Solving for tadpole coefficients in one-loop amplitudes

0.17 MB·English

by  Ruth Britto

Solving for tadpole coefficients in one-loop amplitudes
Ruth Brittoc, Bo Fenga,b
aCenterof Mathematical Science, Zhejiang University, Hangzhou, China
bDivisionof Applied Mathematical and Theoretical Physics,
China Institute for Advanced Study,
Central Universityof Finance and Economics, Beijing, 100081, China
cInstitut de Physique Th´eorique, CEA/Saclay, 91191 Gif-sur-Yvette-Cedex,France
9
0 Abstract
0
2 One-loop amplitudes may be expanded in a basis of scalar integrals multiplied by rational coefficients. We
relatethe coefficientofthe one-pointintegralto the coefficients ofhigher-pointintegrals,by consideringthe
r
p effects of introducing an additional, unphysical propagator,subject to certain conditions.
A

7
1 1. Introduction

] One-loop scattering amplitudes may be expanded in a sum of scalar integrals [1, 2, 3] multiplied by
h
rational coefficients. This expansion arises explicitly in typical computational approaches, reviewed for
t
- example in [4]. The coefficients may be derived directly by reduction of Feynman integrals [1, 2], or they
p
maybesoughtassolutionstolinearequationstakenfromvarioussingularlimits,suchasunitaritycuts,inan
e
h on-shell formalism [5]. Within the second approach, the coefficients can be found by applying “generalized
[ unitarity” multi-cuts [6, 7, 8]. Alternatively, since the master integrals are known explicitly and feature

unique (poly)logarithms, they can also be distinguished by the usual unitarity cuts, which are double-cuts
1
v [9]. Onewaytodothis isbyrewritingthe measureofthe cutintegralinspinorvariables,andthenapplying
6 the residue theorem [10].
6 This procedure of spinor integration has been carried out in generality for theories with at most scalar
7
masses,andanalyticexpressionsforthecoefficientsofthescalarpentagon/box,triangle,andbubbleintegrals
2
have been given [11]. However, the tadpole coefficients are missing, simply because they are obviously free
.
4 of cuts in physical channels. Our note addresses this point.
0
We findthatwecansolveforthe tadpole coefficientsintermsofthe coefficients ofhigher-pointintegrals
9
afterintroducinganauxiliary,unphysicalpropagator. Theauxiliaryloopintegralthenhastwopropagators,
0
: sowecanapplyunitaritycutsformally. Thetadpolecoefficientisaccordinglyrelatedtothebubblecoefficient
v
oftheauxiliaryintegral. Ourresultisasetofrelationsgivingthetadpolecoefficientsintermsofthe bubble
i
X coefficientsofboththeoriginalandauxiliaryintegrals,andthetrianglecoefficientsoftheauxiliaryintegrals.
r It is interesting to consider whether this construction might have other applications.
a
To derive the relations between tadpole coefficients and the others, we make use of work of Ossola,
Papadopoulos, and Pittau (OPP) [7], which gives the result of one-loop reduction at the integrand level,
buildinguponanalysisoftheirtensorstructure[12]. Inadditiontotheintegrandsforscalarboxes,triangles,
bubbles,andtadpoles,thereareanumberof“spuriousterms”whichvanishafterintegration. Thecomplete
decompositionand classificationgiven by OPP allows us to relate the originalloop integralto the auxiliary
integral including the unphysical propagator. We then derive relations among their respective coefficients,
and identify conditions that almost completely decouple the effect of the unphysical propagator.
We note that on-shell approachesto loop amplitudes face important subtleties in seeking tadpole coeffi-
cientsanalytically. Theoperationofmakingasinglecutrelatesann-pointloopamplitudetoan(n+2)-point
treelevelquantity,whichshouldbe consideredasanoff-shellcurrent. These arethe samestartingpoints as
inproposalsto reconstructfullamplitudes entirely fromsingle cuts [13, 14]. Another cut-freeintegralis the
0-mass scalar bubble. In this note, we assume that this contribution is known. For applications to physical
Preprint submitted toElsevier April 17, 2009
amplitudesusing unitaritymethods, itwillbe necessaryto accountforcuts ofself-energydiagrams[8]. The
procedure for finding massless bubble coefficients is not known in general, analytically, and merits further
exploration;here, we will comment on the possibility of taking a limit of vanishing mass. Another proposal
[15] is to fix the tadpole and massless bubble contributions by universal divergent behavior, once all other
integral coefficients are known.
