Higgs boson decay to two photons and the dispersion relations

0.15 MB·

by  Kirill Melnikov

FTPI-MINN-15/49
TTP16-001
Higgs boson decay to two photons and the dispersion relations
Kirill Melnikov∗
Institute for Theoretical Particle Physics, Karlsruhe Institute of Technology, Karlsruhe, Germany
Arkady Vainshtein†
6 William I. Fine Institute for Theoretical Physics and School of Physics and Astronomy,
1 University of Minnesota, Minneapolis, MN 55455, USA
0
2

n We discuss the computation of the Higgs boson decay amplitude to two photons through the
a W-loop using dispersion relations. The imaginary part of theform factor FW(s) that parametrizes
J this decay is unambiguous in four dimensions. When it is used to calculate the unsubtracted

4 dispersion integral, the finite result for the form factor FW(s) is obtained. However, the FW(s)
obtained in this way differs by a constant term from the result of a diagrammatic computation,

] based on dimensional regularization. It is easy to accommodate the missing constant by writing
h
a once-subtracted dispersion relation for FW(s) but it is unclear why the subtraction needs to be
p
done. The goal of this paper is to investigate this question in detail. We show that the correct
-
p constant can be recovered within a dispersive approach in a number of ways that, however, either
e require an introduction of an ultraviolet regulator or unphysical degrees of freedom; unregulated
h and unsubtracted computations in the unitary gauge are insufficient, in spite of the fact that such
[ computations give a finiteresult.

