QED Vertex Correction

Transcription

1 QED Vertex Correction In these notes I shall calculate the one-loop correction to the PI electron-electron-photon vertex in QED, ieγ µ p,p) = ) We are interested in this vertex in the context of elastic Coulomb scattering, e X e X ) so we take the incoming and the outgoing electrons to be on-shell, p = p = m, but the photon is off-shell, q. Moreover, we put the vertex in the context of the complete electron line including the external line factors, thus ūp ) ieγ µ up). As discussed in class, this simplifies the Lorentz and Dirac structure of the vertex and allows us to write it as Γ µ p,p) = F el q ) p +p) µ m + F mag q ) iσµν q ν m = F q ) γ µ + F q ) iσµν q ν m. 3)

2 Working Through the Algebra At the one-loop level of QED, the PI vertex correction comes from a single Feynman diagram 4) Using the Feynman gauge for the internal photon s propagators, this diagram evaluates to ieγ µ loop p,p) = where and reg = e 3 d 4 k ig νλ π) 4 k +i ieγ i i ν p + k m+i ieγµ p+ k m+i ieγ λ reg = e 3 reg d 4 k π) 4 k +i γν d 4 k N µ π) 4 D p + k +m p +k) m +i γµ p+ k +m p+k) m +i γ ν N µ = γ ν k+ p +m)γ µ k+ p+m)γ ν 6) D = [ k +i ] [ p+k) m +i ] [ p +k) m +i ]. 7) 5) The purpose of this section of the notes is to simplify these numerator and denominator. Using the Feynman parameter trick, we may combine the 3 denominator factors as D = dxdydzδx+y+z ) [ xp+k) m ) + yp +k) m ) + zk ) + i ] 3. 8)

3 Inside the big square brackets here we have [ ] = x p+k) m ) + y p +k) m ) + z k = k x+y +z = ) + k µ xp+yp ) µ + xp m ) + yp m 9) = k +xp+yp ) where = xp+yp ) xp yp + x+y)m = xy p p = p +p p p) ) x x) p y y) p + x+y) m = xy q x x y) p y x y) p + x+y) m = xy q xz p yz p + z) m ) For the on-shell electron momenta, p = p = m, we may further simplify z) m xz p yz p = m z) x+y)z = z) ) ) which gives = z) m xy q. ) Let us also define the shifted loop momentum l = k + xp + yp, 3) then we can rewrite the denominator as D = dxdydzδx+y +z ) [l +i] 3. 4) As usual, we plug this denominator into the loop integral 5), then change the order of integration over the loop momentum before over the Feynman parameters, and then shift 3

4 the momentum integration variable from k to l, thus Γ µ loop p,p) = ie dxdydzδx+y +z ) reg d 4 l N µ π) 4 [ l +i ] 3. 5) But to make full use of the momentum shift, we need to re-express the numerator N µ in terms of the shifted momentum l. It would also help to simplify the numerator 6) in the context of this monstrous integral. The first step towards simplifying the N µ is obvious: Let us get rid of the γ ν and γ ν factors using the γ matrix algebra, eg., γ ν aγ ν = a, etc.. However, in order to allow for the dimensional regularization, we need to re-work the algebra for an arbitrary spacetime dimension D where γ ν γ ν = D 4. Consequently, γ ν aγ ν = a + 4 D) a, γ ν a bγ ν = 4ab) 4 D) a b, 6) γ ν a b cγ ν = c b a + 4 D) a b c, and therefore N µ def = = γ ν k+ p +m)γ µ k+ p+m)γ ν = m γ µ + 4mp +p+k) µ p+ k)γ µ p + k) 7) + 4 D) p + k m)γ µ p+ k m). The second step is to re-express this numerator in terms of the loop momentum l rather than k using eq. 3). Expanding the result in powers of l, we get quadratic, linear and l independent terms, but the linear terms do not contribute to the d D l integral because they are odd with respect to l l while everything else in that integral is even. Consequently, in the context of 4