2. Relations among cuts and coefficients
We adopt the notation of OPP [7, 12]. The D-dimensional loop momentum is denoted by q¯, whose 4-
dimensionalcomponentisq. ThedenominatorfactorstaketheformD¯ =(q¯+p )2−M2,wherei=0,1,2,....
i i i
Thetadpoleofinterestshallbeassociatedtothefactorwithi=0. Wedefineℓ=−q−p .andK ≡p −p .
0 i 0 i
Expanding the loop momentum variable into its four-dimensional component plus the remaining part q
satisfying q2 =−µ2, the denominators can be rewritten as D¯ =(ℓ−K )2−Me 2−µ2.
i i i
We are interested in the effect of including an auxiliary denominator factor, which we write as e
e e
D¯ = (ℓ−K)2−M2 −µ2. [K-prop] (1)
K K
At this point, K and M2 are variables unrelaeted to the physical amplitude. Later, they will be chosen
K
subject to conditions that minimize the effect of this auxiliary factor.
The one-loop integrand is
N(q)
I = [true-integrand] (2)
true D¯0D¯1···D¯m−1
where,following[7],weuseN(q)todenotethenumerator,whichisapolynomialinq. Wecalltheintegrand
I the “true” integrand to distinguish it from the “auxiliary”integrand, which we construct by inserting
true
the auxiliary factor D¯ , as follows.
K
N(q)
I = [aux-integrand] (3)
K D¯KD¯0D¯1···D¯m−1
Consider the single-propagatorcut of the tadpole of interest. It is the result of eliminating the denomi-
nator factor D¯ from the integrand:
0
N(q)
Atree = [1-cut-amp] (4)
1−cut D¯1···D¯m−1
ThisintegrandistheanalogoftheproductoftreeamplitudesAtreeAtree obtainedfromastandardunitarity
Left Right
cut. However, from a single cut, we obtain a tree amplitude at a singular point in phase space, since two
external on-shell momenta are equal and opposite. This singularity can create difficulties that we do not
address generally here. It is probably best considered as an off-shell current.
In the OPP method [7], the integrand is expanded in terms of the master integrals multiplied by their
coefficients in the amplitude, plus additional “spurious terms” which vanish upon integration. The uninte-
grated expansion is
m−1 m−1 m−1
I = [a(i)+a(q;i)]I(i)+ [b(i,j)+b(q;i,j)]I(i,j)+ [c(i,j,r)+c(q;i,j,r)]I(i,j,r)
true
Xi Xi<j i<Xj<r
e e e
m−1 m−1
+ [d(i,j,r,s)+d(q;i,j,r,s)]I(i,j,r,s)+ e(i,j,r,s,t)I(i,j,r,s,t) [true-opp-expan(s5io)n]
i<jX<r<s i<j<Xr<s<t
e
wherea(i),b(i,j),c(i,j,r),d(i,j,r,s),e(i,j,r,s,t)arethecoefficientsofthemasterintegrals;a(q;i),b(q;i,j),
c(q;i,j,r), d(q;i,j,r,s) are the spurious terms which integrate to zero; and the master integrals are
e e
Ie(i) = 1 , eI(i,j) = 1 , I(i,j,r) = 1 , I(i,j,r,s) = 1 , I(i,j,r,s,t) = 1 .
D¯ D¯ D¯ D¯ D¯ D¯ D¯ D¯ D¯ D¯ D¯ D¯ D¯ D¯ D¯
i i j i j r i j r s i j r s t
2
Notice that we have included the pentagon explicitly. We shall perform our analysis of the coefficients in
D =4−2ǫ dimensions.
