A Complete Diagrammatic Implementation of the Kinoshita-Lee-Nauenberg Theorem at Next-to-Leading Order

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by  Abdullah Khalil

A Complete Diagrammatic Implementation of the
Kinoshita-Lee-Nauenberg Theorem at Next-to-Leading Order
Abdullah Khalil∗
Department of Physics, University of Cape Town,
Private Bag X3, Rondebosch 7701, South Africa and
Department of Physics, Cairo University, Giza 12613, Egypt
W. A. Horowitz†
Department of Physics, University of Cape Town,
Private Bag X3, Rondebosch 7701, South Africa
We show for the first time in over 50 years how to correctly apply the Kinoshita-Lee-Nauenberg
theorem diagrammatically in a next-to-leading order scattering process. We improve on previous
works by including all initial and final state soft radiative processes, including absorption and an
7 infinite sum of partially disconnected amplitudes. Crucially, we exploit the Monotone Convergence
1 Theorem to prove that our delicate rearrangement of this formally divergent series is correct. This
0 rearrangement yields a factorization of the infinite contribution from the initial state soft photons
2 that then cancels in the physically observable cross section. We derive the first complete next-to-
leading order, high-energy Rutherford elastic scattering cross section in the MS renormalization
n
scheme as an explicit example of our procedure.
a
J
PACSnumbers: 11.10.Jj,11.15.Bt,13.60.Fz,13.60.Hb
3
Keywords: Infrareddivergence,elasticscattering,Rutherfordscattering

]
h
t I. INTRODUCTION tipliedbyalogarithmofthescatteredparticle’smass;for
- asymptotically large energies, the finite coupling times
p
e Infinities are ubiquitous in perturbative quantum field the logarithm becomes larger than 1, and the perturba-
h theoretic derivations of cross sections beyond leading or- tive expansion is uncontrolled.
[ der. In order to make any beyond leading order re- 25 years after BN, Kinoshita [1] and Lee and Nauen-
1 sult sensible, one must render these infinities harmless. berg[2]provedthatthecollinearinfrareddivergencesdue
v Renormalizationeliminatesultraviolet(UV)divergences. to a massless radiating particle cancel when one sums
3 And the Kinoshita-Lee-Nauenberg (KLN) theorem [1, 2] over both degenerate final and, crucially, initial states.
6 assures that a summation over degenerate initial and fi- In their paper, Lee and Nauenberg (LN) included an ex-
7
nalstatesremovesallinfrared(IR)divergences. Weshow ample demonstrating the infrared finite cross section for
0
here for the first time a complete diagrammatic applica- elasticRutherfordscatteringcomputedatNLOwhenone
0
. tion of the KLN theorem at next-to-leading order accu- integrates over a degenerate final-state energy window
1 racy. and degenerate initial and final state angular windows.
0
Infrared divergences come in two flavors: soft, due to Since the publication of the KLN papers in the early
7
the massless nature of the radiation (e.g. the massless 1960’s, there has been confusion regarding the proper
1
: photon in QED), and collinear, which comes from treat- treatment of the degenerate, soft initial state radiation.
v
ingtheradiatingparticleasmassless(e.g.theelectronin LN correctly identified the need to include partially dis-
i
X QED). Even before quantum field theory existed, Bloch connected amplitudes in which a flying radiation does
r andNordsieck(BN)[3]proved,inmodernlanguage,that notinteractwithanyotherpartoftheamplitude. These
a thesoftinfrareddivergencesduetothemasslessradiated partiallydisconnectedamplitudescaninterferewithfully
photon cancel when one sums over the degenerate final connected amplitudes in which a radiation is both ab-
states of the system. To compute an elastic Rutherford sorbedandemittedbytheradiator,formingaconnected
scatteringcrosssectionforanelectrontoscatteroffanin- cut diagram. However, LN did not include degenerate
finitely massive particle at next-to-leading order (NLO), initial state soft photons in their treatment. Subsequent
invoking the BN theorem implies including the 1 → 2 works [4–8] summed over some, but not all, possible de-
process at leading order (LO) with the 1→1 process at generate initial and final states. Ito [9] and Akhoury,
NLO so long as the photon emitted in the 1→2 process Sotiropoulos,andZakharov[10]includedallpossibledis-
possesses an undetectably small energy E <∆. connected amplitudes, including an infinite series of dis-
γ
One is interested in the massless radiator limit of a connected radiations, but, as was pointed out in [6], the
theorybecauseatNLOandbeyond, thecouplingismul- IASZrearrangementoftheformallydivergentinfinitese-
ries yields a NLO result that is exactly 0 at tree level.
