Towards an Effective Field Theory on the Light-Shell

0.21 MB·

by  Howard Georgi

Towards an Effective Field Theory on the Light-Shell
Howard Georgi,1 Greg Kestin,2 Aqil Sajjad,3
Center for the Fundamental Laws of Nature
Jefferson Physical Laboratory
Harvard University
Cambridge, MA 02138
4 Abstract
1
0 Wediscussourworktowardtheconstructionofalight-shelleffectivetheory(LSET),aneffectivefieldtheory
2 for describing the matter emerging from high-energy collisions and the accompanying radiation. We work in

n thehighlysimplifiedvenueof0-flavorscalarquantumelectrodynamics, withagaugeinvariantproductofscalar
a fields at the origin of space-time as the source of high-energy charged particles. Working in this simple gauge
J theory allows ustofocus on theessential features of LSET. Wedescribe howtheeffectivetheory isconstructed

9 and argue that it can reproduce the full theory tree-level amplitude. We study the 1-loop radiative corrections
2 in theLSET and suggest how theleading double-logs in thefull theory at 1-loop order can bereproduced by a
purely angular integral in theLSET.

]
h
p
-
p
e
h
[

1
v
7
6
6
7
.
1
0
4
1
:
v
i
X
r
a
[email protected]
[email protected]
[email protected]
1
1 Introduction
Previously,[1] we expressedhope of constructing an effective field theory on the 2-dimensionallight-shell emerging
fromahigh-energycollision. Theideawasmotivatedbyaclassicalpictureofaveryhighenergyhadroniccollisionin
whichcoloredparticlesareproducedfromaninitiallycolor-neutralstateatt=r=0andinstantaneouslyaccelerate
outward to the speed of light. At the same time, the collision event produces a pulse of color radiation that also
movesoutatthespeedoflight. So,classically,intheveryhighenergylimit,everythingliesonanexpandingsphere
at t = r, which we call the light-shell. In section 2 the classical calculation is discussed in more detail. There we
observe that to have the vector potential Aµ confined to the light-shell requires a special gauge - v Aµ =0, where
µ
vµ is a light-like vector pointing away from the origin. We call this light-shell gauge (LSG).
Inthispaper,webegintheexplicitconstructionoftheeffectivefieldtheory,incorporatingtheintuitiongainedfrom
the classical picture by studying the quantum mechanics of particle production from a gauge invariant source at
the originof space-time. We will see how a gaugeinvariantproduct ofscalarfields atthe originof space-time gives
rise to an effective field theory of the high energy physics that depends only on the angles of the momenta of the
high energy particles and fields. This 2-dimensionaleffective theory is our light-shell effective theory (LSET). Here
we present it in the simplified venue of 0-flavor scalar quantum electrodynamics (sQED). This strips away most of
the physics so that we can focus on the construction of the LSET.
The reader might wonder why we are investing so much effort in a new effective theory so novel that we have to
restrict ourselves to studying it in a toy model when there is a well-developed theory, SCET [2, 3] that is already
being successfully applied to QCD processes at high energies [4]. The real answer, of course, is that we find it
fascinating because our approach is very different. So, while we aim eventually to generalize our construction to
QCD, evenbefore then we hope thatinsightcanbe takenfromthe picture we beginto describe here. For example,
an interesting feature we will discuss is that all our calculations can all be reduced to purely angular integrals.
In the classical picture, the starting point is a very high energy event in which charged particles are produced
at the origin. Appropriately translating this classical setup into quantum field theory suggests a gauge invariant
source at t = r = 0. All of the physics in the effective theory will come from such a source term in our LSET
Lagrangian. Since LSG is undefined at r = 0, we construct the source on a small sphere around the origin whose
size we eventually shrinkto zero. Inthis introduction,we willalsobe ignoringthe initialstate andfocusingjust on
the physicscomingfromoursource. Inleading order,the formofthe sourceis fixedby gaugeinvariance;insection
4 we obtain the explicit form of the source.
The EFT requires that we set an energy scale to define what we mean by “high energy”. In the spirit of HQET
E
[5] (for a recent and comprehensive review see [6]) we scale out the large momenta associated with the energetic
outgoing particles. The associated decomposition into fields above and below we call the large radial energy
E
(LRE) expansion,andthe relatedfields aretermed LRE fields. The LRE fields correspondto high-energyparticles
produced by the source carrying large energy E > outwards from t = r = 0 into the bulk space. We will see
E
that to leading order in 1/E, the direction rˆof propagationaway from the origin is a classical variable and we can
label the LRE fields by rˆ. But in the presence of interactions, the directions of the LRE fields cannot be specified
precisely. So to each charged LRE field we assign an “angular size”.
