a scalar chain of length one. Scalar chains of odd length will have D’s entering t’he blob, scalar chains of even length, (T’s. As an example of these diagrams we write out in Fig. 4 all t#heterms which arise in the reduction of t,hree axial currents to D’s and ETC’s. It is convenient to characterize the kinds of diagrams that will appear in the reduction of n axial currents by the number and length of the scalar and vector chains entering the central blob. To each partition of n: P = (nl , . . . , G ,??~l, . . . , ~1,) with CT=, ini + c&1jmi = n, there corresponds a diagram in which there are n, scalar chains of length i and ‘IX, vector chains of length j and altogether s scalar chains and 2’vector chains. Diagrams of a given type (that is,

GRAPHICAL

METHOD

permutations

FOR

MULTI-SOFT-PION

471

PROCESSES

1

tpermutations I

bc

+

l

t

1

permutations

+,k(Pc-Pb)p
+

permutations

2 FIG.

4. The

diagrams

appeawitlg

in the

reduction

1 .

of ip~~ip?~ip~h(~ll$)p,

partition of n) which differ only in the order of lines entering a chain or in the permutation among chains should not he counted more t,han once. In performing the differentiations we can construct a scalar chain in one way only since the consecutive Ao’s are always to be commuted with the u or D operator available. In construct’ing a vector chain of lengthj the A0 beginning the chain can begin it in j - 1 ways by being commuted with any of the other j - 1 axial currems which will enter the chain. The number of distinct, ways of getting a diagram corresponding t,o the partit,ion P via the multiple use of (1) is N, = II 1B (j - l)““/

>g (j !)mi(jtlj !) 9

(i !)“i(n;

!).

(8)

To secure proper counting and to t*reat all the mesonson an equal footing one proceeds by constructing a particular diagram of kind P, t,hen he symmetrizes with respect t,o meson indices muhiplies by N, and finally sums over all parti-

472

ABARBAISEL

ANTI

NUSSINOV

tions. This completes the rethMion of (n/z::: :z)~~ into (dl’“” “““)pa and ETC terms. Next we want t’o remove the pion poles from ( J1~~:::~;)B, . This is accomplishcd by making the substitution in each axial vector current ~‘l;l:(~j)

+ ile;(Xj)

-

ipjp,Da’(Sj)/JILT2.

(9)

Because the coefficient1of pj,,, is a D we see that only t,he scalar chains will be affected by this operat’ion. A scalar chain of lengt,h 1 ending in Da”(2;) will be multiplied by (pi” + ~,~)/tt~.‘. In a scalar chain of even length k, t,he line ent,ering t,he central blob is a (Tand thus carries no isospiu index. In the symmetrization over isospin indices in the terms arising from the original reduction of and in the ext’ra terms arising from the reduction of iPI,, . . ip,,,(N;::::;::)@,

one finds the same number of terms, namely

Since these are the only kinds of term which can contribute to a scalar chain of even length which has A”’ entering it,, one must multiply a scalar chain of even length by

1+ entering

chain

In a scalar chain of odd length y # 1 the line entering the central blob is a D and thus carries the isospin index of the last axial current entering the chain. In the reduction of ipI,,, * . . z$,,,,(ilr~~:::~~),, to the scalar chain of length q one has q choices for the last index to enter the chain. In the reduction of (10) to the scalar chain of length q, one has q - 1 choices since the D beginning the chain has been specified. Thus one must multiply a scalar chain of odd length p by 1+y

q---l

c

all axial8 entering the chain except the last

GRAPHICAL

METHOD

FOX

MULTI-SOFT-PION

TABLE

PHOCESSES

473

I

THE SUM OF DI.LGR~LMS WITH ASSOCUTED MOMENTUM FSZTORS WHICH EQUALS

ipl,ip2”ip,h(~~;uyh~)pol’ Diagram

Expression

NP

1

‘I The symmetrization

over isospin fact,ors in the third column has been deleted

With these additional multiplicative factors we will have removed all pion poles from (M~~:::~;)ga. Let us call this result (~::::~:)~o . The terms equaling @lP,. . . z&,,,(Asf:: ::z;)sa when 12.= 3 are shown in Table I. The factors corresponding to symmetrization in isospin indices have been delet,ed for simplicity. The notation used in t,he third column is iO*(p1), --.O,(pn>f!sm =

s dx1 . .. d4x,L ill)

.exp (--i

g

[Jj'x.j)

(0 I T(Ol(:rI)

. ‘. On(JA))l a).

Going to the massshell, p, = --HL~~is now without any dangers and one may freely do it. Of course, one may also evaluate the pi” at any point he pleases,and, in particular, an attractive choice may be that, which eliminates all the g terms.

474

ABARBANEL

III.

AND

SOME

NUSSINOV

REMARKS

(1) If one knew all the matrix elements of t,he ETC terms between cx and fl, then by dropping the iplpl. . .ipl,,,(NE:: 1:::)~~ term, the amplitude has been determined for o( -+ 0 + nx to 0( p”) in an expansion in powers of pion momenta. If there should arise pole terms in N, as would be the case if (Y or /3 contained baryons, they must first be subtracted out before the O(p”) expansion is comp1et.e. In dropping the non-pole terms of N, we are again invoking PCAC in the sense that it encourages us to hope that for soft pions the amplitude a! -+ p + na is slowly varying in the momenta of these pions. Alternat,ively one may use the result for ,@I,,; . .~~np,(N~:l::~:)~, to estimate, in the spirit of Adler and Dothan (9), N itself to O(p”). (2) The generalization to the cases where other operators appear in the T product is quite straightforward when one knows their commutation relations with A,“(X) at equal times. One only needs to define new fundamental vertices, write out the appropriate diagrams and perform the necessary combinatorics. IV.

