Trace anomaly and mass independent rational terms

6 February 1997

PHYSICS

LETTERS B

Physics Letters B 393 (1997) 79-83

ELSES’IER

Trace anomaly and mass independent rational terms Guang-Jiong Ni a,b, Ji-Feng Yang b a ICTP f!O. Box 586, 34100 Trieste, Italy h Department of Physics, Fudan University. Shanghai 200433, China ’

Received 12 September 1995; revised manuscript received 4 June 1996 Editor: M. Dine

Abstract We perform a new investigation to show that the trace anomaly in Minkowskian spacetime is originated by a kind of mass independent rational term which is definite and nonlocal. Hence it is regularization independent and may provide more physical insight. A simple way for calculating trace anomalies is proposed and several examples are given.

Just like chiral anomaly is related to the phenomenological physics of PCAC [ 11, trace anomaly of energymomentum tensor is related to the phenomenological physics of PCDC [ 21, i.e. they are physical. But there is one thing one might feel unsatisfied that the known conventional derivations and interpretations of anomalies is not clearly regularization independent, in other words (at least technically), the anomalies are taken as regularization effects. Fortunately, the chiral anomaly conveys topological information [ 31 which enables us to understand it beyond the limitation of regularization methods. But for the trace anomaly in flatspace, we are not so lucky, (though in curved space, the trace anomaly is somehow related to the Euler characteristic class f4] ), since the Lagrangian of gauge fields can not be written as a topological invariant [ 51. Thus, it seems necessary for us to find out the explicit and definite origin of trace anomaly in Minkowskian spacetime. That might also provide us deeper insight into the structures of the quantum field theories ( QFTs). Similar efforts have just been made on chiral anomaly by us [6]. It is found that a kind of mass ’ Permanent address. 0370-2693/97/$17.00 Copyright PII SO370-2693(96)01570-S

0 1997 Published

independent rational terms is responsible for the appearance of anomaly. Basing on the experiences obtained there, we first manipulate a formal analysis of the problem. We consider the case of QEDr+s for simplicity. The amplitude is ( 10,VJaJpl) where OpLyis the energy-momentum tensor [ 71 and J, is the usual fermion current. We write the amplitude in the following general form (consider the zero momentum transfer case from now on):

= $&‘,,(P,

-p,m)

+ r!$,,(p,

-p)

where, IFVyUP(p, -p, m) refers to the mass dependent and nonlocal part which is definite and regularization independent, I’$ap(p, -p) denotes the mass independent rational part which is also definite (nonlocal) and regularization independent, and I$$s (p, -p) represents the possible polynomial part (local) and it is regularization “covariant” if it exists. It is easy to show that l$$, (p, -p, m) must obey the canonical Ward-Takahashi identities. The other two parts would

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G.-J. Ni, J.-F. Yang/ Physics Letters B 393 (1997) 79-83

80

comprise the sources of trace anomaly. So the trace anomaly has two possible origins, one is the mass independent rational term while another is the polynomial term. Since rflvtap(P, -P, m) is coming from fermionloop, it should define the fermion field part of 8,“. But the local term in TILVinpwould (in operator form) contribute to the gauge field part of ecL,,,that is somewhat the operator mixing phenomenon of the renormalization of composite operators [ 81. Thus the local term (if exists) in I’LLy:+(P, -P, m) would lead to the redefinition of the tree level form of the gauge field part in t?,, and can be absorbed away. [If we define the tree level form of the BLtptensor (including the local terms from loop contributions) so that the normal trace identity holds, then either the trace anomaly from the would-be local terms is not a physical one, or the local terms are traceless, i.e., cause no effect of trace anomaly. We need not worry about it. ] Then we need only care about the rational part. More precisely, if

rpv;(la(P, -P, m) = r$& +~:;,,(P,-P)

(p, -p,

+r;;p(~,-~)

= f~“;a/dP)fl(*)(p*/m*, +

m)

Ly)

- S~~P*)P~P”~~‘*‘(p*/~*,

2(P,PP

a) , (5)

where

1

x(1 -

2a -- li- Jd_xx(l-x)ln (I-

x)p2

1112

(a = e2/(47T)). ‘Y~~;,p(P,

-P)