To obtain the LSET Lagrangian, we apply the LRE expansion to sQED, and expand in orders of 1/E. We note
that the gauge interactions at leading order (in 1/E) are proportional to vµA . So choosing LSG eliminates the
µ
gauge interactionsof the LRE fields, showing that these interactionsare just gaugeartifacts, andthat as expected,
all the physics at high energies is occurring at the origin and depends only on angles. This suggests that a source
at the origin is sufficient to describe the physics in LSET. Because different configurations of LRE fields (different
energies and directions) do not interfere, each such configuration is associated with its own sector, and the source
in the EFT is a sum over all such sectors, separated by superselection rules.
As with any EFT, one must match to the full theory. In section 5 the LRE fields are matched to the full theory
fields. Then in section6.1 we confirmthe theory’s structure by showingthat the source in the EFT reproducesthe
tree-level amplitude for the source in the full theory in the appropriate limit.
Finally, we calculate the LSET 1-loop correction for production of a scalar/anti-scalar pair. The relevant photon
propagator is the LSG bulk propagator in the limit that the end points go to the light shell. We will show that
for physically sensible relations between the cut-offs and , the effective field theory can reproduce the leading
E
2
double-logs of 0-flavor sQED.
2 Background
LSETis motivatedby asimpleclassicalelectrodynamicscalculation. ConsiderNchargedparticlescreatedat~r=0
and t=0, whichthen instantaneouslyaccelerateto the speedoflightwith velocities crˆ . The resulting electric and
j
magnetic fields are zeroeverywherein spaceexcept onan infinitesimally thin sphericalshell thatexpands out from
the origin at light speed [1].
The starting point for our quantum mechanical version is a gauge invariant source at the origin of space-time that
produces charged particles moving away from the origin. Gauge invariance of the source requires that the total
charge vanishes,
q =0 (1)
j
j
X
Because we consider a point source at the origin, our source does not conserve energy and momentum and can
produce final states with any physical energy and momentum. We focus on the final states with some large energy
of order E.
The classical observation motivates the picture of LSET, while also motivating the gauge in which we will work.
If one makes an appropriate gauge transformation, not only will the electric and magnetic fields lie on a spherical
shell, butthe potentialwillaswell. Forthe generalchargeconfigurationdescribedabove,inthe appropriategauge,
our potentials are
N
A0(t,~r)= q δ(t r) log(t+rˆ ~r) (2)
j j
− ·
j
X
N
A~(t,~r)= rˆ q δ(t r) log(t+rˆ ~r) (3)
j j
− ·
j
X
Thepotentialsareproportionaltoaδ-functionthatsetst=r,sotheylieonthe“light-shell”. Animportantfeature
is that we can write A~(t,~r)=rˆA0(t,~r) satisfying the constraint
v Aµ =0 where v0 =1 and ~v =rˆ (4)
µ
The vector vµ is a light-like vector that varies with position. (4) defines light-shell gauge (LSG), an essential
component of the LSET effective theory.
Moving to a theory involving quantum effects, we discuss the formulation in the simplest theory, 0-flavor scalar
quantum electrodynamics. In the subsequent sections, we will combine sQED with new ingredients in order to
constructthetheeffectivetheory. Beforedoingso,wewritedowntheimportantpiecesofthefulltheoryLagrangian4
in order to refer to it later on as we mix in these new elements.
1
= ∂µφ2+ e2AµφAµφ∗ ieAµ((∂µφ)φ∗+(∂µφ∗)φ) F2 (5)
Lfull | | − −4 µν
Lφ (cid:0) Lint (cid:1) LA
| {z } | {z }
and we are interested in the functional integral | {z }
S eiRLfulld4x[dφ][dA] eiRLfulld4x[dφ][dA] (6)
full
Z (cid:30)Z
Where S is whatever gauge invariant source at the origin we wish to consider. The source describes the fields
full
which are produced at the origin; so for example if we wish to consider the creation of a scalar/anti-scalar pair,
then S =φ∗(0)φ(0).
full
The restriction to “0-flavor” allows us to ignore matter loops and (at least at one loop) the scalar self-couplings
(for details see Appendix A) in our radiative corrections, which makes the physics simpler and allows us to focus
4Wedonotwritethescalarself-interactionsbecausetheyplaynoroleinthelimitweconsider.