We report in this Letter a systematic treatment of
∗ Email: [email protected] Rutherford scattering at NLO in which we include all
† Email: [email protected] degenerate initial and final states. We show for the first
2
time how to rigorously perform the delicate rearrange-
ment of the resultant formally divergent series. The re-
sultisanIRsafeNLOcrosssectionwhosesmallcoupling k
limit is that from LO. Although an alternative approach p p(cid:48) p p(cid:48)
based on a coherent initial state exists [11–14], we find
that the initial state soft degeneracy fully factorizes and q q
cancelswhenusingtheusualparticlephysicswavepacket
initial state. Even though we only use our procedure in
(a) (b)
QED, we fully expect it to generalize to all other theo-
ries, thus explaining why all previous calculations that
use the usual particle physics wavepacket initial state k
k
that neglected the soft initial state radiation degener-
p p
acyhaveneverthelessyieldedcrosssectionsinagreement
with data. p(cid:48)−k p(cid:48)−k
q q
II. NLO RUTHERFORD
(c) (d)
Consider an electron with inital four momentum p =
k
(E,p(cid:126))thatscattersoffastaticpointcharge(equivalently k
an external potential, as in LN [2]) into a state with fi- p(cid:48) p(cid:48)
nal four momentum p(cid:48) =(E,p(cid:126)(cid:48)). Define the momentum
transfer q ≡ p(cid:48) −p. To regulate the various IR diver- p−k p−k
gences, we introduce a fictitious photon mass m and q q
γ
keep the electron mass m non-zero for now. To apply
e
the BN theorem to the NLO cross section we need to (e) (f)
sumtheLOFig.1(a)andNLOvertexcorrection(b)for
the 1→1 process as well as the indistinguishably soft fi- FIG. 1. The (a) leading order (LO) and (b) next-to-leading
order (NLO) vertex contributions to the 1 → 1 process in
nalstateradiativeprocesses(c)and(d). Summingthese
Rutherfordscattering. The(c)and(d)potentiallydegenerate
processes yields
final state radiative processes. The (e) and (f) potentially
degenerate initial state radiative processes.
(cid:26) α (cid:20) (cid:18)E2(cid:19)(cid:18) (cid:18)−q2(cid:19)(cid:19)
dσ =dσ 1+ e log 1−log
BN 0 π2 ∆2 m2
e
3 (cid:18)−q2(cid:19) (cid:21)(cid:27) toconsider thedegenerate softinitialphotons (whichin-
+ log +O(1) , (1)
2 m2 cludes photons that are both soft and collinear), Fig. 1
e
(e) and (f). Including the contributions from the soft
which is indeed free of soft IR divergences. (In the
photons in these two diagrams, however, introduces new
MSrenormalizationschemeusedhere,thelog(−q2/µ2 )
IR divergences, seemingly in contradiction of the KLN
MS
terms from the vertex correction and the wavefunction
theorem.
renormalization exactly cancel.)
What we will show is that the resolution is to include
Inordertoremovethe(potentiallylarge)collinearlogs,
all possible contributions from degenerate initial and fi-
log(−q2/m2),wemustinvoketheKLNtheoremandsum
e nalstates,includinginterferencetermsfrompartiallydis-
over the physically indistinguishable hard collinear final
connected amplitudes, i.e. amplitudes with disconnected
emission Fig. 1 (d) and initial absorption (e) processes.
sub-components. Fig. 2 (a) is the simplest partially dis-
(Soft collinear emission processes were already summed
connectedamplitude,andthispartiallydisconnectedam-
over in the application of the BN theorem.) Notice that
plitude yields a fully connected cut diagram when inter-
for the collinear divergence, requiring that the final elec-
fering with the fully connected amplitude (b) that has
tron three momentum p(cid:126)(cid:48) is at a physically distinguish-
both an absorption and emission process. We must even
able angle φ > δ from the incoming electron three mo-
include cut diagrams that have sub-components that are
mentum p(cid:126) means that these are the only two additional
notconnectedtoeithertheincomingoroutgoingelectron
diagrams we need to consider for indistinguishable hard
lines as shown in the generic cut diagram Fig. 2 (c).
collinear processes. Adding these additional two pro-
cesses to dσ yields a result in which one simply re- (Notethatinthesepartiallydisconnectedcutdiagrams
BN
placesm2withδ2E2,whichisfreeofallsoftandcollinear the incoming particle does scatter, p(cid:48) (cid:54)=p, and therefore
e
divergences and is the stopping point of the original LN these cut diagrams are not a part of the trivial 1 in the
paper [2]. S matrix.)
