Axial vector current divergences and the pseudoscalar meson spectrum

Volume 97B, number 1

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17 November 1980

AXIAL VECTOR CURRENT DIVERGENCES AND THE PSEUDOSCALAR MESON SPECTRUM

Norman H. FUCHS Physics Department, lh~rdue University, West Lafayette, I N 4 7907, USA Received 9 June 1980

The strong anomaly in the axial vector current divergence is simply related to the mass of the ninth pseudoscalar meson in an SU(3) × SU(3) quark model. A discussion is given of the dependence of the pseudoscalar meson masses u p o n quark masses as well as u p o n the strong coupling; the SU(3) character of the states is also treated.

It is generally believed that in the limit of vanishing bare mass for the up, down and strange quarks (mu, rod, ms) , the 7r, K and r/would become massless Goldstone bosons. In this context the nonexistence of a ninth "light" pseudoscalar meson has appeared to be a problem for many years [ 1 ]. In quantum chromodynamics (QCD) for example, there is a nonet of axial currents; eight of them are conserved in the massless limit. The ninth, the U(1) current, fails to be conserved in this limit due to the strong anomaly. Nevertheless, there is controversy as to whether or not one should conclude that a ninth Goldstone boson exists in QCD [21. In any event, the r/' definitely does exist. If it were not for the relatively large mass of the r/, it is doubtful that one would have hesitated to place it into a nonet of Goldstone bosons. In fact, standard (and successful) phenomenology [3] has always taken the r / t o have a quark content corresponding to the singlet member of the nonet to which the n, K and r/belong. We will show in the present work that the strong anomaly does in fact lead to a nonvanishing mass for the ninth pseudoscalar meson in SU(3) X SU(3) even ip the limit of zero quark masses. We discuss the dependence of the pseudoscalar mesons' properties on quark masses as well as on the strong coupling. We consider matrix elements of axial-vector currents and their divergences for the case in which the quark masses mu, r o d , m s , ... are not degenerate. The axialvector currents themselves are expected to transform just as they appear to transform. Therefore, the matrix

element (rr0 [~7,'},5u + d'),u3'5 d [0) = 0 ,

(1)

as a result of the isospin transformation properties of lr0 U = 1) and the current (I = 0). But the current has a strong anomaly, so

0 = (nO li(mu~Tsu + mdd75d ) + (g2/8$r2) Tr GGI0). (2) Furthermore, (n o 1fiV~VsU - dvu75 d[ 0} = frrP~ ,

(3)

so

2i (rr01 rnu~75 u - ruddy 5 d 10) = J'~rrn2 •

(4)

Now, ifu75u, d75d transformed "naively" we would conclude m u - md ~

2

-2<,~°1 (a2/8~2) Tr a ~ I0> - ~uu; ~ d , ' ~ m = ,

(5)

which we (correctly) pointed out [41 to be disastrous, since it implies that rn u = m d in the limit g2 _+ 0. However, we have successfully [5] treated fi75u/rnd and d75d/md as simple SU(3) objects; therefore, we would conclude that 2 m2 _ 2 (Tr01(g2/Scr2) Tr G~ i O) m u d 2 ,) - 2+m2f~rGr, (5 mu

which is equally disastrous, for the same reason. So the question arises as to where the argument breaks down. One of the results of the present work is an answer to this. First let's consider the simpler case of two quarks - SU(2) X SU(2).

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The point is that the ~r0 is only an eigenstate of isospin approximately, and that this approximation becomes worse as g2 becomes small. This is easy to see in the extreme case o f g 2 = 0, since then the eigenstates are Pu ~ pseudoscalar bound state of u and g ,

(6)

l+e (1.+ 42) 1/2 f=frr"

7r = (Pu - @Pd)/( 1 + 42) 1/2,

(7)

o = (Pal + @Pu)/( 1 + 42)1/2 ,

(8)

where @~ 1 for physical states, but @= O(g 4) and so vanishes for g2 _+ 0. Returning to the original argument we see that it must be modified as follows (we use a formalism [5] which gives quark masses squared, but this is not essential)

(12)

Consequently, m 2 -@m 2 • 2 + 2<7rl g2 TrGGI0) f~rm. m2u + ;C'nd ..2 87r2

Pd -= pseudoscalar bound state of d and d , with masses proportional (respectively) to m u2 and m2d. In other words, in terms of states,

17 November 1980

I-@ (1 + 42) 1/2

(13)

which should be compared to (5') or (5) above. As can be seen, the disaster has been averted - i.e., the right hand side is not zero. Of course, in the limit @-+ 1, the right hand side does vanish, but this clearly implies that g2 is not small; for g2 _+ 0, we have @-+ 0 and then (g2-+ 0) f ~ = f ,

