Goldstone bosons generate peculiar conformal anomalies

Volume 221, number 3,4 PHYSICS LETTERS B 4 May 1989

GOLDSTONE BOSONS GENERATE PECULIAR CONFORMAL ANOMALIES ~

H. LEUTWYLER and M. SHIFMAN lnstttute for Theorettcal Phystcs, Umverstty of Bern, SMlerstrasse 5, CH-3012 Bern, Swttzerland

Received 13 February 1989

We reexamine the low energy theorem for the matrix element (0 [ 0~, ~ 177), where Ou, is the energy-momentum tensor Virtual p~ons are shown to generate anomalous contributions The consequences for the decay of a hght Hlggs particle into two photons are discussed

1. The low energy theorems of quantum chromo- dynamics are a useful - in some cases the only avail- able - source of information about the behaviour of hadromc amplitudes at low energy. The theorems are consequences of the spontaneous breakdown of chiral symmetry, a property of the strong interaction which was established even before the discovery of QCD. There is however a class of low energy theorems which is specific to this theory. We are referring to the low energy predictions concerning the matrix element of the operator G2, where G~, is the gluon field strength. The interest in these matrix elements derives from the fact that low energy transitions generated by virtual heavy quarks can often be analyzed by means of a multlpole expansion in which G2~ represents the leading term. A rather detailed discussion and a rep- resentative list of references is given in ref. [ 1 ]. This paper, in particular, also contains a prediction for the amplitude (01G2~ 137) which describes the emis- sion of a pair of soft photons by the operator 2 G/~V" Unfortunately, in the calculation presented in ref. [ 1 ], the contributions generated by virtual Goldstone bosons were overlooked. The purpose of the present paper is to show that the occurrence of light, charged bound states significantly affects the low energy properties of the matrix element (01G2u, 177). In particular, in the Chlral limit, pion loops generate a peculiar contribution to this matrix

Permanent address: Institute of Theoretical and Experimental Physics, 117259 Moscow, USSR Work supported by Schwelzenscher NaUonalfonds.

element which is related to the fact that the effective chiral lagrangian is anomalous under conformal transformations.

2. We make use of the fact that, at low energies, the standard model reduces to an effective gauge field theory (gluons, photons and Nf hght quark flavours; we disregard the leptons as they do not contribute to the matrix element (01G~, 177) at order e2). De- noting the symmetric, conserved energy-momentum tensor of the system by 0u~, we have

Ou~'=flJgq-b( as /8n )G~,GaU~-~( a/8n)A~,,A ~ ,

(1)

where Au~ is the field strength of the electromagnetic field. The first term represents the breaking of conformal invanance generated by the quark mass matrix ~/, while the remaining two terms originate in the conformal anomalies of the gauge couphngs. The constants b and/~are the first coefficients of the Gell- Mann-Low functions of QCD and QED, respectively.

b = ] ( l l N c - 2 N f ) E = - 4 , gNc tr(Q 2) , (2)

where Q ~s the quark charge matrix. With three col- ours and three light flavours u, d, s, we have b= 9, t7=--~.

Lorentz- and gauge-invariance imply that the two- photon matrix element of a scalar operator contains a single invariant amplitude

( O I S [ 3 7 ' ) = A s F ~ , F ~ ' , Fu~=que~--q~E u. (3)

384


Here q, q' are the momenta of the two photons (q2 = q,2 = 0 ) and e, E' are their polarization vectors. The invariant amplitude depends on the mass of the state only: A s = A s ( p 2 ) , with p = q + q ' . In this nota- non, the relation ( 1 ) reads

Ao(p 2) =Aq(p 2) +AG(p 2) +ab- /4n , (4)

where Ao, Aq and A~ are the invariant amplitudes associated with the operators Ou u, Cl~q and -boq X

a a , u v G~,,G /8n, respectively. To calculate the amplitude A~, we therefore need to analyze the two-photon matrix elements of 0u u and of ClJ/q.