Now, consider the auxiliary integrand I . On one hand, it is simply the true integrand divided by the
K
auxiliary propagator. Thus, using (5), we get the following expansion in master integrals:
m−1 m−1 m−1
I
I = true = [a(i)+a(q;i)]I(K,i)+ [b(i,j)+b(q;i,j)]I(K,i,j)+ [c(i,j,r)+c(q;i,j,r)]I(K,i,j,r)
K D¯
K Xi Xi<j i<Xj<r
e e e
m−1 m−1
+ [d(i,j,r,s)+d(q;i,j,r,s)]I(K,i,j,r,s)+ e(i,j,r,s,t)I(K,i,j,r,s,t) [divided] (6)
i<jX<r<s i<j<Xr<s<t
e
Noticeherethatthe“spurious”termssuchas b(q;i,j)arenolongerspuriouswiththefactorD¯ included.
K
Forexample,while b(q;i,j)I(i,j) =0byconsRtruction,ingeneral b(q;i,j)I(K,i,j) 6=0. Ontheotherhand,
e
theauxiliaryintegraRnedIK hasitsownOPPexpansion,wherewelaRbeeltheauxiliarycoefficientsandspurious
terms by the subscript K, and we separate the auxiliary propagator explicitly, so it is not included in the
summation indices:
m−1 m−1 m−1
I = [a (i)+a (q;i)]I(i)+ [b (i,j)+b (q;i,j)]I(i,j)+ [c (i,j,r)+c (q;i,j,r)]I(i,j,r)
K K K K K K K
Xi Xi<j i<Xj<r
e e e
m−1 m−1
+ [d (i,j,r,s)+d (q;i,j,r,s)]I(i,j,r,s)+ e (i,j,r,s,t)I(i,j,r,s,t)
K K K
i<jX<r<s i<jX<r<s,t
e
m−1 m−1
+[a (K)+a (q;K)]I(K)+ [b (K,j)+b (q;K,j)]I(K,j)+ [c (K,j,r)+c (q;K,j,r)]I(K,j,r)
K K K K K K
Xj Xj<r
e e e
m−1 m−1
+ [d (K,j,r,s)+d (q;K,j,r,s)]I(K,j,r,s)+ e (K,j,r,s,t)I(K,j,r,s,t) [aux-opp] (7)
K K K
j<Xr<s j<Xr<s,t
e
WiththesubscriptK,termssuchasb (q;i,j)andb (q;K,j)aretrulyspurioustermsin(6),e.g. b (q;K,j,r)I(K,j) =
K K K
0. R
e e e
ThepurposeofintroducingtheauxiliaryintegrandI istogiveinformationaboutthetadpolecoefficient
K
by cutting two propagators. So, we will always choose to cut the auxiliary propagator D¯ along with D¯ .
K 0
Restricted to the terms involved in this cut, the auxiliary integrand (7) is
m−1
I | = [b (K,0)+b (q;K,0)]I(K,0)+ [c (K,0,i)+c (q;K,0,i)]I(K,0,i)
K C0K K K K K
Xi
m−1 e me−1
+ [d (K,0,i,j)+d (q;K,0,i,j)]I(K,0,i,j)+ e (K,0,i,j,s)I(K,0,i,j,s) [aux-(op8p)-restricted]
K K K
Xi<j i<Xj<s
e
Similarly, we can restrict our attention to the corresponding subset of terms in (6):
m−1 m−1
I | = [a(0)+a(q;0)]I(K,0)+ [b(0,i)+b(q;0,i)]I(K,0,i)+ [c(0,i,j)+c(q;0,i,j)]I(K,0,i,j)
K C0K
Xi Xi<j
e e e
m−1 m−1
+ [d(0,i,j,r)+d(q;0,i,j,r)]I(K,0,i,j,r)+ e(0,i,j,r,s)I(K,0,i,j,r,s) [divided-restri(c9te)d]
i<Xj<r i<jX<r<s
e
Our plan is to find the tadpole coefficient, a(0), by imposing the equivalence of (8) and (9) after com-
pleting the cut integral. After integration, the spurious terms of (8) simply drop out, as they are designed
3
to do so:
m−1
I = b (K,0) I(K,0)+ c (K,0,i) I(K,0,i)
K K K
Z Z Z
C0K C0K Xi C0K
m−1 m−1
+ d (K,0,i,j) I(K,0,i,j)+ e (K,0,i,j,s) I(K,0,i,j,s) [int-aux-(o1p0p-)restricted]
K K
Z Z
Xi<j C0K i<Xj<s C0K
Here, the cut integral is denoted by , which indicates that we use the Lorentz-invariant phase space
measure including the factor δ(D¯ )δ(DR¯C0K).
