Feynman parameter integrals

Transcription

1 Feynman parameter ntegrals We often deal wth products of many propagator factors n loop ntegrals. The trck s to combne many propagators nto a sngle fracton so that the four-momentum ntegraton can be done easly. Ths s done commonly usng so-called Feynman parameters. We rewrte the product of propagators A + ɛ)a 2 + ɛ) A n + ɛ), ) where A has the form of p 2 m 2. The sgn of A s not fxed, but the magnary part has the fxed sgn because of ɛ. Ths turns out to be useful. A sngle propagator can be rewrtten usng a smple ntegral A = dte ta+ɛ). 2) Thesurfacetermatt vanshes because of ɛ term. Usng ths multple tmes, we fnd A + ɛ)a 2 + ɛ) A n + ɛ) = )n dt dt n e n t A +ɛ). 3) Here comes another trck. We use the dentty dλ = λ δ ) t. 4) λ Ths can be shown usng the general formula δfx)) = δx x )/ f x ) where x s the zero of fx),.e., fx ) =. The delta functon n the above ntegral therefore can be rewrtten as δ ) t = δλ n t ) λ n = λδλ t ), 5) t λ 2 and the ntegraton can be done trvally to yeld unty. We used the fact that all of t n Eq. 3) are postve to lmt the λ ntegraton above zero. Insertng the unty Eq. 4) nto Eq. 3), we fnd dλ A + ɛ)a 2 + ɛ) A n + ɛ) = )n dt dt n λ δ λ ) t e n t A +ɛ). 6) Now we change the varables from t to t = λx, and fnd ) = ) n dx dx n dλλ n x e n λt A +ɛ). 7) Now the λ ntegral can be done by usng the ntegral representaton of the gamma functon Γn) =n )! = dt t n e t. 8)

2 Ths expresson can be generalzed to the followng one n )! X = dt t n e tx 9) n as long as ReX) > so that the ntegral converges. Because Re A + ɛ)) = ɛ>, we fnd ) n )! A + ɛ)a 2 + ɛ) A n + ɛ) = )n dx dx n n t A + ɛ)) δ x n. ) Fnally, note that all x are postve whle the sum of x must be unty. Therefore the ntegraton regon can be lmted to <x < : A + ɛ)a 2 + ɛ) A n + ɛ) =n )! The smple cases of n =2, 3are dx dx n n t A + ɛ)) δ n ) x. ) A + ɛ)a 2 + ɛ) = dx xa + ɛ)+ x)a 2 + ɛ)), 2 2) A + ɛ)a 2 + ɛ)a 3 + ɛ) =2 dx dy dz δ x y z) xa + ɛ)+ya 2 + ɛ)+za 3 + ɛ)). 3 3) 2

3 General Ideas on the Self-energy Dagrams We calculate the two-pont functon G 2 x y) = Ω ψx) ψy) Ω n perturbaton theory. An mportant pont s to realze s that the full two-pont functon can be obtaned once one has all the PI one partcle rreducble) dagrams Σ p) computed because G 2 p) d 4 xg 2 x y)e p x y) = + Σ p)) + Σ p)) Σ p)) + p m p m p m p m p m p m = p m Σ p). 4) Here, m s the bare mass n the Lagrangan whch s dfferent from the physcal knetc) mass. To correctly dentfy the mass of the partcle, we look for the zero of the denomnator n the two-pont functon. We defne the physcal mass m of the partcle by the equaton p m Σ p) p=m = m m Σm) =. 5) Then we expand the self-energy dagram Σ p) around p = m as Σ p) =δm Z 2 ) p m)+z 2 Σ R p), 6) wth δm =Σm), Z 2 ) = Σ p)/ p p=m,andσ R p) behaves as O p m) 2 when p m. Then the two-pont functon becomes G 2 p) = p m δm +Z2 ) p m ) Z2 Σ R p) = Z 2 p m Σ R p), 7) wth m = m + δm. Therefore, δm s nterpreted as the correcton to the mass of the partcle due to nteractons, and ths s why these dagrams are called self-energy dagrams. The factor Z 2 s called the wave-functon renormalzaton factor whch descrbes the strength square root of the probablty) of the feld operator ψ creatng the one-partcle state. At the lowest order n perturbaton theory Z 2 =, but s dfferent from unty once hgher order correctons are taken nto account. Ths s the factor that appears n the LSZ reducton formula. 3

