hep-ph/ Sep 1998

Transcription

1 Some Aspects of Trace Anomaly Martin Schnabl Nuclear Centre, Faculty of Mathematics and Physics, Charles University, V Holesovickach 2, C-8000 Praha 8, Czech Republic July 998 hep-ph/ Sep 998 Abstract We review briey some well known facts about trace anomaly and then concentrate on its 'infrared' manifestation. Among other things we show by means of dispersion relations that dilatations and translations are conicting symmetries. We discuss the occurrence of some subtraction constants in convergent dispersion relations and illustrate it on the process H! in the electroweak theory. Physical aspects From the beginning of 970's we know that dilatation and conformal symmetries of quantum eld theories in at space-time are in general broken by quantum eects [, 2, 3, 4]. The corresponding Noether currents thus possess anomalous divergences. This phenomenon has some important physical consequences. First it explains naturally fairly large hadronic masses (see e.g. [5]). It also allows one to make interesting low-energy predictions for various matrix elements [6]. Another physical case [2], known for a long time, concerns the asymptotic behaviour of the basic QCD observable R(s) which in the lowest order is simply given by the quark-parton model (QPM) R(s) tot(e + e?!! hadrons) tot (e + e?!! +? ) s! X i e 2 i () where s = q 2 and e i are quark charges. Before QPM was established it was quite unexpected behaviour since tot (e + e?! +? ) s but tot (e + e?! pp) s 5 (2) 0

2 Nonvanishing asymptotic limit of R(s) can be understood [2] as a physical consequence of the trace anomaly. Let us start with the anomalous Ward identity (p) = (p) + R 6 2(p p? g p 2 ) (3) where (p) = i (p) = d 4 xe ipx h0jt (J (x)j (0))j0i (4) d 4 xd 4 ye ipy h0jt ( (x)j (y)j (0))j0i (5) For large momentum p we can neglect the rst term on the right-hand side (RHS) of (3), solve that asymptotic equation and nd the imaginary part Im (p) ' R 2 (p p? g p 2 ) (6) Using unitarity we nally see that the asymptotic ratio R is nothing else but the trace anomaly tot (e + e?! hadrons) =? p 4 Im ' 42 3p 2 R (7) 2 Dispersive derivation of trace anomaly Soon after the discovery of axial anomaly based on some ultraviolet (UV) regularization, Dolgov and akharov [7] presented an 'infrared' (IR) derivation. From their point of view the anomaly appeared as a (q 2 ) singularity in the imaginary part of some formfactor of the triangle diagram or as a pole in the real part. Using dispersion relations and unitarity they completely avoided any UV regularization. It turns out that this sort of derivation is not conned only to the axial anomaly but the trace anomaly can obtained in a similar manner as well [8]. Let us start with the naive dilatation Ward identity Current conservation implies the following form! (p) = (p) (8) (p) = (p 2 )(p p? p 2 g ) (p) = (p 2 )(p p? p 2 g ) (9) We can recast the naive Ward identity in terms of these formfactors? 2(p2 ; m 2 ) = (p 2 ; m 2 ) (0)

3 Let us dene (p 2 ) and (p 2 ) by means of once subtracted and unsubtracted dispersion relations respectively (p 2 ) = c + p2 (p 2 ) = 4m 2 Im(t) t(t? p 2 ) dt Im(t) dt () 4m 2 2 t? p The imaginary parts are well-dened nite quantities and should therefore satisfy the canonical Ward Im(t; m2 ) = Im (t; m 2 ) (2) Plugging the dispersive representation () into the left-hand side of the Ward identity (0), using the naive Ward identity for imaginary parts (2) and again () we get? 2 ) = (p 2 )? Im(t) dt 2(p2 4m 2 t Explicit calculation at one loop level yields 4m 2 Im (t) t dt =? 6 e2 (4) The infrared manifestation means that the anomaly is given by the zero momentum value?(0; m 2 ) and also that has a singularity in the IR domain. 3 QED - general case lim m!0 t Im(t; m2 ) =? e2 (t) (5) 6 An obvious drawback of the preceding derivation is its rather special kinematical conguration which forbids both the discussion of the anomalous pair of translation and scale symmetries and the use of unitarity in dierent channels. In analogy with the work [9] in scalar 4 eld theory, we will study the following Green's functions T = T = dx dy e?iqx+iky h0j (x)j (y)j (0)j0i dx dy e?iqx+iky h0j (x)j (y)j (0)j0i (6) where q = k + p. Let us decompose them in the basis of Bose symmetric tensor structures combined out of k; p and g T = F (q 2 )T () + : : : + F 23(q 2 )T (23) T = G (q 2 )T () + : : : + G 4(q 2 )T (4) (7) 2