0 K
However,theintegrationoftheformula(9)isnotsostraightforward,becausetheoriginalspuriousterms
no longer correspond to the structures of the denominators they multiply. So, we shall view the expression
(9) as a function of the loopmomentum q, and find the coefficients of master integralsanalytically,for each
of the spurious terms classified by OPP.
Keepinginmindthatourtargetis thesinglenumbera(0),whichappearsaspartofthe auxiliarybubble
coefficient in (9), we begin by extracting only the auxiliary bubble contributions of the various spurious
terms, divided by their denominators as well as D¯ . (The other non-spurious terms, with b(0,i), c(0,i,j),
K
and d(0,i,j,r), clearly belong entirely to coefficients of other master integrals, of which the 4-dimensional
pentagon is a linear combination of five boxes.)
Ourresultisthatthereareconditionsunderwhichmostofthespurioustermshavenoeffect. Specifically,
for all the propagator momenta K inside D¯ , we would like to take
i i
K·K =0, ∀i; M2 =M2+K2. [KM-cond] (11)
i K 0
We are free to take (11) as a definition of M2, while the condition for K is clearly nontrivial to satisfy
K
physically. Letusconsiderthis conditionmoreconcretely. Forthe purposesofdefining ourconstruction,we
perform a formal reduction. Any integrand having five or more propagators has at least four independent
momenta K that can be used to expand any external momentum vector appearing in the numerator to do
i
the reduction. For integrands with at most four propagators, there are no more than three momenta K ,
i
and the condition (11) can be satisfied, for example by the construction K =ǫ KνKρKσ.
µ µνρσ 1 2 3
If conditions (11) are satisfied, then we find that only one of all the spurious terms contributes to the
auxiliary bubble coefficient. Specifically,
1
b (K,0)=a(0)+ (K2−M2+M2)b [K ], (12)
K 12 i i 0 00 i
Xi
e
where b [K ] is the coefficient of one of the spurious terms defined in [7].
00 i
A convenient way to constrain b [K ] is to identify the effect of the spurious term on the auxiliary
e 00 i
triangle coefficient. Still imposing the conditions (11), we repeat our analysis of all the OPP spurious terms
e
in (9), this time isolating the contributions to triangle coefficients. Fortunately, we find that only this same
single spurious term has a nonvanishing effect, if we focus on the terms with µ2-dependence. (We assume
that explicit µ2-dependence in the numerator N(q) has been set aside.) The result is
K2
c (K,0,i)| = b(0,i)| + i b [K ], (13)
K µ2 µ2 3 00 i
e
where | means the coefficient of µ2.
µ2
Now we propose the following procedure for finding tadpole coefficients.
1. Find the single-cut expressionAt1r−eecut obtained by cutting the propagatorD¯0. Expand the numerator
in µ2, and work term by term, setting aside these explicit factors of µ2.
2. Construct the true integrand I = At1r−eecut/D¯0 and the auxiliary integrand IK = At1r−eecut/(D¯KD¯0). It
may be convenient at this stage already to choose K and M to satisfy the conditions (11). Alterna-
K
tively, they can be taken as arbitrary variables until the final step.
4
3. Use the cutintegral I to evaluatethe auxiliarybubble coefficientb (K,0)andallthe auxiliary
C0K K K
triangle coefficients cR (K,0,i).
K
4. Use the cut integrals I to evaluate all the true bubble coefficients b(0,i).
C0Ki
5. The tadpole coefficienRt is given by imposing the conditions (11) in the following expression.
K2−M2+M2
a(0)=b (K,0)+ i i 0 [c (K,0,i)−b(0,i)]| . (14)
K 4K2 K µ2
Xi i
Thisformulaisvalidtermbyterm,havingsetasidetheoriginalexplicitfactorsofµ2 inthenumerator
N(q).
3. Contributions to bK(K,0) from spurious terms
In this section we will discuss the contributions to b (K,0) of the auxiliary integrand I from the
K K
expression(9), wherethe termshavebeenseparatedintothescalarintegralcoefficients,plusspuriousterms
as classified by OPP [7]. As we have discussed, these terms are no longer “spurious” in the same sense,
once D¯ is included. (N.B.: OPP write the expansion with all denominators multiplied through, so that
K
the “spurious terms” for them are the polynomial numerators. Here, we use “auxiliary spurious terms” to
refer to the correesponding terms with all denominators present, including D¯ .)
