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Radiative corrections to the photon-graviton vertex
PHYSICS LETTERS
Volume 41B, number 4
RADIATIVE
CORRECTIONS R. DELBOURGO Physics Department,
16 October 1972
TO THE PHOTON-GRAVITON
VERTEX
and P. PHOCAS-COSMETATOS Imperial College, London SW7 2BZ, UK
Received 1 August 1972 The lowest order electromagnetic corrections to the yTg vertex due to a charged boson or fermion loop are evaluated and correspond to the effective quadrupole interaction ~~(-1)2J(2J+l)g~UV~~~~~~~~/720~~z where J and Mare the spin and mass of the particle in the quantum loop. Unfortunately the chances of subjecting this result to experimental test, either at present accelerator energies or by astronomical observation, are remote.
Because of gauge invariance, there is a strong resemblance between the kinematic structure of reaction amplitudes in electrodynamics and in gravitation [ 11. This note is a result of an investigation into the nature of the quantum corrections to the classical gravitational vertices. The electromagnetic corrections to the electron-graviton vertex (effectively the electron stress tensor elements) have already been examined by Pagels [2], and here we study the counterpart problem for the photon which, at first glance, holds greater promise of being experimentally verifiable. We find that the order (Ycorrection to the photon-graviton vertex due to a massive charged particle loop (mass M, spin .I) has the momentum space structure (-1)2J(2.Jt1)
e2fk’.E(k)k.f’(k’)(ktk’)~(kt~‘)“E,.(k-k’)/144o~2~2
(1)
which can be reexpressed in terms of the total effective Lagrangian, = -$gp”[gKhFpKFvA + (w(-1)2J(2J+l)
Ftia; TV FK,/720S2
+.. .]
(2)
to this order in the gravitational p = 8rrG) and electromagnetic couplings (e2 =471(y). We also discover, as expected, that the infinite kinetic wavefunction and vertex renormalizations cancel exactly, so that (1) is a completely clean result, on a par with the calculation of the anomalous magnetic moment of the electron. In that sense it is rather a pity that we have been unable to find any way of confronting (1) against experiment, either through astronomical observations or in the laboratory - the basic reason is that, for the correction to stand out, one needs momentum transfers of the same order as 1OS M2 and these are simply not available with known sources of photons. All the electromagnetic and gravitational interactions of the lightest particles with the lowest spin, the pion and electron, follow from the Lagrangian [3],
I-gl1/2 L(cp,J/, g, A) = -?igKhgpvFK,& +!I.T~H~) +gK”(aK-ieAK)~(a,+ieA,)cp+ - ~1~~s’
where we choose the electron and pion fields to have canonical weight 0. As usual, g“ = L$,LK qmn where L is the symmetric tetrad field which defines the graviton field to first order via L = 1 tfh t . . . We can tabulate the Feynman rules provided by L to order f with on-mass shell gravitons: a _d- -
1jy@+p’),,
f
--f- - - + 2e2q,@ ,
d
W
; -1- -y-(p,P:+P,P;) 3
533
Volume
41B. number
4
PHYSICS
;--1 V
LETTERS
16 October
1972
PV
gsft@+z-O,v,,+(P+P’),~~,J,
a
+ - a
a_n,#-+f k;+k&
f
+ 2e2f(v,,~p"+9,,8pr),
-4
PV
k
k'
a
P
b-/,@,,
-
kp k;vpv W
. P
5
v-rat
P
. r
P
4,
2&f
b+p’),y, +(P+P'),Y,~
ef(y7,,+~,17,,)
1 ++
a!
and by sewing the vertices together we are able to construct the entire set of diagrams for the -rTg vertex to order cr. For the n+ loop they are drawn in fig. 1. For the e- loop, the last three diagrams are absent.
Fig. 1. Pion loop contributions
to the 7”yg vertex.