3
narrowly on the construction of the effective field theory. Similarly, the restriction to “scalar” QED allows us to
simplify the discussion of the matter in the theory. So, in theories of this kind, the only important physics is
associated with the source and the radiation generated by it, and we can focus on the construction of the novel
effective theory. The details of the source in LSET will be described in section 4.
3 Constructing the light-shell effective theory Lagrangian
In this section we construct the terms of the LSET Lagrangian that correspond to the terms in equation (5), ,
φ
L
, ,andthesource. Aswithanyeffectivetheory,thesoftphysicsisleftunchanged,sowefocusonthephysics
int A
L L
associated with hard particles, and we distinguish the part of the LSET Lagrangian involving hard particles by
referringto itas . This constructioninvolvestwo new andessentialingredients: a field decompositionwe will
LRE
L
refer to as a large radial energy (LRE) expansion and light-shell gauge.
The large radial energy expansion is reminiscent of the field decomposition of HQET [5] and LEET (the precursor
of SCET [7] that sums soft logs but not collinear logs). We scale out the uninteresting large momenta associated
with the energetic particles, but as its name suggests the LRE expansion involves scaling out by a spherical wave.
In order to do this, we set an energy scale , that determines which fields are large radial energy fields Φ(∗), and
E E
which fields are soft φ . The decomposition is
s
e−iE(t−r) eiE(t−r)
φ=φ + Φ + Φ∗
s √2E E,+q √2E E,−q
EX>E(cid:18) (cid:19) (7)
e−iE(t−r) eiE(t−r)
φ∗ =φ∗+ Φ + Φ∗
s √2E E,−q √2E E,+q
E>E(cid:18) (cid:19)
X
where Φ ( Φ∗ ) annihilates (creates) high energy outgoing scalars with charge q. In the following, we will
E,±q E,±q ±
focus on the particles with charge +q and drop the q subscripts to simplify the tableaux. As usual in such an
±
effective field theory decomposition, the x dependence of the EFT field is assumed to be slow compared to the t
and r dependence of the exponential factor eiE(t−r), and derivatives of Φ are assumed to be small compared to
E
E
in the effective theory.5 The 1/√2E is a normalization, the reason for which will soon be apparent.
Applyingthisexpansion,theLSETLagrangiancanbewrittenasanexpansioninthesmallparameter(1/E),where
E is the energy scaledout of the energetic field at hand. Let’s begin to look at by examining , the kinetic
LRE φ
L L
energyofour matter field, to leadingorderin1/E. Usingour expansion(7), focusing onthe LRE terms,andusing
∂(t r)
− =vµ (8)
∂x
µ
we get
1
(Dµφ)∗D φ (Dµ+iEvµ)Φ∗ (Dµ iEvµ)Φ (9)
µ → 2E E − E
(cid:16) (cid:17)(cid:16) (cid:17)
Thecrosstermsareleadinginthe1/Eexpansion,andhaveafactorofE fromthederivativesactingonthespherical
wave, which cancels the normalization from (7), giving
1
=iΦ∗ ∂ +(rˆ ~ + ~ rˆ)/2 Φ + Φ∗ L˜2Φ (10)
E t ·∇ ∇· E 2Er2 E E
(cid:16) (cid:17)
where the L˜2 = r2(~T iqA~ ) (~ iqA~ ) and we omit terms that vanish by the zeroth-order equations of
motion. While th−e L˜2∇t⊥er−m is o⊥f o·rd∇er⊥1−/E, it⊥also has rapid r dependence as r 0, which we do not want. We
→
can make the following field redefinition to eliminate it:
L˜2
Φ˜ (x) exp i Φ (x) (11)
E E
≡ " 2Er#
5Thisisabittrickierthanitsounds. See[8].
4
NotethatsincethederivativesinL˜2areallcovariant,Φ˜ (x)transformsjustlikeΦ (x)undergaugetransformations.
E E
In terms of Φ˜ (x), and ignoring interaction terms, the kinetic energy becomes
E
=iΦ˜∗ (∂ vµ+vµ∂ )/2 Φ˜ =iΦ˜∗ ∂ +(rˆ ~ + ~ rˆ)/2 Φ˜ (12)
Lφ E µ µ E E t ·∇ ∇· E
(cid:16) (cid:17) (cid:16) (cid:17)
The kinetic energy term (12) looks very much like the corresponding terms in HQET [5] and LEET [7], but there
the analog of the vector vµ is a constant, time-like in HQET and light-like in LEET. The fact that vµ varies with
rˆis responsible for unique properties of the LRE expansion. For example, the LRE decomposition (7) is invariant
under rotations about the origin, not just covariant like HQET or LEET.