Consistency with the principle of summing over all in- Following [9, 10], we consider a general form of our
distinguishable initial and final states, though, leads one process with m incoming soft photons and n outgoing
3
possibilityofaphotonconnectingtotheelectronlinesas
k1 k2 m−i well as any number of disconnected photon lines; these
partially disconnected cut diagrams are given by one of
the four probabilities multiplied by a number of δ func-
p−k p(cid:48)−k
1 2 tions according to the number of disconnected photons
in the cut diagram.
q n−j
Define the function D(m−i,n−j) that describes the
i
number of m−i incoming and n−j outgoing soft pho-
(a) tons in P disconnected from the electron lines. It is
mn
straightforward to see that D(0,0) = 1 and D(a,b) = 0
j
k for a (cid:54)= b. One may show that [9, 10] the transition
2
k
1 probability at NLO of m incoming and n outgoing soft
p(cid:48)−k
2 photons is
p−k1 D(m,n) (cid:88) D(m−i,n−i−1)
q P = P + P
mn m!n! 00 (m−i)!(n−i−1)! 01
i=0
(b) (c) (cid:88) D(m−i−1,n−i)
+ P
(m−i−1)!(n−i)! 10
FIG. 2. A disconnected photon amplitude such as (a) might
i=0
interfere with an amplitude with a photon absorption and (cid:88) D(m−i−1,n−i−1)
emission process such as (b). (c) is a generic cut diagram + (m−i−1)!(n−i−1)!P(cid:101)11. (5)
withmincomingandnoutgoingsoftphotonsthatispartially i=0
disconnected. (Note that the blobs include the possibility of
connectingincomingphotonlinesfromtheleftwithconjugate Since D(0,0) = 1 for every m and n there is al-
incoming photon lines from the right.) ways a term unsuppressed by any factorials; hence P =
(cid:80)
P formally diverges.
m,n mn
In order to manipulate Eq. (5) in a controlled way, we
soft photons:
introduce a convergence factor that becomes small for
e−+mγ (soft) → e−+nγ (soft), (2) large i: we take
withanamplitudeM . Thenthe“transitionprobabil- D(m−i,n−j) → D(m−i,n−j)e−(i+j)/Λ (6)
mn
ity” for the process becomes
with Λ (cid:29) 1. Let us examine the partial sum of P up
mn
P = 1 1 (cid:88)|M |2, (3) totheemissionofN softphotons. SinceD(m,n)=0for
mn m!n! mn m(cid:54)=n, we may simplify our manipulations by replacing
i,f
the double sum over m and n with a single sum over n.
where P contains contributions from both fully con- With the above convergence factor, we are guaranteed
mn
nected and partially disconnected cut diagrams and is that P =lim lim P (Λ) converges.
Λ→∞ N→∞ N
summed over initial i and final f states. The total Lee- One may rearrange the partial sum of Eq. (5) up to N
Nauenberg probability will be soft photons to find
∞
(cid:88) N N n
P =m,n=0Pmn, (4) PN(Λ)= (cid:88) D((nn!,)2n) (cid:104)P00+e−Λ1P01(cid:105)+(cid:88)(cid:88)
n=0 n=1i=1
where the KLN theorem ensures that the quantity P is D(n−i,n−i) (cid:104) (cid:105)
free of soft or collinear IR divergences. [(n−i)!]2 e−2iΛ+1P01+e−2iΛ−1P10+e−2ΛiP(cid:101)11 ,
It is shown in [9, 10] that any cut diagram from P
mn (7)
at NLO can be constructed from four essential probabil-
ities: P00, the cut diagram with no photons in the initial As it must, the above equation yields the same result as
andfinalstates(whichmayincludetheleadingterm,the Eq. (5) as well as the rearranged series IASZ consider
vertexcorrection,thevacuumpolarization,etc.);P10and [9, 10], when all series are given a similar convergence
P01,whichincludesallcutdiagramswithonesoftphoton factor.
in the initial state or final state, respectively; and P(cid:101)11, When the order of limit is switched, however, our re-
which includes all fully connected cut diagrams with a sult is very different from IASZ. We will show that, like
single photon in both the initial state and final state. At IASZ, the disconnected contribution completely factors
NLO only one photon may attach to either the incom- out, allowing for an IR finite cross section; unlike IASZ,
ingoroutgoingelectronline, andthefullyconnectedcut crucially, ourresultstillhastheP term. Thereforeour
00
diagramsaregivenbyoneofthefourpreviousbasicprob- NLO expression correctly reduces to the LO expression
abilities. Partiallydisconnectedcutdiagramsincludethe as the coupling goes to 0.
4
Swapping limits in an infinite series is a delicate pro- TheLorentzinvariancebreakingfinalterminEq.(13)is
cedure. We are guaranteed from the Monotone Conver- fromthe“box”correctioninwhichtheelectroninteracts
gence Theorem that P (Λ) converges to the same result twice with the infinitely massive particle (recall that dσ
N 0
independent of the order of limits taken should our par- also, necessarily, breaks Lorentz invariance).
tial sum 1) monotonically increase in N for each Λ and
2) monotonically increase in Λ for each N [15].