There is no point in giving the details of the calculation which are straightforward enough. The significant feature is the exact cancellation of all the corrections to the kinetic part of the bare vertex (-yiB,,ly), when the graphs are gauge-invariantly regularized. In particular, the logarithmic infinities associated with wave-function and vertex renormalizations are equal and opposite as prescribed by gauge invariance. We are left with a finite quadrupole-type correction, and for the pion and electron loops this can be expressed in the forms (1) or (2). In fact we expect the result to be true for any spin (not just J = 0, M = m, and J = 3, M = me) because (-1)2J is the spin-statistics factor and (2Jt 1) is simply the multiplicity of helicity-conserving gravitational couplings to the intermediate particle- anti-particle pair *. It is quite fair to draw a parallel between ‘y-ygand ee7 since in the latter case there is also cancellation of vertex and wave-function renormalizations, leaving one with the finite anomalous magnetic moment correction. Obviously it is of interest to see if the frequency dependent effects implied by (1) can be tested experimentally. The first place which comes to mind is the bending of light from the sun, with the interaction governed by the electron loop. In order to arrive at a classical particle trajectory from (2), we compare the classical and quantum-mechanical (Born) cross-section formulae, do/da
= lT/8nMo12 = -3 db2/d(cos 8)
which provides a differential relation between the impact parameter b and the scattering angle 0 that carries all the classical dynamical information. At small 0 the amplitude T is predominantly no flip and *It is amusing to note that (1) becomes singular when M = 0, but that, on the other hand, neutrinos are uncharged. tion neutral currents which can couple to neutrinos would be very undesirable from our point of view.
534
Weak interac-
Volume 41B, number 4
PHYSICS LETTERS
16 October 1972
The correction term being of order 0.1 X (Einstein value)2 e 10-11, we see that there is not the very faintest hope of verifying the (w) frequency dependence even with the best X-ray sources! A heuristic argument due to Salam confirms this conclusion: since (1) has the form at/l 80nm2 in the helicity conserving amplitude, it effectively corresponds to a correction a/l 80rrm2r3 to the Newtonian potential. So even if it were possible to disentangle this from inhomogeneities ’ f the sun, we would get an effective force on the photon of 2GM,ERi2 X (1 +a/ 180nm2R$) and the resulting deflection carries many of the marks of (3). Presumably other cosmological tests of (1) also lie in the realms of fantasy. Instead, therefore, let us try to look for repercussions of (1) in high-energy physics. The only possibility for making contact with elementary particles is to liken the 2+ mesons to gravity [4]. Here too, however, we have drawn a blank: (i) The minuscule decay rate I’+27 is hardly likely to be measured in the forseeable future. (ii) The 3”/f coupling cannot be obtained from 2n photoproduction in the same way that y7n0 is found from the Primakoff effect. The reason is that the background amplitude which mterferes with the photon-exchange graph is not precisely known even at small momentum transfers. (iii) If the Pomeron trajectory, or more loosely diffraction scattering, has similar characteristics [5] as graviton or f-exchange (one unit down in angular momentum) then we can look for possible effects of s-channel helicity nonconservation caused by (1). It is a simple matter to obtain the helicity amplitudes for high energy Compton scattering, T++ +fls”p’f)[ 1 - (~(-1)~~(2J+l)t/180r&f~], T+- + /?PP(t)[- a(-1)2J(2J+ 1)t/180rrM2] . The polarization effects will be maximized if we take the lightest hadron loop that couples to the Pomeron, i.e. the pion. Even so, they are tiny, being of order 0.0005t in GeV/c units, and giving only 1% effects in cross-sections even when t = 10 (GeV/c)2. We are forced to conclude that in spite of its simple nature, the radiative correction is just a nice academic exercise, because it evades any kind of experimental test. We hope we are wrong. We thank Professor Abdus Salam for many pertinent
remarks.
References [l] S. Weinberg, Phys. Rev. 138 (1965) B988, and 135 (1964) B1049; R. Delbourgo and A. Salam, Phys. Lett. 40B (1972) 381. [2] H. Pagels, Phys. Rev. 144 (1966) 1250. [3] R. Delbourgo, A. Salam and .I. Strathdee, Nuovo Cim. Lett. 2 (1969) 354; C.J. Isham, A. Salam and J. Strathdee, Phys. Rev. 3 (1971) D1805. [4] R. Delbourgo, A. Salam and J. Strathdee, Nuovo Cim. 49 (1967) 593; P.G.O. Freund, Phys. Rev. Lett. 16 (1966) 291,424. [5] H.F. Jones and A. Salam, Phys. Lett. 34B (1971) 149.