The Φ˜ propagator associated with the kinetic energy term (12) is directional and has the form6
E
1
0 TΦ˜ (x)Φ˜∗(x′) 0 = θ(t t′)δ(t r t′+r′)δ(rˆ rˆ′) (13)
E E rr′ − − − −
D (cid:12) (cid:12) E
Onecancheck(13)easilyand(cid:12)itcanbeformall(cid:12)yderivedusingcanonicalquantization,asweshowinappendixB.The
(cid:12) (cid:12)
propagator (13) describes radially outgoing particles and this form establishes the connection between the spatial
direction ofthe coordinatex and the directionofpropagationofthe particle, whichdetermines the directionofthe
momentum of the LRE particle far away from the source. This connection between position space and momentum
space for the high-energy particles is one of the crucial components of our construction. We will return to this and
seetheconnectionveryexplicitlyinsection5. Butwhiletheconnectionisexactinthefreetheory,wewouldexpect
that quantumeffects make itimpossible to specify the momentumdirectionprecisely. This expectationis reifiedin
the calculationofquantum loopswhere specifying the directionsprecisely leads to divergences[9]. We assumethat
this is associatedwith the physicalimpossibility ofmeasuring a jet directionexactly. Thus we associateanangular
size with each LRE particle quantifying the uncertainty in direction.
We now return to equation (9) to explore the consequences of the LRE expansionfor . It can be written in the
int
L
suggestive form
i 1
( ∂µΦ† )v Φ +vµΦ† ∂ Φ + v AµΦ† Φ + (DµΦ )†D Φ (14)
2 − E µ E E µ E µ E E 2E E µ E
h i
In this form, it is clear that in LSG our interactions vanish atleading order. The removalof the gauge interactions
with LREscalarssimplifies calculations,anditmakesitclearthatthe essentialphysicsofthe high-energyparticles
is associated with the source at the origin. This is consistent with the expectation of a purely angular theory on
the light-shell.
Lastly, consider the purely gauge part of the Lagrangian. In LSG, becomes7
A
L
LA =−14Fµ2ν =−21(cid:18)Ar,A~T⊥(cid:19) (cid:18)(∂t+∇~∇~(∂·trˆ+)(∂rˆt·+∇~)rˆ·∇~) (∂∇~t+∇~T∇~+·rˆ✷)∇~IT(cid:19) (cid:18)AA~⊥r(cid:19) (15)
Here we have used A and A~ , defined by
r ⊥
A =rˆ A~ (16)
r
·
A~ =A~ rˆ A~rˆ (17)
⊥
− ·
Keep in mind that in LSG the temporal component of Aµ is equal to A . An LRE expansion, similar to that of
r
the scalars, holds for the gauge field. This expansion is in terms of longitudinal and perpendicular components of
the gauge field, which is appropriate in light-shell gauge. Again, the rescaling of each field is determined by the
canonical form of the kinetic energy term.
1 1
A~ =A~ + eiE(t−r) ~∗ + e−iE(t−r) ~ +eiE(t−r) ∗ rˆ+e−iE(t−r) rˆ+ (18)
s √2E AE⊥ √2E AE⊥ AEr AEr ···
E>E(cid:18) (cid:19)
X
After applying this expansion to (15) and considering the LRE terms, we can redefine the gauge field as
~AEr = 01 (∂t+Rˆ·∇~)1−1∇~T/√2E AA~Er (19)
(cid:18)AE⊥(cid:19) (cid:18) (cid:19)(cid:18) E⊥(cid:19)
6When rˆappears as an argument, it refers to dependence on angles θ and φ. Likewise rˆj refers to the angles θj and φj. So, here
δ(rˆ−rˆ′)isequaltoδ(z−z′)δ(φ−φ′),withz=cos(θ).
7Thisformisnotcompletelyobvious,atleasttous. Detailsofthederivationcanbefoundin[10].
5
Where (∂ +~ Rˆ)−1 is the inverseof a differential operatorthat is non-localin space and time and givenexplicitly
t
∇·
by
1
(∂ + ~ Rˆ)−1(x′,x)= θ(t′ t)δ(t′ r′ t+r)δ(rˆ′ rˆ) (20)
t ∇· (r′)2 − − − −
Operatorsof this sortappear frequently in our LSET analysis(though we will sparethe readerof this introduction
most of the gory details — these will appear in [10] and [9]). These operators are treated on the same footing as
linear operators, so for example, the first row of (19) could be written more explicitly as
(x)=A (x)+ (∂ +Rˆ ~)−1(x,x′)~′T/√2EA~ (x′)d4x′ (21)
Er Er t E⊥
A ·∇ ∇
Z (cid:16) (cid:17)
In section (6.1) we will see that the quanta of the ~ field can be directly related to those of the full theory. Also,
⊥
A
this allows us to write the LRE photon kinetic energy in the following diagonal form.