K
Thefirstcontributionisobviouslya(0),whichisthetadpolecoefficientthatinterestsus. Nowwediscuss
the possible contributions from the terms with a,b,c,d in (9). We will see why we choose the decoupling
conditions (11). They arise naturally by considering the terms of lowest degree. We have proceeded step
by step through all the spurious terms of [7]. Oeuer reeseults have been derived in the formalism of [10] and
verified using Passarino-Veltmanreduction [1] as implemented in FeynCalc [16].
• One-point spurious terms: In the simplest case, all spurious terms of this type are linear in the
numerator I = 2ℓ·R /D¯ . The auxiliary integrand, including D¯ in the denominator, is then
11 1 0 K
ID¯K = 2ℓ·R /(D¯ D¯ ). It is easy to find the scalar bubble coefficient from a standard unitarity cut
11 1 Ke0
(or alternatively, by straightforwardreduction). The result is
e
(K·R )(K2+M2−M2)
C[D¯ ,D¯ ]= 1 0 K . [I11-D0DK] (15)
0 K K2
Therearefourindependent“1-pointlike”spurioustermsasgivenbyOPP,i.e.,fourindependentvalues
of R . We see that we can decouple all their contributions by imposing the condition
1
K2+M2−M2 =0 [alpha=0] (16)
0 K
• Two-point spurious terms: Spurious 2-point terms can be either linear or quadratic in loop mo-
mentum. In the case of linear dependence, the auxiliary integrand with D¯ has the scalar bubble
K
coefficient
−(K·K )(K·R )+K2(K ·R )
C[D¯ ,D¯ ] = i 1 i 1 . [I21-D0DK] (17)
0 K (K ·K )2−K2K2
i i
In the spurious terms, R takes three possible values of vectors, called ℓ ,ℓ ,n. These vectors are
1 7 8
definedin[7];hereweonlyneedtousesomeoftheirproperties. (WeuseK astheauxiliarymomentum
in the OPP construction of these vectors.) In each of these cases, we have R ·K = 0. Moreover,
1 i
K·ℓ =0,butK·n6=0. Tomakethislastspuriouscontributionvanish,weenforceanewdecoupling
7/8
condition:
K·K =0. [KK1=0] (18)
i
5
Nowwemoveontothequadraticspurious2-pointterms. Therearefivesuchterms. Forfourofthem,
the auxiliarybubble coefficient vanishes under the twodecoupling conditions. The fifth spurious term
is K(q;0,i), which can be written (ℓ·n)2−((ℓ·K )2−K2ℓ2)/3. Its coefficient in the OPP expansion
i i
is denoted b (0,i). After imposing the decoupling conditions, the auxiliary bubble coefficient from
00 e e e
this term is
e
K2+M2−M2
Ce = i 0 i . (19)
b00(0,i) 12
Because this spurious term gives a nonzero contribution under the decoupling conditions, we must
calculate it and subtract its contribution when we calculate the tadpole coefficient. For this reason,
we will turn to the auxiliary triangles c (K,0,i) in the following section.
K
• Three-point and four-point spurious terms: All of the auxiliary three-point spurious terms
decouple after imposing (11). There is just one auxiliary four-point spurious term, and it gives no
bubble contribution at all, because its numerator is linear in the loop momentum.
To summarize, have seen that if we impose the conditions (11), then all contributions from spurious
terms will decouple, except one, whose coefficient is b . We have
00
e K2+M2−M2
b (K,0)=a(0)+ b (0,i) i 0 i [a0-eq-1] (20)
K 00
12
Xi
e
where b (0,i) is the coefficient of the spurious term K(q;0,i) as defined by OPP.