535
Volume 41B, number 4
RADIATIVE
CORRECTIONS R. DELBOURGO Physics Department,
16 October 1972
TO THE PHOTON-GRAVITON
VERTEX
and P. PHOCAS-COSMETATOS Imperial College, London SW7 2BZ, UK
Received 1 August 1972 The lowest order electromagnetic corrections to the yTg vertex due to a charged boson or fermion loop are evaluated and correspond to the effective quadrupole interaction ~~(-1)2J(2J+l)g~UV~~~~~~~~/720~~z where J and Mare the spin and mass of the particle in the quantum loop. Unfortunately the chances of subjecting this result to experimental test, either at present accelerator energies or by astronomical observation, are remote.
Because of gauge invariance, there is a strong resemblance between the kinematic structure of reaction amplitudes in electrodynamics and in gravitation [ 11. This note is a result of an investigation into the nature of the quantum corrections to the classical gravitational vertices. The electromagnetic corrections to the electron-graviton vertex (effectively the electron stress tensor elements) have already been examined by Pagels [2], and here we study the counterpart problem for the photon which, at first glance, holds greater promise of being experimentally verifiable. We find that the order (Ycorrection to the photon-graviton vertex due to a massive charged particle loop (mass M, spin .I) has the momentum space structure (-1)2J(2.Jt1)
e2fk’.E(k)k.f’(k’)(ktk’)~(kt~‘)“E,.(k-k’)/144o~2~2
(1)
which can be reexpressed in terms of the total effective Lagrangian, = -$gp”[gKhFpKFvA + (w(-1)2J(2J+l)
Ftia; TV FK,/720S2
+.. .]
(2)
to this order in the gravitational p = 8rrG) and electromagnetic couplings (e2 =471(y). We also discover, as expected, that the infinite kinetic wavefunction and vertex renormalizations cancel exactly, so that (1) is a completely clean result, on a par with the calculation of the anomalous magnetic moment of the electron. In that sense it is rather a pity that we have been unable to find any way of confronting (1) against experiment, either through astronomical observations or in the laboratory - the basic reason is that, for the correction to stand out, one needs momentum transfers of the same order as 1OS M2 and these are simply not available with known sources of photons. All the electromagnetic and gravitational interactions of the lightest particles with the lowest spin, the pion and electron, follow from the Lagrangian [3],
I-gl1/2 L(cp,J/, g, A) = -?igKhgpvFK,& +!I.T~H~) +gK”(aK-ieAK)~(a,+ieA,)cp+ - ~1~~s’
where we choose the electron and pion fields to have canonical weight 0. As usual, g“ = L$,LK qmn where L is the symmetric tetrad field which defines the graviton field to first order via L = 1 tfh t . . . We can tabulate the Feynman rules provided by L to order f with on-mass shell gravitons: a _d- -
1jy@+p’),,
f
--f- - - + 2e2q,@ ,
d
W
; -1- -y-(p,P:+P,P;) 3
533
Volume
41B. number
4
PHYSICS
;--1 V
LETTERS
16 October
1972
PV
gsft@+z-O,v,,+(P+P’),~~,J,
a
+ - a
a_n,#-+f k;+k&
f
+ 2e2f(v,,~p"+9,,8pr),
-4
PV
k
k'
a
P
b-/,@,,
-
kp k;vpv W
. P
5
v-rat
P
. r
P
4,
2&f
b+p’),y, +(P+P'),Y,~
ef(y7,,+~,17,,)
1 ++
a!
and by sewing the vertices together we are able to construct the entire set of diagrams for the -rTg vertex to order cr. For the n+ loop they are drawn in fig. 1. For the e- loop, the last three diagrams are absent.
Fig. 1. Pion loop contributions
to the 7”yg vertex.