LA,LRE = A∗Er A~∗E⊥ (cid:18)−(∂t+∇~ ·rˆ0)(∂t+rˆ·∇~) (∂t+rˆ·∇~/02+∇~ ·rˆ/2)(cid:19)(cid:18)A~AEE⊥r(cid:19) (22)
(cid:0) (cid:1)
The final piece of the LSET Lagrangianis the source. This is where the physics of our theory lies, and we describe
it in the following section.
4 LSET source
So far we have constructed the LSET Lagrangian by bringing the large radial energy expansion and light-shell
gauge to the full theory. In doing so we have removed all of the interactions of the LRE particles except for those
directly associated with the point source at the origin in the full theory. The full-theory source is proportional to
a gauge invariantproduct of localfields at the origin. Thus we also expect the correspondingsource in the EFT to
be gauge invariant. The conventions for the gauge transformations of our fields are listed in appendix C.
Whilethefull-theorysourceisattheorigin,light-shellgaugeisill-definedthere,sowebeginbyconsideringasource
in the EFT that is “spread out” about the origin. We also expect from our classical picture that as the energy in
the event goes to infinity, all of the physics goes onto the light shell, at t = r. Thus in our quantum version, we
spread out our source onto a surface r =s surrounding the origin, near the light-shell, with t r 0 as E .
− → →∞
To understand the symmetry of the spread-out source, it is convenient to write ϕ(x)=ϕ(t,r,rˆ) and to let
ϕ(r,r,rˆ) ϕ(x) (23)
≡ |t=r
represent either an LRE field or a soft field on the light shell. When we eventually write down the full source,
the LRE fields will be evaluated at particular values of rˆ, while the soft fields will be integrated over rˆ. But this
notation will allow us to focus on the symmetries for both types of fields simultaneously. In this notation, a term
in the source spread out over S appears as
ϕ†(s,s,rˆ ) (24)
O∝ j j
j
Y
This is not gauge invariant, but transforms as
exp iq Λ(x ) (25)
O→O − j j |tj=rj=s
Yj h i
To maintain gauge invariance we construct a compensating exponential on the light-shell
e
exp i ℓ(rˆ,rˆ ) ∂ Aµ(x)dS (26)
j µ
 2π   
Z j
X
   
where dS is our Lorentz covariant surface element on the small sphere.
dS =θ(t)rδ(r s)δ(r2 t2)d4x (27)
− −
6
and assuming zero net charge
ℓ(rˆ,r )=q log(1 rˆ rˆ) (28)
j j j
− ·
Putting all the pieces together, our gauge invariant source on the light-shell, call it , is of the form
S
m+n
e
C lim exp i ℓ(rˆ,rˆ ) ∂ Aµ(x)dS
j µ
s→0  2π   
Z Z j=1
X (29)
   
m m+n m+n
r−1Φ† (x ) r−2φ(†)(x ) dS
 j j,Ej j  j j j  j
j=1 j=m+1 j=1
Y Y Y
   
where there are n soft scalarsφ(†), m LRE scalarsΦ, and dS refers to dS with xµ xµ. Also, eachLRE scalar Φ
j → j
will have an energy associatedwith it E , this is the energy scaled out by the LRE expansion. Notice that there is
i
a constant C, which must be determined by matching.
Assuming gauge transformations on the light-shell, our compensating exponential is unique. Also, one cannot help
but notice the connection to the classical potential (3).