4 Explct Calculaton of the One-loop Self-energy Dagram At the one-loop level, the self-energy dagram s gven by Σ p) = eγ µ ) d 4 k 2π) 4 p + k m + ɛ g µν k 2 + ɛ eγν ). 8) Ths ntegral, however, s dvergent both n the ultravolet k and the nfrared k. To deal wth these dvergences, we regulate the ntegral,.e., ntroduce parameters whch make the ntegral formally convergent, but the parameters must be taken to zero or nfnty at the end of calculatons. There are many ways of regularzng the ntegrals, but we adopt the followng prescrpton for ths purpose: k 2 + ɛ k 2 µ 2 + ɛ k 2 Λ 2 + ɛ = µ 2 Λ 2 k 2 µ 2 )k 2 Λ 2 ). 9) In the end we take the lmt µ andλ to recover the orgnal expresson, but the ntegral becomes fnte as long as we keep both µ and Λ fnte. They are called nfrared or ultravolet cutoffs. Below, we often drop ɛ terms but t s understood that they are always there. The regularzed form s then Σ p) = e 2 d 4 k 2π) 4 γ µ p + k + m )γ µ p + k) 2 m 2 µ 2 Λ 2 k 2 µ 2 )k 2 Λ 2 ). 2) The next step s to use Feynman parameters to combne three propagator factors and smplfy the numerator by the dentty γ µ aγ µ = 2 a, Σ p) = e 2 2 dxdydzδ x y z) d 4 k 2 p + k)+4m )µ 2 Λ 2 ) 2π) 4 [xp + k) 2 m 2 )+yk 2 µ 2 )+zk 2 Λ 2 )]. 2) 3 The denomnator can be smplfed usng x + y + z =to denom = [k 2 +2xk p + xp 2 xm 2 yµ2 zλ 2 ] 3 = [k + xp) 2 + x x)p 2 xm 2 yµ 2 zλ 2 ] 3. 22) By shftng the ntegraton varable k µ k µ xp µ, we fnd Σ p) = e 2 2 dxdydzδ x y z) d 4 k 2 x) p + k)+4m )µ 2 Λ 2 ) 2π) 4 [k 2 + x x)p 2 xm 2 yµ 2 zλ 2 ]. 23) 3 4

5 The k term n the numerator can be dropped because t s an odd functon of k µ and vanshes upon d 4 k ntegraton. The next step s to perform d 4 k ntegraton. We need to work out a general formula d 4 k 2π) 4 k 2 M 2 + ɛ) = 3 4π) 2 2 M 2 ɛ. 24) Here we recovered ɛ term because t s mportant. To show ths, we frst focus on k ntegraton. The trple) poles are located at k = ± k2 + M 2 ɛ and hence the pole wth + sgn s below the real axs, and that wth sgn s above the real axs. Snce the ntegrand goes to zero suffcently fast at the nfnty k on the complex k plane, the ntegraton contour can be rotated from k, ) tok, ) wthout httng the poles. Ths s called Wck rotaton. Then we can change the ntegraton varable k = k 4 such that the new varable ranges n, ). Then the above ntegral s rewrtten as d 4 k d 4 2π) 4 k 2 M 2 + ɛ) = k E 3 2π) 4 ke 2 M 2 + ɛ). 25) 3 Here, d 4 k = dk d k, d 4 k E = dk 4 d k,andke 2 =k 4 ) 2 + k 2 = k ) 2 + k 2 = k 2. Now that the denomnator s postve defnte no poles along the ntegraton contour), we do not have to worry about ɛ n performng ntegraton. We then use the polar coordnates for fourdmensonal k µ E and the ntegraton volume s d 4 k E =2π 2 ke 3 dk E = π 2 ke 2 dk2 E after ntegratng over three angle varables. In general, d n x = 2πn/2 Γn/2) xn dx.) Then the ntegral becomes qute smple, = π2 dk 2 2π) 2 E k2 E ke 2 + M 2 ɛ. 26) The ke 2 ntegraton can be done n parts and we fnd Eq. 24). Now gong back to Eq. 23), we can perform d 4 k ntegraton usng Eq. 24) and fnd Σ p) = e 2 dxdydzδ x y z) 4π) 2 2 x) p +4m )µ 2 Λ 2 ) xm 2 + yµ 2 + zλ 2 x x)p 2 ɛ. 27) The ntegraton over z can be done trvally usng the delta functon, x Σ p) = e 2 dx dy 2 x) p +4m )µ 2 Λ 2 ) 4π) 2 xm 2 + yµ 2 + x y)λ 2 x x)p 2 ɛ. 28) Note that the ntegraton regon of y s now lmted as [, x] because of the delta functon constrant x + y + z =andz>. The ntegraton over y s also a smple logarthm and we fnd Σ p) = e2 dx 2 x) p +4m 4π) 2 )log xm2 + x)λ 2 x x)p 2 ɛ xm 2 + x)µ 2 x x)p 2 ɛ. 29) 5