00
In this analysis, we are assuming a renormalizable theory. We have assumed that the power of ℓ in the
e
numeratoris equal to or less than the number of propagatorsin the denominator. In those terms where the
e
power of ℓ is strictly less than the number of propagators, then we have b (0,i) = 0, ∀i. Thus we have
00
b (K,0) = a(0), i.e., we get the tadpole coefficient a(0) immediately by calculating the bubble coefficient
K e e
under the decoupling conditions. From terms where the power of ℓ is equal to the number of propagators,
b (0,i)6=0, and we need to compute it. We have found that we can use a similar decoupling approach to
00 e
calculate the triangle coefficient C[D¯ ,D¯ ,D¯ ] and extract the corresponding b (0,i). This procedure will
e 0 K i 00
be discussed in the next section.
e
4. The calculation of b00(0,i)
Recalltheexpansion(e9),whereweaugmentedthe OPPexpansionwiththe extrafactorD¯ ,sothatthe
K
spuriousterms no longerintegrateto zero. We see that the term b(q;0,i)I(K,0,i) contributesnot only to the
coefficient of the bubble I(K,0), but also to the coefficient of the triangle I(K,0,i). Thus it is possible to find
e
b (0,j) from the evaluation of the coefficient of triangle I(K,0,i) within I | .
00 K C0K
Justasintheprevioussection,wherewestudiedallcontributionstocoefficientofI(K,0) fromI | ,so
e K C0K
c (K,0,i) in (8) also receives contributions from the original spurious terms in (9). Thus we carry out the
K
corresponding analysis in this section. We impose the decoupling conditions from the start. Then we find
that three of the auxiliary spurious terms still give auxiliary triangle contributions. The first nonvanishing
contribution comes, indeed, from the term we want, namely K(q;0,i). Its auxiliary triangle contribution,
after having applied the decoupling conditions, is
(K2+M2−M2)2−4K2(M2+µ2)
C[D¯0,D¯K,D¯i]eb00 =− 1 0 i 12 i 0 . [tri-22] (21)
The second and third nonvanishing contributions come from the spurious 3-point terms with quadratic
dependence on the loop momentum. The auxiliary integrand is
ID¯K = (2ℓ·R1)2 , (22)
32 D¯ D¯ D¯ D¯
0eK i j
6
where R takes two values, called ℓ . After applying the decoupling conditions, we find that the triangle
1 3,4
coefficient is
(K·R )2((K ·K )(K2+M2−M2)−K2(K2+M2−M2))
1 i j i 0 i i j 0 j
−
K2((K ·K )2−K2K2)
i j i j
Thisquantitydoesnotvanishidentically,soourdecouplingmightseemtobeinadequate. Butthereisgood
news here: the contribution does not depend on µ2, while the contribution from b does depend on µ2.
00
Thus we can use the µ2-dependence to find exactly the term we need.
e
Ourplanisnowclear: (1)calculatethebubblecoefficientb(0,i)fromtheintegrandI ,whichdoesnot
true
contain D¯ ; (2) calculate the triangle coefficient c (K,0,i) ≡ C[D¯ ,D¯ ,D¯ ] of the integrand I , which
K K 0 K i K
does contain D¯ ; (3) find the µ2-dependent terms in b(0,i) and c (K,0,i), and then solve the following
K K
equation to find b (0,i).
00
e K2
c (K,0,i)| = b(0,i)| + i b (0,i), [c-K-i] (23)
K µ2 µ2 3 00
e
where| meansthe coefficientofµ2. After computingb (0,i)foreveryi,wesubstitute backinto (20)and
µ2 00
finally find a(0), the tadpole coefficient.
e
5. Examples and discussion
Finally, we list some formulas we obtained from our algorithm and comment on their properties. We
denote a general integrandterm by two indices n,m, writing I [{K },{R }]= m (2ℓ·R )/( n−1D¯ ).