Wewillfinditeasiertoprovethatourrearrangedsum
satisfiesthesetwopropertiesifweexploit[5]thefactthat III. CONCLUSIONS
P01+P10 =−P(cid:101)11, (8) A self-consistent application of the KLN theorem re-
quires a sum over all degenerate initial and final states
where the probability to emit a soft radiation given a
to arrive at an IR safe cross section. We derived the
scattering occurred is the usual
NLO correction to the Rutherford scattering cross sec-
P01 =e2|M0|2(cid:88)(cid:12)(cid:12)(cid:12)(cid:12)pp··(cid:15)kλ − pp(cid:48)(cid:48)··(cid:15)kλ(cid:12)(cid:12)(cid:12)(cid:12)2. (9) tinioitni,alinancldudfiinnaglstthaetefsu.lTl hsuismsmumatmioantioonveirsfaolrlmdaelglyendeirvaetre-
λ gent; after introducing a convergence factor, we proved
Then we have that our rearrangement of this summation allows one to
safely exchange taking the limit of the convergence fac-
PN(Λ)= (cid:88)N D((nn!,)2n) (cid:104)P00+e−Λ1P01(cid:105)+ ttohretcoonzveerrogepnrcioerfatoctothretoinzfienroit,ethsuerme wlimasita.cAomftperlettaekfiancg-
n=0 (10) torization of the disconnected, soft initial state radiation
(cid:88)N (cid:88)n D([n(n−−i,in)!]−2 i)2P01e−2Λi (cid:20)cosh(cid:18)Λ1(cid:19)−1(cid:21). afunldlyacocnonmecptleedtesocaftncineliltaiatilosntaotfe aralldioatthioern peffaretcitasl.lyCaonnd-
n=1i=1 sistent with intuitive reasoning, the infinite factor from
Since P , P , Λ, and D(n,n) are all strictly positive, thesoftinitialstateradiationwillcancelinthephysically
00 01
Eq.(10)clearlyincreasesmonotonicallyinN forfixedΛ. observable cross section.
To show that PN(Λ) increases monotonically in Λ for Wethereforearrivedattheextremelynontrivialresult
fixed N, we take the derivative with respect to Λ: that for NLO Rutherford scattering the summation over
all indistinguishable initial and final states is equivalent
N (cid:20) (cid:21)
dPdNΛ(Λ) = (cid:88) D((nn!,)2n) Λ12 e−Λ1 P01 +O(cid:0)Λ13(cid:1). (11) tfionathlesosfutm, hmaardtiocnololivneeraor,nalyndthseoifntiatinadlhcaorlldinceoalrlindeeagrenaenrd-
n=0
ate states.
Although one finds that the higher order in 1/Λ correc-
OurresulttriviallyextendstoallNLOQEDandQCD
tion term is negative, for any N we can find a Λ large
processes since the crucial IASZ [9, 10] insight to write
enoughsuchthatthefirstterm,whichisstrictlypositive,
the sum of all soft contributions in terms of four ba-
dominates.
sic building blocks, Eq. (5), will hold for any QED or
We have thus proved that we may exchange limits for
QCD process at NLO accuracy, and the critical identity,
our rearranged formula Eq. (10), and we may evaluate
Eq. (8), for the factorization, Eq. (12), after the rear-
the Λ→∞ limit first, yielding our main result:
rangement, Eq. (10), also holds for both QED and QCD
∞
(cid:88) D(n,n) [5]. Presumably our work can be further generalized to
P =(P +P ) . (12)
00 01 (n!)2 all theories and orders.
n=0
As was noted by Weinberg [17], “no one has given a
We find that all the soft initial state physics of the
completedemonstrationthatthesumsoftransitionrates
infinite number of undetectably soft photons completely
that are free of infrared divergences are the only ones
factorizes. When the cross section is computed, one sim-
that are experimentally measurable.” We believe that
plydividesoutbythisunobservedinfinity. Wehavethus
our work here can be expanded in the future to provide
rendered all soft and collinear IR divergences harmless.
such a proof.
The final fully IR safe NLO Rutherford cross section
including vacuum polarization [16] is
(cid:26) α (cid:20) (cid:18)E2(cid:19)(cid:18) (cid:18) −q2 (cid:19)(cid:19)
dσ =dσ 1+ e log 1−log ACKNOWLEDGEMENTS
NLO 0 π2 ∆2 δ2E2
3 (cid:18) −q2 (cid:19) 2 (cid:32)−q2(cid:33) 28(cid:35)(cid:41)
The authors thank the South African National Re-
+ log + log −
2 δ2E2 3 µ2 9 search Foundation and the SA-CERN consortium for
MS
πα3E (cid:18) 1 1 (cid:19) their support. AK also thanks the African Institute
+ e − . (13) for Mathematical Sciences (AIMS) and the University of
q2 |(cid:126)q| |p(cid:126)|
Cape Town for their support.
5
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