+ 07

g’““B,, = rrr,,~,& + CF. F ,

Then, it is easy to see that

r;$,(p,

-P, m> + rE;,(p,

= t,,;,p(p)fl’*)(p*/m*,

-P> a)

4cum* +- “p2 P,PY(PcYPP - &%pP2) I X

dx J 0

(3)

with cF . F as the counterterm for canceling the anomaly induced by (2). Then, after quantum correction is included, one gets gll”@,V =m+C1,,

(7)

(2)

then, we can not remove it, since it is definite and regularization independent. If one wants to introduce a tree level counterterm to cancel it, this term would in turn lead to a different trace identity at classical level already, then the trace identities from quantum level and classical level again differ in form, anomaly still shows up. Or, that amounts to starting from the following tree level trace identity (the would-be local term already treated as above)

>

0

r~;,,(p,

x2 m* -

(X -

x

’

X*)p*

-PI = -

Now we have

(4)

which is different from Eq. (3). That is still a reflection of trace anomaly. Therefore, through formal analysis, we see that the definite and hence physical origin of trace anomaly is the existence of a kind of mass independent rational term. (Of course, gw$,, (p,p) is local.) To ascertain the above analysis, we list the result of explicit calculations as follows (see Adler et al. in Ref. [7]):

I

4am2 = --+PaPp

- $PaPp = mr$ where

- &PP2)

s 0

dx,*

_;:_;2)p2

- &q?P2) (P. -p, m> + ayiap ,

(10)

G.-J. Ni, J.-F, Yang/Physics

terters B 393 (1997) 79-83

81

+ possible remaining So the rational

local terms.

term rRa. PV,ap is linearly

m + cx; limit of the mass dependent

(12) So, the result of explicit calculations confirms the formal analysis given above. We stress again that the trace anomaly in ( 12) is originated by a kind of mass independent rational term which is definite and regularization independent as shown in Eq. (9). It is the definite intrinsic (physical) structure of QFT (QED), as is in the case of chiral anomaly [ 61. Though the above investigation is carried out at the one-loop level, but the key structure is already revealed and will survive in higher loop corrections. The coefficients of the rational terms define the magnitude of the trace anomaly (see Eqs. (9)) ( 12) >. Conventionally, the trace anomaly is derived via renormalizationgroup equation technique, the magnitude of the trace anomaly is proportional to the beta function which is stemming from the renormalization structure of the QFT (QED), i.e.

/3((Y) = z.

(13)

Thus, we see an interesting correspondence between the definite property (mass independent rational term) of QED and the renormalization effect of QED: two apparently different quantities become correlated. It suggests that /3(a) may be a special way of expressing the inherent properties of QED. We think that further study on this subject would be fruitful. Moreover, we hope this investigation would add to the discussion of the trace anomaly and chiral anomaly in supersymmetric gauge theories [ 91, as we have known that both trace anomaly and chiral anomaly originate from unambiguous rational terms in corresponding amplitudes 161. Now, we which will easier. In the m polynomial

which in turn is related to mr$, relation,

= g~~r;&,(p, = -auLnp(p)

-+ co limit, rPViap (p, -p, (local) term, then

m) must be a

+

related to the part of rE:alj,

because of the WT

ml Loo

possible remaining

local terms , t 15)

that is, the trace anomaly is linearly related to the m -+ 03 limit of the amplitude m(&+kl,Jp). (We can also see that the renormalization property of trace anomaly should be the same as the composite operator m$t,b.) Thus, we find an easy way to calculate the trace anomaly without resorting to the conventional approaches. As to the possible remaining local terms in Eq. ( 15), we may afford a reasonable argument that they are zero. This is because we are working with the renotmalizable QED,+3 (no chiral anomaly etc. which violates the renormalizability), then, according to Appelquist and Carrazone [ 101, in the m + cxi limit, the fermion is decoupled, no dynamical effects can survive. In other words, the whole loop contribution from the heavy fermion (Eq. ( 1) ) vanishes: rPL,;,p (P, -P,

m) Lo3

(16)

= 0,

or, gfi~r,u;ap(p,

-P,

= mr$(p,

m> Loo

-p,m&-+,

+

Utr:& = 0 .