6 Even the ntegraton over x can be done wth elementary functons only, but the expresson becomes lengthy and not very nsprng and therefore we keep x ntegraton. Fnally usng α = e 2 /4π, Σ p) = α 4π dx 2 x) p +4m )log xm2 + x)λ 2 x x)p 2 ɛ xm 2 + x)µ 2 x x)p 2 ɛ. 3) Followng the general dscussons, we now dentfy δm and Z 2. Snce Σ p) and hence δm = m m are Oe 2 ), I can replace m n Σ p) bym by neglectng Oe 4 ) correctons. Frst, δm =Σ p) p=m s gven by δm = Σ p) p=m = α dx 2 x)m +4m)log xm2 + x)λ 2 x x)m 2 ɛ 4π xm 2 + x)µ 2 x x)m 2 ɛ = m α dx2 + x)log x2 m 2 + x)λ 2 ɛ 4π x 2 m 2 + x)µ 2 ɛ. 3) The argument of the logarthm s manfestly postve, and we can safely drop ɛ. Moreover, we take the lmt Λ and µ n the end, and we can neglect x 2 m 2 n the numerator and x)µ 2 n the denomnator. Then the expresson becomes drastcally smpler and the end result s δm = m α [ 3log Λ2 4π m + 3 ]. 32) 2 2 Ths s the correcton to the mass of the electron. The nterpretaton of ths self-energy s qute nterestng. In classcal electrodynamcs, we actually have a lnearly-dvergent self-energy. An electron creates a Coulomb feld around t, and t feels ts own Coulomb feld. If, for nstance, one magnes the electron to be a sphere of radus r e wth a unform charge densty, the total potental energy s V = 3 e 2 5 r e. The total energy of the electron s the sum of the rest energy m c 2 and the potental energy V and hence the total mass of the electron we observe s gven by mc 2 = m c ) 5 4πr e In the lmt of r e, the bare mass m needs to be sent negatve to cancel the lnearly dvergent Coulomb self-energy to obtan the observed mass of the electron. In the quantum mechancal language, cleary ths s an ultravolet dvergence as t corresponds to short- physcs r e. If we magne the electron to be as small as the Planck sze r e = dstance hg N /c 3 =.6 33 cm, where presumably the quantum gravty takes over physcs, we need to make the bare mass as negatve as MeV whch s cancelled by the self-energy for 2 dgts to get.5 MeV. Ths s absurd. The classcal electrodynamcs therefore breaks down at the dstance scale where the rest energy and the self-energy become comparable, r e e 2 /mc 2 3 cm, and a better deeper) theory needs to take over the classcal electrodynamcs below ths dstance scale. 6 e 2

7 What we have learnt here s that the stuaton n the QED s much better. The total mass of the electron s mc 2 = m c 2 + α [ 3log Λ2 4π m + 3 ]). 34) 2 2 The ultravolet cutoff Λ corresponds to the nverse sze of the electron n the classcal language. Agan f we magne the electron to be as small as the Planck sze, the correcton to the electron mass s only 8.%. The crucal dfference from the classcal theory s that ) the dependence on the sze of the electron s only logarthmc nstead of power, and 2) the correcton s proportonal to the electron mass tself and hence can never be much larger than the bare mass. Compared to the classcal case Eq. 33), the quantum result Eq. 34) corresponds to the cutoff r e h/mc whch s nothng but the Compton wave length. Below ths dstance scale, quantum effects are essental and the classcal theory does not apply any more. There are several lessons to be learnt from ths dscusson. Frst, the QED knows ts own lmtaton by the fact that t requres the ultravolet cutoff to regulate the theory. The theory does not apply beyond certan energy scale. Such a theory s called an effectve feld theory, and s true to almost all quantum feld theores. However, the cutoff can be extremely large unlke n the classcal electrodynamcs. Even though the QED should be regarded as a theory whch should be taken over by a yet deeper theory at extreme hgh energes, ts applcablty s practcally nfnte. Second, even though there s an ultravolet dvergence n the electron mass, what we observe s the total of the bare mass and the self energy. Therefore, any calculatons should be expressed n terms of the observed mass of the electron nstead of the bare mass. The same comment apples to the fne-structure constant as we wll see later. It turns out that any physcal quanttes are fnte once expressed n terms of the observed mass and fne-structure constants despte the fact that there are dependence on the ultravolet cutoff as well as bare parameters at the ntermedate stage of calculatons. Ths s a general property of renormalzable quantum feld theores: all physcal quanttes are fnte once expressed n terms of observable quanttes. Because of ths property, one can use the QED to do precse calculatons wthout worryng about what physcs s there at the energy scale of the ultravolet cutoff. Next topc s the wave-functon renormalzaton factor. Followng the general dscussons, we calculate Z2 = Σ p). 35) p p=m Usng Eq. 3) agan, and payng attenton to the fact that p 2 = p) 2, we fnd Z2 = α [ dx 2 x)log xm2 + x)λ 2 x x)p 2 ɛ 4π xm 2 + x)µ 2 x x)p 2 ɛ x x)2 p + 2 x) p +4m) xm 2 + x)λ 2 x x)p 2 ɛ 7

8 )] x x)2 p. 36) xm 2 + x)µ 2 x x)p 2 ɛ p=m We can drop the frst term n the parenthess because t vanshes n Λ lmt, and also m 2 µ 2 ) n the numerator denomnator) n the logarthm can be set zero. After a smple ntegraton, we fnd Z 2 = α 4π [ 2log m2 Λ2 log µ 2 m ]. 37) 8