5 eq. 5) we may neglect the linear terms, thus N µ = m γ µ + 4mp +p+l xp yp ) µ p+ l x p y p )γ µ p + l x p y p ) + 4 D) p + l x p y p m)γ µ p+ l x p y p m) skipping terms linear in l = m γ µ + 4mp+p xp yp ) µ 8) lγ µ l p x p y p )γ µ p x p y p ) + 4 D) lγ µ l + 4 D) p y p x p m)γ µ p x p y p m). Next, we make use of p p = q and x y = z to rewrite xp + yp = x+y) p+p ) + x y) p p ), p + p xp yp = z p +p) + x y) q, p xp yp = z p y q = z p x) q, 9) p xp yp = z p + x q = z p + y) q, and consequently N µ = m γ µ + 4mzp +p) µ + 4mx y)q µ + +4 D) lγ µ l z p +x ) q)γ µ z p+ y) q) ) + 4 D)z p +x q m)γ µ z p y q m). The third step is to make use of the external fermions being on-shell. This means more than just p = p = m : We also sandwich the vertex ieγ µ between the Dirac spinors ūp ) on the left and up) on the right. The two spinors satisfy the appropriate Dirac equations pup) = mup) and ūp ) p = ūp )m, so in the context of ūp )Γ µ up), A p = A m and p B = m B ) for any terms in Γ µ that look like A p or p B for some A or B. Consequently, the terms on 5

6 the last two lines of eq. ) are equivalent to z p +x ) q)γ µ z p+ y) q) = zm+x ) q)γ µ zm+ y) q) = z m γ µ x) y) qγ µ q + zx y)m {γ µ, q} = q µ) ) + z x y)m [γ µ, q] = iσ µν q ν z p +x q m)γ µ z p y q m) = z )m+x q)γ µ z )m y q) ) = z) m γ µ xy qγ µ q z)x y)m {γ µ, q} = q µ) ) + z)x+y)m [γ µ, q] = iσ µν q ν. Let s plug these expressions back into eq. ), collect similar terms together, and make use of x y = z. This gives us N µ = D ) lγµ l + 4mzp +p) µ + m γ µ z + 4 D) z) ) + qγ µ q ) z +xy) 4 D)xy ) + mq µ x y) 4 z 4 D) z) + imσ µν q ν z+z) 4 D) z) ). 3) Furthermore, in the context of the Dirac sandwich ūp )Γ µ up) we have qγ µ q = q µ q q γ µ = q γ µ 4) because ūp ) qup) =, and also p +p) µ = mγ µ iσ µν q µ 5) the Gordon identity). Plugging these formulae into eq. 3), we arrive at N µ = D ) lγµ l + m γ µ 8z +z )+4 D) z) ) ) ) q γ µ z +xy) 4 D)xy imσ µν q ν z) z +4 D) z) ) + mq µ x y) 4 z 4 D) z). 6) 6

7 To further simplify this expression, let us go back to the symmetries of the integral 5). The integral over the Feynman parameters, the integral d D l, and the denominator [l ] 3 are all invariant under the parameter exchange x y. In eq. 6) for the numerator, the first two lines are invariant under this symmetry, but the last line changes sign. Consequently, only the first two lines contribute to the integral 5) while the third line integrates to zero and may be disregarded, thus N µ = D ) lγµ l + m γ µ 8z +z )+4 D) z) ) ) ) q γ µ z +xy) 4 D)xy imσ µν q ν z) z +4 D) z). 7) Finally, thanks to the Lorentz invariance of the d D l integral, and hence lγ µ l = γ λ γ µ γ ν l λ l ν = γ λ γ µ γ ν g λν l l λ l ν = gλν l D, 8) D = D )γµ l D. 9) Plugging this formula into eq. 6) and grouping terms according to their γ matrix structure, we arrive at N µ = N γ µ N iσµν q ν m 3) where N D ) = l + D z +xy) 4 D)xy 8z +z )+4 D) z) ) m ) q D ) = l D ) + z m q ), 3) D ) N = z) 4z +4 D) z) m. 3) Note that splitting the numerator according to eq. 3) is particularly convenient for calculating the electron s form factors: Γ µ loop = Floop q ) γ µ + F loop q ) iσµν q ν m, 33) 7

8 F loop q ) = ie F loop q ) = +ie d D l N dxdydzδx+y +z ) π) D [ l +i ] 3, 34) d D l N dxdydzδx+y +z ) π) D [ l +i ] 3. 35) Electron s Gyromagnetic Moment As explained earlier in class, electron s spin couples to the static magnetic field as Ĥ eg ) q S B where g = F mag = F +F m. 36) e = The electric form factor F F el for q = is constrained by the Ward identity, F tot = F tree + F loops + F counter terms q. 37) Therefore, the gyromagnetic moment is g = + F q = ) 38) where F = F loops because the there are no tree-level or counter-term contributions to the F, only to the F. Thus, to calculate the g at the one-loop level, all we need is to evaluate the integral 35) for q =. Let s start with the momentum integral d D l π) D N [ l +i ] 3 39) where = z) m for q = and N is as in eq. 3). Because the numerator here does not depend on the loop momentum l, this integral converges in D = 4 dimensions and there is no 8