(14)

which is perfectly consistent• Now, a similar discussion may be given for matrix elements of the other state, o. (o1~7u75 u - dTu75 d 10)


= - (Pd Ia7** 75 d ] 0) + @( Pu IU3'**3'5 u I0 )

_ - @(Pdld3'.3' 5 d 10)

(1 + 42) 1/2

(1 + 42) 1/2 _ efu-fd fu -- @fd 1-- @ (1 + 42) 1/2 p** (1+

(1+ @2)17-2p" -

where isospin implies fu =fd =- f Taking the divergence, we find

2 i ( 2 m 2 - 2m2@)H (1 + @2)1/2 _

+2
152

1-@

; 2

(16)

(l+@2)l/2Jmo •

_ fd + @fu f P . ,(17) (1 + 42)1/2

so 2i(2mu2g + 2 m 2 ) H + 2

fu + @fd -

(1 + 42) 1/2 p " ' (10)

(1--I-t) f = f o " (1 + @2)1/2

SO

and

2i(2mu@ - 2m~)H (1-42)1/2

(9)

where H is the "reduced" matrix element (Pu lu3'5u/ 2mul0), which, of course, is the same as
2i(2m 2 + 2m2~) H=/~vrn 2 ' (1 + @2)1/2

Taking the divergence, we find

= foP.

(1--~2)l/2fm 2 ,

(15)

Furthermore,

1--4

= f=p**

1--g (1+ 42) 1/2 p**f"

(11)

(19)

Consequently,

@m2 + m 2 1-@ fm 2 + 2
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Again, there is no difficulty with taking 4 -+ 1 ; for the limit g2 --, 0 we have 4 ~ 0; and then

fo = f -

(g2 ~ 0)

(21)

However, if we put together the results for 7r and for o we get interesting conclusions. First, from (12) and (19),

f~=fo-

1+4 f. (1+42)1/2

(22)

17 November 1980

mate eigenstates of SU(3), and this approximation becomes worse as g2 becomes small. In the extreme case o f g 2 = 0, the eigenstates are r/N S -- pseudoscalar bound state o f u, ~, d, d , r/s - pseudoscalar bound state of s, g ,

(26)

with masses proportional (respectively) to rk 2 and rk~. In other words, in terms of states r/= (r/N S -- e~TS)/(1 + e2) 1/2 ,

(27)

8im2n= fm 2 ,

r/' = (r/S + e//NS)/(1 + e2) 1/2 ,

(28)

(Tr[(g2/8 rr2) Tr GG 10) = 0 ,

we proceed just as before. The results are

-~g2 for 8irh2H+ 2(o I TrGSlO)=fom2 f 87r2 '

8 ~ i ( rh2~

It is easy to see that in the limit m u = m d - m

1- 4

1- g

4im2(l+42)l/2H-(l+42)l/2

tim 2

,

w +ems

(23)

SO

1 4i[2th2_em2 (1 + e2) 1/2 X/r3 ~ X/2 _ X/~ e X/3

2 2 + ; ( 0 [ g 2 TrGOIO) m° = mrr Jrr 8rr 2 4= 1.

(24)

(Assuming 4 > 0, which is simply convention.) That is, for equal quark masses, the rr is purely isovector and its mass vanishes if the quark mass vanishes; i.e., it is a Goldstone boson. The o is purely isoscalar, and in the limit of vanishing quark mass its mass is nonzero, being proportional to the matrix element of the axial anomaly. There is another limiting case of interest; namely, m u, m d -+ 0 but mu/m d fixed and not unity. In this case, the results are the same as for m u = m d taken first are then allowed to go to zero: 2

2

g2

m2=O, mo=~(oI - - TrGG[O), 4 = 1 .

(25)

8rr 2

The preceding discussion can be generalized to the case of three flavors without much difficulty. We will sketch what happens for m u = m d = rh 4= ms, which bears on r / - r / m i x i n g . After that, we have some words to say about the U(I ) problem. First, however, the isoscalar sector in the three flavor system is treated (isospin is exact). For SU(3) X SU(3), the r / a n d 7?' are only approxi-

) H - l + "v/2 e f m

2 ~ H+x/~_(rllg- TrGG[O ~ 8rr 2

f m2

(30)

(1 + e2) 1/2 '

8i /(eFn2 - m2s ) H _ e - x/2 frn2 ,/x

1

(29)

4i ~2Un2 + m 2 ) / / . + X/j (r/, I

(l + 7 2 ) 1 / 2 ~ ' ~

= V"2e + 1 x/3

(31)

,

-

fm 2

g2

Tr GGI0)