5 Matching LRE fields
The simplest non-trivial matching to consider is that of LRE scalars. For this, we will match the amplitude of a
source creating a one-particle state in the full theory to the corresponding amplitude in the effective theory. Of
course, for this source to be gauge invariant, the particle must be neutral. This allows us to focus on the LRE
matchingallby itself. Intheprocess,wewilldefine creation/annihilationoperatorsinthe EFTbyrelatingthemto
the familiar creation/annihilation operators in the full theory. This construction can then be carried over trivially
to interesting sources involving charged particles. Let the matching condition be
~k FullSource0 ma=tch ~k EFTSource0 (30)
h | | i h | | i
~k is a one particle state for a scalar with momentum kµ = (k,~k) as defined in the full theory. This matching
h |
will connect the position space of the effective theory to the momentum space of the full theory, as well as fix the
coefficient of the effective theory source. The full theory source is just φ(0). The EFT source for a high-energy
particle, to leading order, has the form
c dΩ s(rˆ )Φ† (s(rˆ ),s(rˆ ),z ,φ ) (31)
1 1 1 1,E1 1 1 1 1
Z
where c is the coefficient we will determine herein. The matching condition is then
1
~k φ(0)0 ma=tch ~k c dΩ s(rˆ )Φ† (s(rˆ ),s(rˆ ),z ,φ ) 0 (32)
h | | i 1 1 1 1,E1 1 1 1 1
(cid:28) (cid:12) Z (cid:12) (cid:29)
(cid:12) (cid:12)
(cid:12) (cid:12)
The LHS is 1, and the RHS of (32) is (cid:12) (cid:12)
0 √2ka dΩ c s(rˆ )Φ† (s(rˆ ),s(rˆ ),z ,φ ) 0 (33)
k 1 1 1 1,E1 1 1 1 1
(cid:28) (cid:12) Z (cid:12) (cid:29)
(cid:12) (cid:12)
Making the commutation of opera(cid:12)tors involved above well defined requires a few(cid:12) steps. First, define a full theory
(cid:12) (cid:12)
annihilation (a ) operator in spherical coordinates by relating it to a standard full theory operator. The familiar
s
commutation relation is
a ,a† =(2π)3δ(3)(p~ p~′) (34)
p p′ −
h i
which can be expressed in spherical coordinates as
(2π)3
a ,a† = δ(p p′)δ(z z′)δ(φ φ′) (35)
p p′ p2 − − −
h i
7
The spherical creation (a ) and annihilation (a†) operators we define by
s s
(2π)2
a ,a† a (p,z,φ),a†(p′,z′,φ′) (36)
p p′ ≡ pp′ s s
h i h i
Notice that we have
p p′ p′ p
a a† a† a =a (p,z,φ)a†(p′,z′,φ′) a†(p′,z′,φ′)a (p,z,φ) (37)
2π p 2π p′ − 2π p′ 2π p s s − s s
(cid:16) (cid:17)(cid:18) (cid:19) (cid:18) (cid:19)(cid:16) (cid:17)
so the relations between conventional and spherical creation and annihilation operators are
p
a (p,z,φ)= a (38)
s p
2π
p
a†(p,z,φ)= a† (39)
s 2π p
In the EFT we can write our fields in terms of creation/annihilationoperators as
1 dk
Φ†(t,r,rˆ)= eik(t−r) a† (k,rˆ) (40)
r LRE 2π
Z
which is described in detail in appendix B. Using the above two relations, (33) becomes
2π dk
= dΩ 0 √2k a (k,kˆ)c a† (k ,rˆ ) 1 0 (41)
1 k s 1 LRE 1 1 2π
Z (cid:28) (cid:12) Z (cid:12) (cid:29)
(cid:12) (cid:12)
The final and crucial step is to notice tha(cid:12)t the commutation relations of a and(cid:12)a (in appendix B) look the
(cid:12) s (cid:12) LRE
same,buthavetwoimportantdifferences: a involvesanglesinmomentumspaceandtheenergyinvolvedisthefull
s
energy k, whereas a involves angles in position space and the residual momentum k. So we identify the angles
LRE
in momentum space and position space and set
a†(k +k,kˆ)=a† (k ,rˆ) (42)
s 1 LRE 1
This allows us to turn our a† into a†. We find that the RHS of (32) becomes
LRE s
2
=2π c (43)
1
k
r
1 k
So, we have c = and our full theory scalars relate to our LRE scalars as
1
2π 2
r
1 k
φ(0)= dΩ s(rˆ )Φ† (s(rˆ ),s(rˆ ),rˆ ) (44)
2π 2 1 1 1,E1 1 1 1
r Z
c is the contribution from one LRE scalar to C in our general source (29), but we will have contributions from
1
matching the other LRE fields involved in the process as well. While this matching procedure is fairly simple, it
is essential for connecting the objects in LSET, which are formulated in position space, to the momentum-space
amplitudes one is accustomed to calculating in the full theory. Relations for LRE photons operators,analogous to
those introduced here, will be described in the following section.