There is no point in giving the details of the calculation which are straightforward enough. The significant feature is the exact cancellation of all the corrections to the kinetic part of the bare vertex (-yiB,,ly), when the graphs are gauge-invariantly regularized. In particular, the logarithmic infinities associated with wave-function and vertex renormalizations are equal and opposite as prescribed by gauge invariance. We are left with a finite quadrupole-type correction, and for the pion and electron loops this can be expressed in the forms (1) or (2). In fact we expect the result to be true for any spin (not just J = 0, M = m, and J = 3, M = me) because (-1)2J is the spin-statistics factor and (2Jt 1) is simply the multiplicity of helicity-conserving gravitational couplings to the intermediate particle- anti-particle pair *. It is quite fair to draw a parallel between ‘y-ygand ee7 since in the latter case there is also cancellation of vertex and wave-function renormalizations, leaving one with the finite anomalous magnetic moment correction. Obviously it is of interest to see if the frequency dependent effects implied by (1) can be tested experimentally. The first place which comes to mind is the bending of light from the sun, with the interaction governed by the electron loop. In order to arrive at a classical particle trajectory from (2), we compare the classical and quantum-mechanical (Born) cross-section formulae, do/da
= lT/8nMo12 = -3 db2/d(cos 8)
which provides a differential relation between the impact parameter b and the scattering angle 0 that carries all the classical dynamical information. At small 0 the amplitude T is predominantly no flip and *It is amusing to note that (1) becomes singular when M = 0, but that, on the other hand, neutrinos are uncharged. tion neutral currents which can couple to neutrinos would be very undesirable from our point of view.
534
Weak interac-
Volume 41B, number 4
PHYSICS LETTERS
16 October 1972
The correction term being of order 0.1 X (Einstein value)2 e 10-11, we see that there is not the very faintest hope of verifying the (w) frequency dependence even with the best X-ray sources! A heuristic argument due to Salam confirms this conclusion: since (1) has the form at/l 80nm2 in the helicity conserving amplitude, it effectively corresponds to a correction a/l 80rrm2r3 to the Newtonian potential. So even if it were possible to disentangle this from inhomogeneities ’ f the sun, we would get an effective force on the photon of 2GM,ERi2 X (1 +a/ 180nm2R$) and the resulting deflection carries many of the marks of (3). Presumably other cosmological tests of (1) also lie in the realms of fantasy. Instead, therefore, let us try to look for repercussions of (1) in high-energy physics. The only possibility for making contact with elementary particles is to liken the 2+ mesons to gravity [4]. Here too, however, we have drawn a blank: (i) The minuscule decay rate I’+27 is hardly likely to be measured in the forseeable future. (ii) The 3”/f coupling cannot be obtained from 2n photoproduction in the same way that y7n0 is found from the Primakoff effect. The reason is that the background amplitude which mterferes with the photon-exchange graph is not precisely known even at small momentum transfers. (iii) If the Pomeron trajectory, or more loosely diffraction scattering, has similar characteristics [5] as graviton or f-exchange (one unit down in angular momentum) then we can look for possible effects of s-channel helicity nonconservation caused by (1). It is a simple matter to obtain the helicity amplitudes for high energy Compton scattering, T++ +fls”p’f)[ 1 - (~(-1)~~(2J+l)t/180r&f~], T+- + /?PP(t)[- a(-1)2J(2J+ 1)t/180rrM2] . The polarization effects will be maximized if we take the lightest hadron loop that couples to the Pomeron, i.e. the pion. Even so, they are tiny, being of order 0.0005t in GeV/c units, and giving only 1% effects in cross-sections even when t = 10 (GeV/c)2. We are forced to conclude that in spite of its simple nature, the radiative correction is just a nice academic exercise, because it evades any kind of experimental test. We hope we are wrong. We thank Professor Abdus Salam for many pertinent
remarks.
References [l] S. Weinberg, Phys. Rev. 138 (1965) B988, and 135 (1964) B1049; R. Delbourgo and A. Salam, Phys. Lett. 40B (1972) 381. [2] H. Pagels, Phys. Rev. 144 (1966) 1250. [3] R. Delbourgo, A. Salam and .I. Strathdee, Nuovo Cim. Lett. 2 (1969) 354; C.J. Isham, A. Salam and J. Strathdee, Phys. Rev. 3 (1971) D1805. [4] R. Delbourgo, A. Salam and J. Strathdee, Nuovo Cim. 49 (1967) 593; P.G.O. Freund, Phys. Rev. Lett. 16 (1966) 291,424. [5] H.F. Jones and A. Salam, Phys. Lett. 34B (1971) 149.
535