3. Lorentz invariance, gauge invariance, Bose sta- tistics and energy-momentum conservation imply that the two-photon matrix element of the energy- momentum tensor is of the form

(01G, I ~ ' )

=o, ( Fuc, F ~' + F,~F~' - ~gu~F,~BF '~' )

+ [02(PuPv -gu, .PZ)+O3duAv]FapF a'~' , (5)

where p = q+ q', A = q - q' (throughout this paper, we restrict ourselves to on-shell photons; the invariants 01, 02, 03 only depend on p 2 = _A2) . The amplitude 0~ receives a contribution of order 1 from the energy- momentum tensor of the electromagnetic field, 0~ = 1 + O ( e 2) while 02 and 03 are of order e 2. The two-photon matrix element of OuU is of order e 2,

A o = ( - p : ) ( 3 0 2 +03) , (6)

the invariant 01 drops out in the trace. At low momenta, the matrix element (010u~ IY'/) is dominated by the contribution of order p2 from 01, while 02 and 03 only contribute at order p4. Eq. (6) therefore implies that Ao is of order p2/lt2 , where/~ is some hadronic mass scale, characteristic of the channel in question. For p z <</~2, Ao is negligibly small, while A~ is of O ( 1 ); the low energy theorem given m ref. [ 1 ], AG= -ab- /4n , immediately follows from (4) if the quark mass t e r m Aq lS dropped.

4. As mentioned above, the occurrence of light charged particles invalidates this argument: the scale /t is small, of the order of the pion mass. Indeed, consider the meson loop graphs shown in fig. I. Current algebra implies that the meson couplings both to the

/ ', / \

/ \\ / \

/

e/Le I I I

/ , i I \ \

i / /

I / x\ \

Q

/ x

]11 e

/ e i I \\ ! x

/ x

Fig. 1 Feynman graphs describing the mamx element (0l 0u~ l Y'/) to order e 2 in scalar electrodynamlcs.

photon field and to the energy-momentum tensor are the same as in scalar electrodynamics,

, ,~= Vu~* VU~_ M2to*~o_ ~Au~,A u~' , ( 7 )

with Vu= O~,-ieA u. Since this lagrangian specifies a renormalizable theory, the ultraviolet divergences occurring in the graphs of fig. 1 are automatically cancelled by the wave function renormalization of the external photon lines. The contributions to the amplitudes 01, 02, 03 generated by the meson loops are therefore unambiguous.

In massless electrodynamics, 01 contains a loga- rithmic divergence which stems from the fact that it is impossible to distinguish a photon from a pair of massless, collinear charged particles. In the case of the amplitudes 02, 03 and Ao, this problem only shows up at order e 4 - it does therefore not concern us here. On dimensional grounds, the loop contributions to 02, 03 are of the form e a M - 2 f ( p 2 / M 2 ) , where M is the mass of the meson running around the loop. In the chiral limit M--,0 both 02 and 03 are proportional to e E / p 2. The coefficients are readily worked out by performing the loop integrals. We find

02 = - - 5a~M/ST~P 2 , 03=OL~M/87~p 2 , (8 )

where we have expressed the result in terms of the first coefficient of the Gell-Mann-Low function generated by a multiplet of spinless charged particles. For a complex meson field of charge e, such as n-+, we have/~M = -- ~. If the meson multiplet is described by a set of real fields with charge matrix Q, the coefficient is given by

~'M = - ~ t r (~2) • (9)

385


Since the charges of the Goldstone bosons are deter- mined by the quark charges, we can alternatively ex- press the coefficient ~M in terms of the quark charge matrix Q

/TM = - ] [Nf t r (Q 2) - (tr Q)2] . (10)

Charged spin ½ particles generate a similar one-loop contribution to the amplitude (0 ] Ou, [~q,). In massless spinor electrodynamics, one finds

0 2 ~--0 3 = - - a ~ f / 1 6 r i p 2 , ( 11 )

where E l = - 4 t r (Q2) /3 is the corresponding Gell- Mann-Low coefficient.