Here the anomalous trace identity Eq. (17) implies that Gr;n@ = --lnr$&, =

demonstrate another interesting property render the calculation of trace anomaly

-P,

(14)

(17)

( 11) is used. Then

-P, m) (Ill--ra

-m(&bJ, Jp)l,,-+,

(18)

Eq. ( 18) is in fact valid at all orders because of decoupling properties and it is an easier calculation. In fact, at, is just the discrepancy between rnIf$ gpvTgv:ap when fermion mass goes to infinity.

and

G.-J. Ni, J.-F. Yung/Physics Letters 3 393 (1997) 79-ll.3

82

As a check, one may calculate the m -+ 00 limit of m(*t,bJaJp) in QED. The result is

X

P2 lQ2-~211(Q+~)2-~211(Q-(-~))z-~21

P-+oo

+ at, A2

=igp+uv=o,

(21)

and so, at,=--=_-,

&;Ccp(P)

=

-$PaPa

(20)

- &q3P2) 9

which coincides exactly with that given in Eq. ( 12). With this decoupling argument, it is easy to see 4 that In QEDI+I, W,+1 and A&+2, there would be no trace anomaly (in Minkowskian spacetime). One only needs to substitute the amplitudes (BPyJ,Jp), m($ljlJaJp), etc. in QEDt+s in Eqs. (14) to (17) with corresponding amplitudes in QEDi+i and &bT+,

(n = 1, 2). That is, (~JJ~)I+~, etc. for QEDI+I; A2(8,d24*)~+,

m(&WJp)~+~ (= A2rpv;ppj,

A2p2(+2q52q52)(= A2p21$242) etc. for A+;+,, n = 1, 2. Since these theories are superrenormalizable, all corresponding high loop corrections (to this issue) are definite and free of anomalies. Hence we can focus on the relevant one loop amplitudes to see whether there is trace anomaly in these theories (in Minkowskian It is easy to check that rnI’$$ of QED,,, z (n = 1, 2) are vanishing in and A2p21$z92 of A#,, the decoupling limit. Then Eq. (17) will tell us that there is no trace anomaly in these theories. But in QEDi+2, things are quite different and the above procedure from Eq. ( 14) to Eq. (20) is no longer valid. This is because the usual decoupling theorem [ lo] fails as a physical Chern-Simons term emerges [ II]. We must do direct calculations to find whether there is any mass independent rational term that originates any trace anomaly. It will be discussed elsewhere. While for A#+s, the procedure from Eq. (14) to . . Eq. (20) remains valid (with TpViap etc. in QED1+3 substituted by A21YPy++z etc. in A&+s). so at one loop level, we have spacetime).

g~“A2r,V:tiz62 (P, -P, = A2&$z62

.A2 = z4 J

d4Q

P(A)

128~~

4!

P(A) = 2,

(22) (23)

or in operator form, (24) Here, p(A) is the usual beta function of Ac$~+~.The above result is also in conformity with that from conventional approaches [ 121. Once again we stress that the above calculation can be done at all orders and the form of Eq. (24) is valid to all orders, as pointed out after Eq. (18). In summary, we have presented a new investigation on the appearance of the trace anomaly in Minkowskian spacetime and the regularization independent reason is found to be the existence of a kind of mass independent rational term. We also note that the beta function which reflects the behavior of running coupling constants in QFT is closely related to this term and this investigation may be relevant to the discussions of anomalies in supersymmetry theories. Then a simple way for calculating trace anomaly is proposed and several examples are given. We think that many problems related with this topic are worthwhile of further study. This work is supported in part by the National Natural Science Foundation of China and the Foundation of State Education Commission of China. One of the authors, Ni, wishes to thank the warm hospitality of ICTI? We are grateful to the referee for his kind comments and suggestions which lead to some improvements of our manuscript.

~2) llr+m

(P, -P,

(2T)4

A2

d)

lcL+oo + at,

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