9 need for dimensional regularization. All we need is to rotate the momentum into Euclidean space, d 4 l N id 4 π) 4 [ l +i ] 3 = N l E π) 4 l E + )3 = in 6π = in 6π dl E l E l E + )3 = i 3π N = 4z z)m for D = 4 = z) m for q = = i 3π 4z z. 4) Substituting this formula into eq. 35), we have F loop q = ) = e 6π dxdydzδx+y +z ) 4z z. 4) The integrand here depends on z but not on the other two Feynman parameters, so we can immediately integrate over x and y and obtain dxdyδx+y +z ) = z dx = z. 4) Consequently, F loop q = ) = e 6π dz z) 4z z = e 6π = α π 43) and the gyromagnetic moment is g = + α π + Oα ). 44) 9

10 Higher-loop calculations are more complicated because the number of diagrams grows very rapidly with the number of loops; at 4-loop order there are thousands of diagrams, and one needs a computer just to count them! Also, at higher orders one has to include the effects of strong and weak interactions because the photons interact with hadrons and W ± particles, which in turn interact with other hadrons, Z, Higgs, etc., etc. Nevertheless, people have calculated the electron s and muon s g factors up to the order α 4 back in the 97s, and more recent calculations are good up to the order α 5. Meanwhile, the experimentalists have measured g e to a comparable accuracy of significant digits and g µ to 9 significant digits g e = ), g µ =.33844). 45) The theoretical value of g e is in good agreement with the experimental value, while for the muon there is a small discrepancy g exp µ g theory µ 59 ± 3 ± ). This discrepancy indicates some physics beyond the Standard Model, maybe supersymmetry, maybe something else. In general, effect of heavy particles on g µ is proportional to m µ /M heavy ), that s why g µ is much more sensitive to new physics than g e. For a recent review of high-precision g e and g µ both the experiments and the theory see I would like to complete this section of the notes by calculating the F loop q ) form factor for q. Proceeding as in eq. 4) but letting = z) m xyq, we have d 4 l π) 4 N [ l +i ] 3 = i 3π 4z z)m z) m xyq 46) and hence F loop q ) = e 6π dxdydzδx+y +z ) 4z z)m z) m xyq. 47) To evaluate this integral over Feynman parameters, we change variables from x,y,z to w = z

11 and ξ = x/x+y), x = wξ, y = w ξ), z = w, dxdydzδx+y +z ) = wdwdξ. 48) Consequently, F loop q ) = e 6π = e 6π = e 8π dξ dww m dξ m ξ ξ)q m dξ m ξ ξ)q 4 w)w m w m w ξ ξ) q = α π 4m q 4m q ) arctan dww 4w w) w q 4m q = α π 4m q ) 4m q ) log 4m q + q m. 49) For q < and q m, F loop q ) α π m q log q m. 5)

12 The Electric Form Factor Now consider the electric form factor F q ). In the first section we have obtained F loop q ) = ie d D l N dxdydzδx+y +z ) π) D [ l +i ] 3, 34) for N µ and = z) m xyq. = D ) D l D ) + z m q ) 3) Let s start by calculating the momentum integral in eq. 34). The numerator N depends on l as al +b, so there is a logarithmic UV divergence for l ; to regularize this divergence, we work in D = 4 ǫ dimensions. Thus, i reg d 4 l al +b π) 4 [l +i] 3 iµ4 D id = iµ 4 D D l E al E +b π) D [l E + ]3 = µ 4 D d D l E π) D = µ 4 D d D l E π) D = = dt d D l π) D al +b [l +i] 3 = [ al E b l E + )3 = a l E + ) a +b l E + )3 dt a t a +b) t) e t +l E) a t a +b) t) e t µ 4 D d D l E π) D e tl e dt a t a +b) t) e t µ4 D 4πt) D/ = µ4 D 4π) D/ = µ4 D 4π) D/ dte t a t D/) a +b) t D/)) { a Γ D 4πµ ) ǫ 6π Γ+ǫ) ǫ ) D a +b) Γ 3 D { a ǫ a +b }. ] } ) D 3 5)