6 Reproducing full-theory results
6.1 Tree-level
We are now preparedto compare amplitudes in the full theory and effective theory. A relevant process to compare
is one with the final state of an energetic photon, scalar (labelled by ‘-’), and anti-scalar(labelled by ‘+’). For this
comparison we will focus on the transverse component. In the full-theory, we have
e pˆ kˆ(pˆ kˆ) pˆ kˆ(pˆ kˆ)
~k ~p p~ φ∗(0)φ(0)0 = −− −· +− +· (45)
h − +| | i |k| 1−kˆ·pˆ− − 1−kˆ·pˆ+ !
8
On the LHS above,~k refers to a transverse final photon state with momentum ~k. In the effective theory we want
to do the calculationwith the same final state, but we now use our EFT source ~k p~ p~ 0 . The dependence on
− +
h |S| i
scalar factors disappears. After integrating by parts and making use of the rescaling for LRE photons, we get
ie 1
~k − ∗ (x)(∂ +rˆ ~)+ ~∗ (x) ~ ℓ(rˆ,rˆ ) dS 0 (46)
* (cid:12)(cid:12) 2π Z (cid:18)AEr t ·∇ √2E AE⊥ ·∇(cid:19)j=+,− j  (cid:12)(cid:12) +
(cid:12) X (cid:12)
(cid:12)   (cid:12)
Note that the expone(cid:12)(cid:12)ntialassociatedwiththe LREexpansionhasgone awaybecause ofthe δ((cid:12)(cid:12)r2 t2) indS. Using
−
the transformation that diagonalizes the kinetic energy (19) gives
ie
= ~k − δ(x x′) ∗ (x′) ~∗ (x′) ~′(∂ + ~ Rˆ)−1(x′,x)/√2E (∂ +rˆ ~)
* (cid:12) 2π − AEr −AE⊥ ·∇ t ∇· t ·∇
(cid:12) Z (cid:16) (cid:17) (47)
(cid:12)
(cid:12)
(cid:12) +δ(x x′) 1 ~∗ (x′) ~′ ℓ(rˆ,rˆ ) dx′dS 0
− √2E AE⊥ ·∇!j=+,− j  (cid:12)(cid:12) +
X (cid:12)
  (cid:12)
Where (∂t + ~ Rˆ)−1 is given in (20) and ~′ involves derivatives with respect (cid:12)(cid:12)to x′. Since the final physical
∇· ∇
photon state is transverse, and the relevant propagator is diagonal, we can remove the term involving ∗ . Then
AEr
simplifying and manipulating our differential operator gives
ie 2
= ~k − ~∗ (x′) ~′(∂ + ~ Rˆ)−1(x′,x)/√2E ∂ + ~ rˆ
* (cid:12)(cid:12) 2π Z (cid:16)−AE⊥ ·∇ t ∇· (cid:17)(cid:18)(cid:16) t ∇· (cid:17)− r(cid:19) (48)
(cid:12)
(cid:12)
(cid:12) +δ(x x′) 1 ~∗ (x′) ~′ ℓ(rˆ, rˆ ) dx′dS 0
− √2E AE⊥ ·∇!j=+,− { j}  (cid:12)(cid:12) +
X (cid:12)
  (cid:12)
(cid:12)
ie 1 (cid:12)
= ~k − ~∗ (x′) ~′(∂ + ~ Rˆ)−1(x′,x) ℓ(rˆ, rˆ ) dx′dS 0 (49)
* (cid:12)(cid:12) π Z (cid:18)AE⊥ ·∇ t ∇· √2Er(cid:19)j=+,− { j}  (cid:12)(cid:12) +
(cid:12) X (cid:12)
(cid:12)   (cid:12)
Now, as in (40) for(cid:12)scalars,can write our transverse photon field as (cid:12)
(cid:12) (cid:12)
~∗ (t′,r′,rˆ′)= eik(t′−r′) 1 ~† (k,rˆ′) dk (50)
AE⊥ r′ aE⊥ 2π
Z
Using this in (49) gives
= 0 2k ~a −ie eik(t′−r′) 1~† ~′ (∂ + ~ Rˆ)−1(x′,x)1 ℓ(rˆ, rˆ ) dk dx′dS 0 (51)
* (cid:12)(cid:12) | | k⊥ π√2E Z (cid:18) r′aE⊥·∇ t ∇· r(cid:19)j=+,− { j} 2π (cid:12)(cid:12) +
(cid:12)p X (cid:12)
(cid:12)   (cid:12)
Again, f(cid:12)or the gauge fields’ creation/annihilation operators, we can use relations analogous to those intro(cid:12)duced in
(cid:12) (cid:12)
the previous section for scalars,
p
~a (p,z,φ)= ~a (52)
s p
2π
Identifying
~a†(k+E,kˆ)=~† (k,rˆ) (53)
s aE
allowsustohavecreation/annihilationoperatorswithwell-definedcommutationrelations. Also,notethatE = ~k,
| |
and (51) becomes
= 0 ~a (E,rˆ )−2ie eik(t′−r′) 1 ~a† (k+E,rˆ′) ~′ (∂ + ~ Rˆ)−1(x′,x)1 (54)
s⊥ k E r′ s⊥ ·∇ t ∇· r