4 assuming for simplicity k 2 = p 2 = 0. The vector, translation and trace Ward identities at the classical level k T = k (p) (8) q T = k (p) + p (k) (9) T = T + (k) + (p) (20) can be recast in terms of the new formfactors. Combining these identities we get q 2 (F F 6) =? 2 G 2 (2) for formfactors standing in front of k p (k k + p p ); k p (k p + p k ); g respectively. Assuming that G 2 (q 2 ) is given by an unsubtracted dispersion relation we can obtain an anomalous contribution to the RHS which can be explicitly calculated in the lowest order of QED A = 2 4m 2 ImG 2 t dt = 2 2 (22) Now we can assign this anomaly to any of the used symmetries. If we choose to x the vector and translation Ward identities we get anomalous trace identity T = T + (k) + (p) + which corresponds to the standard result [3, 4] e2 2 2 (q2 g? 2k p ) (23) anom = e F F (24) Let us note that this anomalous pair can be also studied by various UV techniques, for the case of point-splitting see [0]. 4 Subtraction constants and H! decay When we used unsubtracted dispersion relations (DR) for some of the formfactors we were in fact assuming a vanishing limit for k 2! or q 2!. This is not always correct. However in some important cases it is possible to justify such an assumption proving thus a kind of "No Subtraction Theorem" without loop integrals calculation; it just suces to study the integrands. These ambiguities due to the subtraction constants can be demonstrated on a specic physical process of the decay of Higgs boson to two photons. This one is interesting because the contribution of heavy particles is given precisely by the value of the trace anomaly of the photon eld. Consider just the W contribution. The amplitude can be expressed in terms of dimensionless formfactor F W = F W (m 2 W =m 2 H) M = 2 v F W (kp g? p k )" (k)" (p) (25) 3

5 The formfactor F W is nite and gauge independent. So we can calculate it e.g. in U-gauge. It was observed [] that although the DR for F W converge without any subtraction we do need some subtraction constant. Nevertheless we may note that the o-shell value of F W which we use in dispersion relations is not gauge independent. Analyzing the integrands of loop integrals we see that the asymptotic value for q 2! is nonvanishing in U-gauge due to the presence of =m 2 W terms in W propagators. On the other hand we may prove an opposite statement for the renormalizable gauge and it can be conrmed by a direct calculation. Let us remark that direct calculation in R-gauge is very tedious due to a huge number of diagrams. The easiest way to get the result is to extract the dierence between the o-shell values in U and R gauges which is given by three diagrams with unphysical Goldstone bosons. Thus in a renormalizable gauge we may obtain full trace anomaly immediately without any subtraction. 5 Acknowledgments I wish to thank Professor J. Horejs for guidance throughout the work and reading the manuscript and J. Novotny for valuable discussions. This work was partially supported by the grant GA CR-0506/98. References [] S. Coleman, R. Jackiw: Ann. Phys. 67 (97) 552. [2] M. Chanowitz, J. Ellis: Phys. Rev. D7 (973) [3] S. Adler, J.C. Collins, A. Duncan: Phys. Rev. D5 (977) 72. [4] J.C. Collins, A. Duncan, and S.D. Joglekar, Phys. Rev. D6, 438 (977) [5] J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model, (Cambridge University Press, 992) [6] M. Shifman: Phys. Rep. 209 (99) 34. [7] A.D. Dolgov, V.I. akharov: Nucl. Phys. B27 (97) 525. [8] J. Horejs, M. Schnabl:. Phys. C76 (997) 56. [9] N. Kawka, O. V. Teryaev, O. L. Veretin: preprint hep-ph/ (997) [0] J. Novotny, M. Schnabl, submitted to Fortschr. Phys., hep-th/ (998) [] J. Horejs, M. Stohr: Phys. Lett. B379 (996) 59. 4