AN

APPLICATION

TO

SOFT

r-9

SCATTERING

As an application of these t#echniques we take (Y = /3 = 0 and n = 4 with p, and p, represented as outgoing pions. Ostensibly we ran now determine the on mass shell x-r scattering amplitude to fourth order in pion momenta. In reality, however, we encounter, in the not’ation of Eq. (II), terms like { a(pl + pz), c(p3 + p4)), etc. which we cannot handle. On the mass shell, however, it turns out t,hat, by considering (~4,~(pr)~4,~(pz)a(p~ + p4)} and bringing in twomomenta we can eliminate the terms (u, u) , {D, D) , and {c) , and by considering we can {A,“(n), ~4,b(p2)Vxk’(p3 + ~411 and again bringing in two-momenta, eliminate the terms (V, V), and (A, D). This ehmination of terms involves dropping terms of O(p2) and reduces t’he level to which we can determine the F-P amplitude to second order in the momenta. Writing D”(X) = 8’,n~‘+,~(X) w h ere F, is the ~1~ decay amplitude we have for the r-r scattering amplit,ude (on the mass shell) t’o the announced order of pion momenta nf(T”(pl)

+ . (w&

73232) -

---f

&Ald)92(P2

+ two permutations

~‘(ps‘)

+ -

am)

Pl)%(P4

+ Fr-‘[?$mr2

= -

-F,“[?,i

-

2.ff”(-(p1

+

~2)“)

Pd

-

(12) f”(

-

(PI

+

p2)2)]&b&d

+ two permutations. In Eq. (12) f” and f” are respectively the vector and c form factors of on-shell pions evaluated at the “masses” of the photon or g equal to - (pl + ~2)~ (or its permutations).

GI1APHICAL

METHOI)

F021

MULTI-SOFT-PION

By evaluating (12) at threshold, -(pl 1 = 0 and 1 = 2 P-K scattering lengths

8rgA2nl,az = -4

-

+ p,)”

PHOCESSES

175

= 4rr,‘?, WC may extracat the

2f”(~u,2, ,wL,‘, 0)/m,‘.

(13)

The t,hird argument off” is the d “mass” (2). Khuri (6) has shown that one may evaluutef”(~~~~112,0, TU,‘) using PCAC and our Eq. (5). He finds the value at that point to be --M,‘. If we simply accept his argument,s that S” is such a slowly varying function of all its variables that it may be accurately approximated by (5) results: -lHT2 at the point,s we’re interested in, then we find Weinberg’s and nl,a2 = -0.06. ‘mTaO = 0.2 The dangerous part of Khuri’s extrapolation of j” from Ohe known point to the desired points lies in the changing of the u “mass” which is sensitive to large s-wave int,eractions. It may be worthwhile to point out that even the extreme assumption of the domination of the G~R form factor by a sharp I = 0 P-T resonance at 2.5 m, c 350 meV leads only to a 1040% alteration in Weinberg’s values of a2 and a0 . This is due to the particular forms of (13) and (14) whkh contain t,erms independent of f” and possibly suggest a certain caIiCe&Lti~JIl in t,he t,wo extrapolations from “mc2” = ~~~~’t,o “111,“’ = 0 and ‘L~~z,2” = k,‘. RECEIVED

: November

16, 1966 REFERENCES

1. S. L. ADLER, Phys. Rev. Letters 14, 1051 (1965); Whys. 12ev. 140, B736 (1965); W. I. WEISBERGER, Phys. Rev. Letters 14, 1047 (1965); Ph,ys. Rev. 143, 1302 (1966). 2. M. GELL-MANN, Phys. Rev. 126, 1067 (1962) and Physics 1, 63 (1964). 3. A comprehensive account of the applications of these techniques is given in the rapporteur’s talk by R. F. Dashen at the Thirteenth High-Energy Physics Conference at Berkeley, September, 1966 4. S. WEINBERG, Phys. Rev. Letters 16, 879 (1966); S. WEINBERG AND L. CHANG, to be published. 5. S. WEINBERG, Phys. Rev. Letters 17, 336 and 616 (196G). 6. N. N. KHURI, SLAC. Phys. Rev. 163, 1477 (1967). 7. There is no point in explicitly reducing ext)ernal photons or fermions. The extra ETC’s which would arise in these cases are equivalent to the pole terms necessary if LY or fi cont.ains baryons or to the termswhich appear because D*(z) + P(Z)+ ee&~(z)il~*(z) [Ah(z) is the electromagnetic vector potential] when a or p contains photons. In practice, therefore, the only new commutators one expects are those of the axial current with the Hamiltonian responsible for nonleptonic weak decays. 8. M. GELL-M.ZNN AND M. LEVY, Nuovo Cimento 16, 705 (1960). 9. S. L. ADLER AND Y. DOTHAN, Phys. Rev. X1,1267 (1966).