Inserting (8) in (6), one concludes that, in massless scalar electrodynamics, the two-photon matrix element of 0" ~ is given by

Ao = 7ab-M/4n . (12)

5. Scalar electrodynamics properly describes the low energy couplings of the Goldstone bosons in QCD. Since the meson loop contribution to Aq vanishes in the chiral limit, the relation (4) implies

AG = a (7/~M -- t~)/4~Z. (13)

The factor of 7 looks odd. In fact, if we replace the mesons by particles of spin ½ and use ( 11 ) rather than (8), we instead obtain Ao= ab"f/4n - the standard conformal anomaly. The factor of 7 does not originate in an algebraic mistake, however. In the following, we show that it occurs because spontaneously broken chiral symmetry is inconsistent with conformal lnvarlance [ 2 ], even at the classical level.

The peculiarities of the plon loop contribution to the matrix element of OuU originate in a property of scalar field theories which was discovered long ago [3]: the canonical energy-momentum tensor of massless scalar fields is not traceless - even ffthe field is a free field. Let us again consider scalar electrodynamics, characterized by the lagrangian (7), and let us first treat the fields occurring there as classical fields. The energy-momentum tensor is given by

Ou. = Vu~0* V.~0 + V.~o*VAo-Au~A.~-g . .~ ( 14 )

Using the classical equations of motions, the trace can be brought to the form

OJ' = ( 2 M 2 - [ ] ) ~o*~0. ( 15 )

386

Clearly, 0a" does not tend to zero as M ~ 0 , even at the classical level - the canonical conformal current OF,~x" is not conserved, nor does it generate conformal transformations of the meson field. As pointed out in ref. [ 3 ], the offending term can be eliminated if the canonical energy-momentum tensor 0~,~ is replaced by the "new-improved" quantity

Ou. =0~. + ~ (g~.[~ - 0~0.) ~0"~0. (16)

In the massless theory, the corresponding conformal current Ou~x" does generate conformal transformations and, classically, 0u ~ vanishes.

Why not replace Ou. by the new-improved tensor? The point here is that we are not interested in scalar electrodynamics as such, but only as an effective field theory which describes the degrees of freedom of the Goldstone bosons which occur within the standard model. In this context, the form of the energy-momentum tensor is fixed by chiral symmetry. The modification needed to improve the conformal properties of 0~ ruins its chiral properties. [The modification (16) amounts to adding the term R~o*~o/6 to the lagrangian of the theory. This term breaks chiral symmetry for the same reason as the mass term M2~0*~0 breaks it. ] In fact, in contrast to the lagrangian of massless electrodynamxcs, the effective chlral lagrangian contains dimensionful coupling constants which break conformal invarlance ab initlo. We do not dwell on this further, as the issue is discussed in detail in the literature [ 2 ]. At low energies, the matrix elements of the energy-momentum tensor of the underlying gauge field theory - which is unambiguous - coincide with the matrix elements of the canon- tcal energy-momentum tensor associated with the effective chlral lagranglan, while the improved object is of use only as a technical device.

In the above, we considered the classical theory. In scalar quantum electrodynamlcs, the short-distance singularities of the meson field generate the standard additional contribution given by the first coefficient of the Gell-Mann-Low function. For a single charged scalar meson field, the full expression for the trace of the energy-momentum tensor becomes

OuU=(2M2-[])~o*~o-SM(a/8n)Au.A u" , (17)

with /TM = ---~. At low energies, the matrix elements of 0. ~ thus receive three types of contributions: (i) a


contribution proportional to the square of the meson mass which explicitly breaks conformal invanance; (fi) a contribution from the term-l~(~0*~o) which stems from the fact that the conformal current Ou~x" of QCD does not generate conformal transformations of the meson field and (iii) a contribution due to the short-distance properties of the meson field theory. In the case of the matrix element ( 010uU 177), these contributions are readily evaluated by calculat- ing the amplitude (0 ] ~*~0 [ 77) to one loop. In the choral limit, we find

A~o*~ = 3abM/2np 2 . (18)