13 In light of eq. 3), a = D ) D, b = z m q ) D ), 5) so on the last line of eq. 5) a ǫ a +b = ǫ ǫ zm q ). 53) Consequently, the momentum integral in eq. 34) for the electric form factors evaluates to d ie µ 4 D D l N π) D [l +i] 3 = 54) = α 4πµ ) ǫ { Γǫ) ǫ) Γ+ǫ) z m q } ), π and now we need to integrate this expression over the Feynman parameters. Changing the integration variables from x,y,z to w and ξ according to eq. 48), we have F loop q ) = α π 4πµ ) ǫ dξ ǫ)γǫ) dww [ w,ξ)] ǫ Γ+ǫ) w)m q ) [ w,ξ)] +ǫ 55) where w,ξ) = z) m xyq = w m ξ ξ)q ), 56) or equivalently, w,ξ) = w Hξ) where Hξ) def = m ξ ξ)q. 57) The form 57) is particularly convenient for evaluating the dw integral in eq. 55), which becomes { ǫ)γǫ) dw H ǫ w w ǫ Γ+ǫ) m q H +ǫ w w) } w +ǫ. 58) Near the lower limit w, the integrand is dominated by the second term, which is proportional 3

14 to w ǫ. But for any ǫ i.e., for any dimension D 4 the integral positive dw w +ǫ 59) diverges: For D = 4 the divergence is logarithmic while for D < 4 it becomes power-like. Infrared Divergence. Physically, the divergence 59) is infrared rather than ultraviolet, that s why it gets worse as we lower the dimension D. Indeed, let s go back to the diagram 4) and look at the denominator D in eqs. 5) and 7). Taking the electron s momenta p and p on-shell before introducing the Feynman parameters, we have p+k) m = k + kp and likewise p +k) m = k + kp. 6) Therefore, for k the denominator behaves as D k 4 while the numerator N µ remains finite, which makes the integral d D k Nµ D d D k k 4 6) diverge for k. In D = 4 dimensions, the infrared divergence here is logarithmic, while in lower dimensions D < 4 it becomes power-like, i.e. O /k min ) 4 D) precisely as in eqs. 59) and 58). We can regularize the infrared divergence 6) and also 59) by analytically continuing spacetime dimension to D > 4. Such dimensional regularization of the IR divergences is used in many situations in both QFT and condensed matter. However, taking D > 4 makes the ultraviolet divergences worse, so if some amplitude has both UV and IR divergences, we cannot cure both of them at the same time by analytically continuing to D 4. In particular, when calculating the electric form factor F q ) of the electron, we need D < 4 to regulate the momentum integral d D l, but then we need D > 4 to regulate the integral over the Feynman parameters. A common dirty trick is to first continue to D < 4 and evaluate the d D l momentum integral, then analytically continue the result to D > 4 and integrate over the Feynman parameters, and then continue the final result to D = 4. However, in this kind of dimensional regularization it s 4

15 hard to disentangle the /ǫ poles coming from the UV divergence logλ /µ ) from the /ǫ poles coming from the IR divergence logµ /kmin ), so we are not going to use it here. Instead, wearegoingtousedrfortheuvdivergenceonly, whiletheirdivergenceisregulated by a tiny but not-quite-zero photon mass m γ m e. Strictly speaking, a massive vector particle has three polarization states and its propagator is = i k m γ +i g µν kµ k ν m γ ). 6) However, the longitudinal polarization of the massive but ultra-relativistic photon does not couple to a conserved current, so we are going to disregard the k µ k ν terms in the propagator 6) and use = ig µν k m γ +i. 63) In other words, we use the Feynman gauge in spite of the photon s mass; this is not completely consistent, but the inconsistencies go away in the m γ limit. Using this infrared regulator for the internal photon line in the one-loop diagram 4), we get the vertex amplitude that looks exactly like eq. 5) except for one factor in the denominator, k +i becomes k m. 64) γ +i Interms oftheintegral 5), thischange hasno effect onthenumerator N µ or theloopmomentum l which remains exactly as in eq. 3)), but the in the denominator becomes x,y,z) = x,y,z) + z m γ. 65) Consequently, the electric form factor is F loop q ) = dfp) µ 4 D dd l ie N π) D [l +i] 3, 66) exactly as in eq. 34), except for the instead of the in the denominator. The momentum integral here converges for any D < 4 and it evaluates exactly as in eq. 5). The only subtlety 5

16 here is that in the numerator, the l independent term b involves the un-modified instead of cf. eq. 5)), but we can fix that by writing b = z m e q + ǫ)m γ) ǫ). 67) Hence, instead of eq. 55) we get F loop q ) = α π 4πµ ) ǫ dξ ǫ)γǫ) dww [ w,ξ)] ǫ Γ+ǫ) w)m e q + ǫ)m γ ) [ w,ξ)] +ǫ 68) where w,ξ) = z) m e xyq + zm γ = w Hξ) + w) m γ. 69) Note that the photon s mass is tiny, m γ m e,q ; were it not for the IR divergences, we would have used m γ =. This allows us to neglect various Om γ) terms in eq. 68) except when it would cause a divergence for w ; in particular, we may neglect the ǫ)m γ term in the numerator of the second term in the integrand. As to the denominators, in eq. 69) the second term containing the photon s mass becomes important only in the w limit, and in that limit w)m γ m γ. Thus, we approximate and the dw integral in eq. 68) becomes w,ξ) w Hξ) + m γ 7) dww { ǫ)γǫ) [w Hξ)+m Γ+ǫ) w)m e q } ) γ ]ǫ [w Hξ)+m γ ]+ǫ = ǫ)γǫ) dww H ǫ [w +m γ/h)] ǫ + Γ+ǫ) m e q H +ǫ Γ+ǫ) m e q H +ǫ dww [w +m γ /H)]+ǫ dww [w +m γ/h)] +ǫ. 7) 6