n,m i j j=1 j i=0 i
Further, we define α = K2+M2−M2 and ∆ = (K ·K )2−K2K2. The tadQpole coefficientsQfrom the
i i 0 i ij i j i j e
first few of these integrands are:
a[D¯0]I21 = −RK1·2K1 [I21-tad] (24)
1
a[D¯0]I22 = −α1[(R1·R2)K123−(K4(2R)21·K1)(R2·K1)] [I22-tad] (25)
1
a[D¯0]I32 = P2i,j=1AiKj(12KKi22·∆R112)(Kj·R2), (26)
where
A11 =K22(K1·K2), A22 =K12(K1·K2), A12 =A21 =−K12K22 [I32-tad] (27)
a[D¯0]I33 = Pi=1,2Pj≤kAi3;j(sK(K2Ki·2R)22∆)(2Kj·R1)(Ks·R1), [I33-tad] (28)
1 2 12
where
A1;11 = −2(K22)2(∆12(4α1(K1·K2)−2α2K12)+5K12(K1·K2)(α2(K1·K2)−α1K22))
A2;11 = A1;12 =−2K12(K22)2(α1∆12−5K12(α2(K1·K2)−α1K22)),
A2;22 = A1;11|α1,K1↔α2,K2, A2;12 =A1;22 =A1;12|α1,K1↔α2,K2
a[D¯0]I43 = P3i=1P3t≤Ks,121KA22i;Kts3(2K∆i12·∆R123)∆(K23t∆·R1213)(Ks·R1), (29)
7
where
∆123 = K12K22K32+2(K1·K2)(K2·K3)(K3·K1)−K12(K2·K3)2−K22(K1·K3)2−K32(K1·K2)2
A1;11 = 2∆23K22K32˘(K2·K3)[(K1·K2)2∆13+(K1·K3)2∆12]+(K1·K2)(K1·K3)[K22∆13+K32∆12]¯
A1;12 = A2;11 =2K12K22K32∆13∆23(−K22(K1·K3)+(K2·K1)(K2·K3)), A1;13 =A3,11 =[A1;12]|K2↔K3,
A2;22 = [A1;11]|K1↔K2, A3;33 =[A1;11]|K1↔K3, A3;23 =A2;33 =[A1;12]|K1↔K3, A2;12 =A1;22 =[A1;12]|K1↔K2,
A2;23 = A3;22 =[A1;12]|K1,K2,K3→K2,K3,K1, A3;13 =A1;33 =[A1;12]K1,K2,K3→K3,K1,K2,
A1;23 = A2;13 =A3;12 =−2K12K22K32∆12∆13∆23
We note somepatternsinthese tadpole coefficients: (1)the tadpolecoefficientis independent ofµ2; (2)for
In,n−1, the coefficient is independent of masses; (3) for In,n, the coefficient is of the form iαici.
Finally, we offera commentonmasslesslimits. Inour calculation,wehave assumedthPat K2 6=0. What
i
happens when K2 →0? Let us use I as an example. The full reduction of I is
i 21 21
M2−M2 R ·K R ·K
I = (R ·K ) 1+ 0 1 I − 1 1I [D¯ ]+ 1 1I [D¯ ]. [Exa-1-final](30)
21 1 1 (cid:18) K2 (cid:19) 2,0 K2 1,0 0 K2 1,0 1
1 1 1
Using the expressions for the scalar integrals,
µ2ǫΓ(ǫ)(M2−iε)1−ǫ−(M2−iε)1−ǫ Γ(−1+ǫ)
ID(K2 =0;M2,M2)= 0 2 , I =µ2ǫ (M2−iε)1−ǫ, (31)
2 1 1 2 r (1−ǫ)(M2−M2) 1,0 r 0
Γ 0 2 Γ
itis easyto see that the term with 1/K2 willbe canceledexactly onthe righthandside ofI . Thatis, for
1 2,1
I , knowing the coefficient of the bubble, the smoothness of the limit K2 →0 is enough to fix the tadpole
2,1 1
coefficient. Does this pattern hold in general? Looking back through our examples, we see that for I ,I ,
32 33
and I , the limits K2,K2 → 0 give divergences which should be cancelled elsewhere. It is not yet clear
43 1 2
exactly how generally we can relate the bubble coefficients to various tadpole coefficients by considering
smoothness requirements in various limits, although the general behavior of M2 dependence suggests that
such relations should exist.
Acknowledgments
We thank the participants of the International workshop on gauge and string amplitudes at IPPP
Durham, where this work has been presented. The work of R.B. was supported by Stichting FOM; by
Fermilab, which is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359
with the United States Department of Energy; and by the DSM CEA-Saclay. B.F. would like to thank the
CUFEforhospitalitywhilethefinalpartwasdone. HisworkissupportedbyQiu-ShifundingfromZhejiang
University and Chinese NSF funding under contract No.10875104.
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