In this limit, the three types of contributions mentioned above therefore add up as 0 + 6 + 1 = 7. Alter- natively, one may observe that the difference be- tween Ou~ and tT~,~ only affects the term proportional to p u p ~ - g ~ p 2, i.e., the invariant 02. With (16) and ( 18 ), we obtain

02 = 02 + a~M/2np 2 = -- a~M/8np 2 . (19)

If the canonical tensor 0~ is replaced by the new-improved version, the two-photon matrix element of the trace is therefore given by A0-= a~M/4n - the standard form of the anomaly ~t

6. In the above, we limited ourselves to an analysis of the matrxx elements m the chiral limit. To extend the analysis to nonzero quark masses, we consider the standard chiral expansmn of these matrix elements (expand in powers of the momenta and in powers of the light quark masses and reorder the series, count- ing the quark masses as quantities of order p2 [5 ] ). Chiral perturbanon theory asserts that the leading term in this expansion is given by the one-loop graphs of the effective chlral lagrangian, i.e. by the graphs shown in fig. 1. Evaluating these graphs for M ~ 0, we arrive at the following low energy theorems

A o = e 2 ~,, [ (pZ + 2 M 2 ) H ( p Z , M ) - ( 4 8 n 2) - 1 ] , M

Aq = e 2 E M 2 H ( p 2, M ) , (20) M

~ A sxmflar phenomenon occurs m the analysis of the conformal anomaly generated by second rank antlsymmetnc tensor gauge fields As shown by Gnsaru et al. [4], the energy-momentum tensor of the eqmvalent scalar field theory also faxls to be traceless, even at the classical level.

AG=e 2 ~ [ ( p + M 2 ) H ( p 2 , M ) - (48n 2 ) - ' ]

- a ~ / 4 n . (20 cont 'd)

The formulae are valid up to corrections of order p2. We have made use of the fact that, in the standard model, the charged Goldstone bosons occur in pairs of charge + e. In the sums occurring in (20), a single term accounts for the contribution generated by such a pair. For three light flavours, the sum thus contains two terms, M=M,~ and M = M K . (In this notation, ~M = -- -~ ~M1. ) The function H ( p 2, M ) is a kinemat- ical quantity associated with the triangle graph

H(p 2, M ) = ( 1/8n2p 2) [ ( z - l arcsin z) 2 - 1 ] ,

z= (p2/4M2) 1/2. (21)

On the interval - ~ (p2 ~ 4M 2, the funcUon H is real. Above threshold, it develops an imaginary part

a r c s l n z = ½ n + i l n ( z + z x / ~ - l ) , p2>~4M2.

The real part dominates both at small and at large momenta

H ( p 2, M ) = (96n2M 2) - l p2 <<M2 '

= - (8nZpZ) - l p2 >>M2.

Note that A q ( p 2) and A G ( p 2) do not decouple for M 2 >>p2. I f the quark masses are nonzero, but small compared to the scale of QCD, we have

. 4 . ( 0 ) = - ~ / T M / 8 n ,

A t ( 0 ) = a (~'M - 26 ) /8n , (22)

while, if the quark masses become large, Aq tends to - a ~ / 4 n and AG tends to zero.

7. Finally, we briefly discuss the consequences for the decay of a hypothetical light Higgs particle into two photons. The lagrangian which describes the couplings of this particle to the light degrees of freedom - including light leptons - is given by [6,7 ]

v

+bh(as/8n)G~,~Gau~+~h(a/8n)Au~AU~] ,

(23)

where v= (v/2 G)-1 /2=246 GeV is the weak interaction scale. The constants bh and /Th represent the

387


contr ibut ions of the heavy degrees o f f reedom to the Ge l l -Mann-Low coefficients of Q C D and QED, respectively. In the case of bh, only quark loops contr ibute, bh= - 2 N J 3 . In the s tandard model with three heavy flavours, we have bh = - 2. The QED coefficient is given by ~h = 7 -- 4-- 4 = ~, where the first con- t r lbut ion stems from loops involving the W or the charged componen t of the Hlggs doublet , while the second and third terms are generated by heavy quarks and by the z, respectively. The t rans i t ion ampl i tude for the process H ~ 7Y is therefore given by