(cid:28) (cid:12) Z (cid:18) (cid:19)
(cid:12)
(cid:12)
(cid:12) dk
ℓ(rˆ, rˆ ) dx′dS 0 (55)
j
×j=+,− { } 2π (cid:12)(cid:12) +
X (cid:12)
  (cid:12)
(cid:12)
(cid:12)
9
The relevant commutation relation is
~a (p,z,φ),~a† (p′,z′,φ′) =2πP δ(p p′)δ(z z′)δ(φ φ′) (56)
s⊥ s⊥ ⊥ − − −
h i
where P is a projection operator for the perpendicular components. Using this we obtain
⊥
= −2ie δ(rˆ rˆ′)eik(t′−r′) 1 ~′ (∂ + ~ Rˆ)−1(x′,x)1 ℓ(rˆ, rˆ ) dx′dS (57)
E k − r′∇⊥ t ∇· r  { j} 
Z (cid:18) (cid:19) j=+,−
X
 
which involves
rˆ rˆ(rˆ rˆ )
r~ ℓ(rˆ, rˆ ) = q j − · j (58)
⊥ j j
∇  { }  1 rˆ rˆ
j=+,− j=+,− (cid:18) − · j (cid:19)
X X
 
Using this along with integrating over dx′ and dS in (57) gives
ie rˆ rˆ (rˆ rˆ )
j k k j
q − · (59)
j
− E 1 rˆ rˆ
j=+,− (cid:18) − k· j (cid:19)
X
(59) has the same absolute magnitude as (45), confirming the structure of the effective theory.
6.2 Double logs
The simplest1-loopcorrectiontocalculate isto the productionofascalar/anti-scalarpair,havingenergiesE and
+
E , respectively, and with p~ ~p =E E z ,
− + − 1 2 +−
·
~p p~ 0 (60)
− +
h |S| i
The pointlike source can produce any energy and momentum, so the energies, E and directions of the scalars
±
are unconstrained. As discussed above, the LRE expansion breaks the production process up into independent
sectors, and the matrix element (60) picks out a particular sector in which the only hard particles produced by the
source are the scalar and anti-scalar with the specified energies and angles. In the interactive theory, the radiative
correctionsrenormalizethe sourceinthis sectortoaccountfor(for example)decreasingthe amplitude for exclusive
production of only charged particles with no photons. Formally, the renormalizationfactor is
exp i e ℓ(rˆ,rˆ ) ∂ Aµ(x)dS eiRLLSd4x[dA] eiRLLSd4x[dA] (61)
j µ
 2π    ,
Z Z j=+,− Z
X
   
Where isthegaugeLagrangianinlight-shellgauge. Noticethatxcontributestheonlynon-angulardependence,
LS
L
but dS involves delta functions that leave us with purely angular dependence. This dependence solely on angles
will persistfor any process to any orderin LSET.Evaluating (61) to ordere2 using the methodology introducedin
[10] we arrive at
e2 rˆ1 rˆj rˆ2 rˆk
=1 × · × dΩ dΩ (62)
− 64π4 (1 z )(1 z )(1 z ) 1 2
j=+,− Z −(cid:0) 1j (cid:1)− (cid:0)12 −(cid:1) 2k
kX=+,−
To obtain (62), we have manipulated distributions such that the result is not well-defined without regularization.
For j =k, the only non-integrablesingularity is atrˆ =rˆ . After regulating by taking (1 z ) (1 z +λ) we
1 2 12 12
6 − → −
find the j =k contribution
6 e2
log(λ) log(1 z ) (63)
2π2 − +−
Forj =k in(62),thecalculationismuchmoredelicateanddependsonthedetailsoftheangularcut-offaroundthe
rˆ and rˆ directions. But for physical consistency, the λ and θ dependence must disappear as the hard emission
+ −
becomes neutral. For example, if θ θ θ 1, the λ dependence should cancel as θ θ, because in this
+ − +−
≈ ≈ ≪ →
limit, we have two small, oppositely charged and equal-sized jets sitting right on top of one another to the level of
accuracy to which we know their directions. Furthermore we expect that the j = k contributions should depend
10