8~ (32)

(1 + e2)l/2

For physical states, e ~ x / 2 , but e = O(g 4) and so vanishes for g2 _+ 0. This resolves the problem for the r / - r / s y s t e m just as the discussion above did so for the 7r-o system. Note that e is simply tan SOp in conventional notation [6], so that in terms of the usual octetsinglet mixing angle 0p, x/2 + t a n 0 p e = -

(33)

1 - X/~ tan 0p

Let's look at the limit o f zero quark mass. First, if rh = m s - m (assuming e > 0 as usual), 4im2H=f m2,

2 ( r f l 8 L2 T r G G I 0 ) = 0 ,

4 i m 2 ( e - x/r2)H = (e - x/~) f m 2 ,

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x/2e + 1 3 (7/'[ g 2 Tr GGI0) (i +e2) 1/2 4 i m 2 H + 8rr 2 _ V'2e+l fm2n (1 + e2) 1/2 '

(34)

17 November 1980

for axial coupling constants. After all, fK/frr ~ 1.2, which is significantly different from the unity expected from SU(3). Therefore, we will relax SU(3) symmetry and allow fNS and j's to differ, define )'S = XfNS,

SO

2

2

3

rn n, =rnn + fn (r/'] g2 Tr G(~[0) 8rr2

(35)

e =,/2,

(36)

We note in passing that this is not sensitive to m2/rh 2 (so long as it is large compared to unity) and that in the formalism [7] in which meson masses are linear in quark masses, it is ms/rh which appears, but the consequences are the same. The problem is that (31) is not satisfied by this value of e and the physical r/' mass. The easiest way to see, in fact, that there is something not quite satisfactory is to try to solve (29) and (31) together for e in 2 and ran,. 2 . there is no real solution. terms of mr~ The resolution of this problem lies in the recognition that we have been too demanding on SU(3) symmetry 154

but still assume SU(3) for the corresponding reduced matrix elements of the pseudoscalar densities so that (29) and (31) now become 8i s/ -| /~r h 2 + e m 2 ~ H = l + x / ~ e X ~

That is, for equal quark masses, the r~ is pure SU(3) octet and its mass vanishes if the quark mass vanishes; i.e., it is a Goldstone bosorl. The r/' is purely SU(3) singlet, and in the limit of vanishing quark masses its mass is nonzero, being proportional to the matrix element of the axial anomaly. Furthermore, in the related limiting case of interest, rh and m s -+ 0 but ms/rh fixed and not unity, the results are the same as for rh = m s taken first and then allowed to go to zero. One cannot stop here, because the rT-r/' mixing problem has existed for so long it cries out for attention. That is to say, we are forced to consider the question of how the actual masses of r~ and rT' fit into the framework presented therein. There was no such imperative for the SU(2) X SU(2) case, since it is not supposed to be a good description of real life; however, one should expect to be able to do reasonable phenomenology within an SU(3) × SU(3) framework. We proceed as follows. First, the reduced matrix element H is eliminated in favor ofm ~2,f~ and rh 2 from (11) and (12). The result is substituted into (29)-(32). Assume SU(3) symmetry to relate fn and f; then (29) -~2ie52 from ref . [ 5 ]. g i v e s a v a l u e f o r e " ff w e t"a ke '"s/..This gives e = 1.11 .

(37)

I

2

(38)

-

while (11) is unchanged. From these three equations, 2 , ran, 2 m 2, and the previously using physical values for rnrr derived value for m2/rh 2, one may compute the two parameters e and X, finding e =0.871,

X = 1.258.

(40)

Note that this value of e corresponds to an octet-singlet mixing angle 0p = 13.7 °. The value for X is reasonable, if we recall fK/frr ~ 1.2. (The parameter X corresponds in some sense to Xp of ref. [6], since they both measure SU(3) breaking insofar as they differ from unity.) We now turn to a comparison of our conclusions with some relevant published papers. In a classic paper [1 ], Weinberg showed that under what were at the time, generally accepted assumptions, the mass of the lightest isoscalar pseudoscalar particle must be less than x/3m~r. The results followed from the current algebra expression /t,,2rr mrr2 = o~o, ISs,oFs, a,C-~oo,, 2

(41)

where F~o is the coupling of an isosinglet pseudoscalar particle (o) to the isosinglet axial current ~ sk~ = . V7r~_ "~0 + x f ~ A 8 3~/z

(42)

constructed from (u and d) nonstrange quarks exclusively , a , and c ~ 2o, is the mass-squared matrix for the isoscalar pseudoscalars (of unspecified number). The lightest of these has mass mL, where m 2 ~ (F~r/Z 2 ° Fs,2 o)rnrr2 .