T,_vv = - (1 /v ) (Aq + Ae + xAo -G~h/4zr)F~,,F u"' (24)

where Aq and Ao are the invar iant two-photon ampl i - tudes discussed above, evaluated at p Z = M 2 . The coefficient x stands for - bh/b; numerically, x = 2. The matr ix element An is de termined by the triangle graphs generated by the light leptons. It can be expressed in terms of the funct ion H ( p 2, M ) as

A~=e 2 ~ [ ( P z - 4 m Z ) H ( P 2 , m ) + ( 8 1 r 2 ) - l ] , m

(25)

where the sum contains two terms, m = m e and m = m, . The t rans i t ion a m p h t u d e thus takes the form

T H ~ = -- ( a / 4 ~ w ) C F u . P '~' , (26)

where C is a d imensionless complex number which only depends on the rat io o f the mass o f the Higgs part icle to the meson and lepton masses. In the chiral l imit ( M = m = 0 ) , we find C = - ~ , to be compared with the value C = 29 - ~ which obta ins ff the contri- but ions generated by vir tual Golds tone bosons are omi t t ed [ 7 ]. The t ransi t ion rate is given by

FH~vv = ( M. / zc ) ( aMH/16~zv)21CI 2. (27)

In fig. 2, the numerica l value of I CI 2 is shown as a function o f the mass of the Higgs particle. The peak and the d ip stem from the threshold singulari t ies gen- era ted by the n+rc- and K + K - cuts, respectively. Fo r small Higgs masses, C is p redominan t ly real, while, above nn threshold, the imaginary par t dominates . In the region shown, vir tual muons do contr ibute significantly, while electrons are suppressed - as it is the case with Aq, the amph tude A~ tends to zero for p2 >> m 2. The dashed line shows the behaviour of I C I 2 which results ff processes involving vir tual light quarks or gluons are d ropped (Aq = A o = 0).

3 2 I 0

-1 -2

D 2M~ 2MK MH

Fig 2 Reduced amphtude of the process H-~q, The full hnes correspond to the calculation described m this paper The dashed lines result if the contributions generated by virtual light quarks or by virtual gluons are dropped

Clearly, the low energy expansion underlying our analysis loses its meaning i f the energy l iberated in the decay suffices to excite massive hadronic degrees of f reedom such as scalar qq bound states. For this reason, our analysis of the process H~TT is signif icant only If the mass o f the Higgs part icle is below 1 GeV.

Acknowledgement

We are indebted to Hans Bebie for generous help with the numerica l work. Fur thermore , one o f us (M.S.) thanks the colleagues from the Inst i tute for theoretxcal Physics, Bern Universi ty , for k ind hospxtality.

[ 1 ] V Novlkov et al., Nucl Phys. B 191 ( 1981 ) 301 [2] M B Voloshln and A.D. Dolgov, Yad. FlZ 35 (1982) 213

[Sov J Nuel Phys 35 (1982) 120] [3] C G Callan, S Coleman and R Jacklw, Ann Phys (NY)

59 (1970) 42 [4] M T Gnsaru, N K Nielsen, W Siegel and D Zanon, Nucl

Phys B247 (1984) 157 [5] H Pagels, Phys Rep 16 (1975) 219,

S Wemberg, Physlca A 96 (1979) 327, J. Gasser and H Leutwyler, Nucl. Phys. B 250 (1985) 465.

[6 ] J Ellis, M K Galliard and D V Nanopoulos, Nucl Phys. B 106 (1976) 292, A Vamshtem, V Zakharov and M Shlfman, Usp Fxz. Nauk 131 (1980) 537 [Sov Phys Usp 23 (1980)429]

17 ] M B Voloshm, Yad. Flz 45 (1987) 190 [Sov. J. Nucl Phys 45 (1987) 122]

388

Documents

Goldstone bosons generate peculiar conformal anomalies