17 For < ǫ < i.e., for 3 < D < 4 the integrals on the second and third lines here converge even for m γ =, dww [w ] ǫ = dww [w ] +ǫ = ǫ ǫ for ǫ <, for ǫ <, 7) so we may just as well evaluate them without the photon s mass. Only on the last line of eq. 7) we do need m γ to make the integral converge for some D 4: dww [w +m = γ /H)]+ǫ ǫ [w +m γ /H)]ǫ = ǫ [ H m γ ) ǫ ]. 73) Combining all these dw integrals together, we get { } dw = Γǫ) H ǫ + Γ+ǫ) ǫ m e q H +ǫ Γ+ǫ) ǫ = Γǫ) [ { H ǫ + m e q H ǫ H m γ [ ) ǫ m e q H H +ǫ ] m γ ) ǫ ]} 74) and hence F loop q ) = α 4π 4πµ ) ǫ dξ Γǫ) { + m e q Hξ) Hξ) [ ǫ ) ǫ ]} Hξ) m γ 75) where Hξ) = m e ξ ξ)q. 57) Before we even try to perform this last integral, let s remember that Γ µ net = γ µ tree + Γµ loops + δ γ µ 76) 7

18 and hence F net q ) = tree + F loops q ) + δ. 77) Also, the net electric charge does not renormalize, so we must have F net q ) for q 78) and hence δ = F loops q = ). 79) To calculate the counterterm δ to order α we use eq. 75) for q =, in which case Hξ) m e and the dξ becomes trivial the integrand does not depend on ξ at all). Thus, δ = α 4π Γǫ) 4πµ m e ) ǫ { + ǫ m ) ǫ } e m γ + Oα ). 8) This formula holds for any dimension D between 3 and 4 i.e., < ǫ < ). In the D 4 limit, it becomes δ = α 4π { ǫ γ E + log 4πµ m e } + 4 log m e m + Oα ). 8) γ Now let s go back to the electric form factor F net q ) for q. According to eqs. 77) and 79), at the one-loop level F net q ) = F loop q ) F loop ) + Oα ) 8) where F loop q ) is given by eq. 75). Taking the ǫ limit of that formula, we arrive at F loop q ) = α 4π dξ { ǫ γ E 4πµ + log Hξ) + m e q [ log Hξ) ]} Hξ) m, 83) γ and now we should subtract a similar a similar expression for q =. This subtraction cancels the UV divergence and the associated /ǫ pole but not the IR divergence. Moreover, not only the 8

19 subtracted one-loop amplitude depends on the IR regulators, but the coefficient of the logm γ has a non-trivial momentum dependence. Indeed, F loop q ) F loop ) = = α 4π = α 4π dξ dξ { log m e Hξ) + m e q Hξ) ) { + m e q log m e Hξ) Hξ) + [ log Hξ) ] m γ = α 4π { hq /m e ) + f IRq /m e ) log m e m γ m e q } Hξ) [ ]} log m e m γ ) [ ]} log m e m γ where hq /m e) and f IR q /m e) are finite in the limit m γ ) functions of the q /m e ratio; both of them vanish for q =. Specifically, 84) m f IR q /m e ) = dξ e q Hξ) = q ξ +ξ ) ) m e q, 85) ξ ξ) which happens to be the same function that governs the IR divergence of the the soft-photon bremsstrahlung. In terms of 6. of the Peskin & Schroeder textbook, f IR q /m e ) = Iv,v ) = d Ω n 4π [ p µ ) ] n = n = np ) pµ, 86) np) seetextbookseqs )fortheproof. Note: mydefinitionofthei IR differsfromthetextbook s by a factor of. Altogether, the electric form factor of the electron is F net q ) = α { } 4π gq /m e ) + f IRq /m e ) log m e m γ + Oα ). 87) Implications of this formula will be discussed in class; see also 6.4 of the textbook. 9