(43)

+1 The tilde indicates that the current has had a piece substracted so that its divergence is anomaly-free. See ref. [2] for a discussion of this point.

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Weinberg argued that all Fs, a could not be much smaller than Frr , and with some further symmetry assumptions he obtained the bound m 2 ~< 3m 2 .

(44)

17 November 1980

r = 3.7

(Novikov et al.)

r -- 7.3

(Goldberg)

r = 6.3

(present w o r k ) ,

(48)

Since no such bound appears in the present work, there must be some step in Weinberg's argument that is not valid in the formalism here. In fact, it is just the assumption that all Fs, o are not much smaller than F~ which is questionable. For the case of SU(2) X SU(2), (25) shows that the o meson has a mass ~vhich is determined by the strong anomaly when the limit of zero quark mass is taken. Therefore,

to be compared with experimental values which are not well established. Older experimental numbers [ 11 ] tend to agree with smaller r such as predicted by Novikov et al.; more recent numbers [12] are larger and tend to agree with Goldberg's and our predictions. It is worth mentioning that our prediction for r is independent of the parameter 2, introduced which described SU(3) breaking for certain matrix elements; in fact, one may show that

Fs, o = 0

(01Tr GGIr/> _ m2' x/~ e m 2 + rn 2

(45)

in this limit, contrary to Weinberg's assumption. Simi2 is deterlarly, for SU(3) X SU(3), (35) shows that m n, mined by the strong anomaly in the limit of zero u- and d-quark masses. This argument is easily extended to allow for several isoscalar mesons; in the limit of zero ~S u- and d-quark masses, their couplings to A u all vanish. More precisely, Fs, o/F~ = O(mr~/m,, 2 2) .

(46)

In a recent paper, Witten [8] has considered the U(1) problem from the point of view of the 1/N expansion. He finds that, in the large N limit, the anomaly gives a mass to the 7' which otherwise would have been 2 is order 1/N. The expression massless; explicitly, ran, (35) above indicates that in the 1/N expansion fro 2, is order lINt~2; therefore, since f is order N 1/2, we agree with Witten on this result even though his expression for the 77' mass is not the same as ours. A calculation of the proportionality constant lies outside his paper just as it does ours. It is interesting, however, that the proportionality constant appearing in (35) does enter some phenomenological work on q~ decays. For example, Novikov et al. [9] and Goldberg [10] have assumed that the dominant decay mechanism for q~ -+ 7/(r/')7 involves radiation from a c-quark. The relative branching ratio is then _ E(q, -+ ~ ' 7 ) r= - F - ~ 0.813

(0l Tr GGIr/') [ 2 ~ , (0[Tr GG[rl) j

(47)

including effects of phase space. From this expression and the theoretical estimates of the anomaly matrix elements, one finds

<01TrGGIr/)

m 2 x/2 r n 2 - e r n 2 "

(49)

As a final observation, we would like to note the parallel between the approach of the present paper, which uses axial currents and their divergences, and the work of ref. [6], which discusses the pseudoscalar mesons from the point of view of Hamiltonian dynamics. In this paper, the r/' mass is related to the vacuum to one particle matrix element of the strong anomaly; in ref. [6], the r/' mass is related to diagrams in which a quark antiquark pair annihilates into a pair of gluons. There is a close connection, clearly. Unfortunately, in neither approach has one computed the relevant matrix element ab initio. This is obviously a much more difficult problem. Discussions with M.D, Scadron and H. Goldberg were helpful in clarifying points in this paper. This work was supported in part by the United States Department of Energy. [1] S. Weinberg, Phys. Rev. D l l (1975) 3583. [2] R. Crewther, Phys. Lett. 70B (.1977) 349; Riv. Nuovo Cimento 2 (1979) 63. [3] M. Glfick, Mainz Univ. report MZ-TH 79/5. [4] N.H. Fuchs, Phys. Rev. D20 (1979) 1244. [5] N.H. Fuchs and M.D. Scadron, Phys. Rev. D20 (1979) 2421. [6] H.F. Jones and M.D. Scadron, Nucl. Phys. B155 (1979) 409. [7] S. Weinberg, in: A Festschrift for I.I. Rabi, ed. L. Motz (New York Academy of Sciences, New York, 1978). [8] E. Witten, Nucl. Phys. B156 (1979) 269. [[9] V.A. Novikov et al., Phys. Left. 86B (1979) 347; ITEP report 1979-73. [10] H. Goldberg, Phys. Rev. Lett. 44 (1980) 363. [ 11 ] Particle Data Group, LBL-100 (April, 1978). [12] R. Partridge et al., SLAC report 2430 (November 1979). 155