1
PACSnumbers:
v
6
0 I. INTRODUCTION where β =4m2 /s and
4 W
0 2
1 1+√1 β
0 The decay rate of the Higgs boson to two photons f(β)= ln − iπ . (3)
. through the W-loop was computed in the literature at −4(cid:20) 1 √1 β − (cid:21)
1 − −
least thirteen times [1–13]. The recent flurry of activity
0 TheconstanttermF∞ inEq.(2)isthegistofthecur-
6 around this process, important for understanding Higgs rent discussion: accordWing to Refs.[1–3, 6–12] F∞ = 2
1 bosonproperties,wascausedbythefactthattheoriginal W
andaccordingtoRefs.[4,5,13],F∞ =0. Thetwogroups
: computationsoftheH γγ decayrate[1–3],performed W
v → [4,5,13]thatclaimF∞ =0haveusedtwodifferenttech-
almost forty years ago, were challenged in Refs. [4, 5]. W
i
X Among the follow up computations [6–13], only Ref. [13] niquesintheircomputationsthat,however,havetwoim-
portant features in common. Indeed, both groups refuse
r agreed with the findings of Refs. [4, 5].
a to use the dimensional regularization, so that all the al-
Agoodwaytodescribethecontroversialsituationisas
gebraic manipulations are performed in four dimensions
follows. Consider the H γ(k )γ(k ) decay amplitude,
→ 1 2 and both groups insist on using only physical degrees of
focusing on the W-boson loop, and write it as
freedomin their calculations,i.e. the unitarity gauge for
α
= F (m2 )(kµǫν kνǫµ)(k ǫ k ǫ ). (1) the W-bosons.
M 4πv W H 1 1− 1 1 2µ 2ν− 2ν 2µ The authors of Refs. [4, 5] do this in the context of
Here v=2m /g= G √2 −1/2 is the Higgs field vac- Feynman diagrams and loop integrations. This is a del-
W F
icate matter since all the individual diagrams are diver-
uum expectation val(cid:0)ue and(cid:1)ǫ1,2 are the photon polariza-
gent and need to be combined before the actual inte-
tion vectors. The form factor F (s) reads
W
gration over the loop momentum to ensure the finite re-
FW(s)=FW∞+FWc (s), sult. It is understandable, that this method of calcula-
(2)
Fc (s)=3β+3β(2 β)f(β), tion drew criticism from Refs. [6, 8, 9, 14]. Interestingly,
W − the authors of Refs. [4, 5] recognize this issue and try
to ameliorate it by imposing an additional requirement
on their result. This requirement is the heavy Higgs bo-
∗Electronicaddress: [email protected] son decoupling condition FW(s → ∞) = FW∞ = 0 whose
†Electronicaddress: [email protected] validity was, however, criticized in Refs. [6, 8–10]. In-
2
deed, the decoupling limit, β =4m2 /m2 0, can also computed in a renormalizable theory since each inde-
W H →
be viewed as the limit m 0. It is well-known that pendentsubtractiontermcorrespondstoanindependent
W
→
in the m 0 limit the Higgs boson interaction with renormalization condition that usually are fixed by con-
W
vector boson→s, 2Hm2 W†Wµ/v, does not vanish for the sidering divergent, rather than finite, quantities. Also,
W µ
longitudinal polarizations of the W bosons. This is in the use of unsubtracted dispersion relations should be
contrast to the Higgs interactions with fermions that do possible if one combines an ultraviolet (UV) regulariza-
vanish in the zero fermion mass limit. tion, such as dimensional or Pauli-Villars, with the dis-
Ontheotherhand,thecomputationofRef.[13],based persion relations. Indeed, taking the dimensional reg-
onthedispersiveapproach,iswell-groundedatfirstsight. ularization as an example, any integral over the in-
Ifonewantstousethefour-dimensionalsetupandphys- finitelyremoteintegrationcontourcanbediscardedsince
icaldegreesoffreedom,thebestthingtodoistousedis- F (s) s−ǫ for dimensional reasons and ǫ can always
W
∼
persionrelationsforthe formfactorF (s) whoseimagi- chosen in such a way that such an integral vanishes. In
W
narypartcanbe computedfromtree-levelFeynmandia- caseofthe Pauli-Villarsregularization,F (s)is alsode-
W
grams. As it is seen from Eqs.(2, 3), ImF (s) does not creasing for values of √s that are larger than the ultra-
W
depend on the ambiguity in F∞ and equals to violet cut-off, given by the regulator mass M .
W PV
Combining these observations with the fact that the
3π 1+√1 β ImF (s)inEq.(4)is finite andintegrableinthe disper-
ImF (s)= θ(1 β)β(2 β)ln − . (4) W
W 2 − − 1 √1 β sionintegral,andthat the StandardModelis, obviously,
− −
arenormalizabletheory,weconcludethatsomethingun-
Note, that the imaginary part does vanish in the β 0
usual should occur in ImF (s) in the limit when the
→ W
limit.
regulators are taken to their limiting values (ǫ 0 or
The full function F (s) is then reconstructed using →
W M ). Indeed, as we will see, this is exactly what
PV
the unsubtracted dispersion relation in s, →∞
happensandanadditionalcontributiontotheimaginary
∞ part of the form factor is generated at √s of the order
1 ds1Im[FW(s1)] of the ultraviolet cut-off. This additional contribution
F (s)= . (5)
W π Z s1 s i0 to the dispersion integral changes FW(s) if s is in the
4m2 − − range m √s M , effectively leading to a non-
W W PV
≪ ≪
vanishing “constant” contribution to F .
W
The resultofthe integrationin Eq.(5) is the formfactor
Although this approach may look somewhat unphys-
Fc shown in Eq.(2), which implies that F∞ = 0. The
W W ical because it refers to the behavior of the theory for
authors of Ref. [13] interpret this result as the support-
values of Higgs masses that are larger than the UV cut-
ingevidence forthe computationreportedinRefs.[4,5].
off of the theory, we will see that it is consistent with an
However, it should be recognized that the use of the
infrared condition, e.g. the fixed value of F at s = 0.
W
unsubtracted dispersion relation assumes that the form
We investigate how this happens in detail in this paper.
factor F (s) vanishes at s , i.e. F∞ = 0. In
W → ∞ W
other words, decoupling is assumed, rather than proved
in Ref. [13]. Without such an assumption, one can just II. LONGITUDINAL POLARIZATIONS
add any real constant to the right hand side of Eq.(5).
The constant F∞ then either needs to be computed
W The issue of non-decoupling at m = 0 refers to the
with a method that is different from the dispersion rela- W
longitudinally-polarizedW bosons. Todescribethesepo-
tions or oneshouldhaveaphysicalargumentthatdeter-
larizations at large energies, E m , one can sub-
mines the value of the form factor FW(s) for one value stitute W = ∂ φ/m where φ≫is thWe charged scalar
of s. The most well-known example of the latter is the µ µ W
field; this statement is the essence of the equivalence
requirement that the Dirac form factor of the electron
theorem [15–17]. When written in terms of φ-fields,
equals to one at zero momentum transfer.
the interaction of the W-bosons with the Higgs field
In case ofthe formfactor F (s), the low-energytheo-
W 2(H/v)m2 W†Wµ takes the form
remofRef.[3]fixes its value ats=0 to be the W-boson W µ
contribution to the coefficient of the one-loop QED β- H
function bW Sint =Z d4x v ∂µ∂µ φ†φ . (7)
(cid:0) (cid:1)
limFW =bW =7. (6) Technically,thisinteractionlooksasadimension-five,i.e.
s→0
non-renormalizable,operator. This factaloneshouldact
It is straightforward to check, using Eq.(2), that this like a warning sign for the application of unsubtracted
condition at s=0 implies that F∞ =2. dispersionrelations,eveniftheresultofthecomputation
W
Nevertheless, we can ask under which conditions the turns out to be finite.
dispersionrelationswithouttheintegralovertheinfinitely Wewillstudythecontributionoftheφparticlestothe
remote contour and the subtraction constant can be used formfactorforthetwo-photonHiggsdecayassumingthat
in general. The answer to this question is well-known. Higgs-φinteractionisgivenbyEq.(7)anddenotingtheir
Such a possibility should exist if a finite form factor is masses as m . We will see that this toy model captures
φ
3
all the essential features of the problem discussed in the whereβ =4m2/s. ThisexpressioncoincideswithEq.(9)
φ φ
Introduction. The counter-part of the full form factor atlargesbutitisvalidforalls. Toobtaintheformfactor
F (s) of Eq.(2) in our toy model is denoted by F (s). F (s), we multiply the real and imaginary parts of Φ by
W φ φ
TherearetwowaystodealwiththeoperatorinEq.(7). m2 and identify m2 with s. We find
H H
The first one is based on the observation that for the
purpose of computing H γγ decay amplitude, it is Fφ(s)=sΦ(s),
→
possible to improve the ultraviolet properties of the ac-
1+ 1 β
tion in Eq.(7). To this end, we integrate by parts in ImF (s)= πθ(1 β )β ln − φ , (15)
Eq.(7), use equations of motion for the Higgs particle φ − − φ φ 1 p1 βφ
− −
∂µ∂µH =−m2HH and obtain F (s)=2 1 β f(β ) . p
φ φ φ
−
m2 (cid:0) (cid:1)
Sint =− vH Z d4xHφ†φ. (8) Note that the physical form factor Fφ(s) contains the
constantcontributionin the limit s and, therefore,
This transformation makes the interaction between the does not support the s decoup→lin∞g condition.
Higgs and the φ’s explicitly renormalizable and guaran- Wecannowaskwhat→ist∞hedispersionrelationthatthe
tees that an unsubtracted dispersion relations for suit- formfactorF (s)satisfies,providedthatΦ(s)satisfiesan
φ
ably defined form factor should be applicable. unsubtracteddispersionrelation. Itisstraightforwardto
Toproceedfurther,weparametrizethematrixelement answer this question. We start from the unsubtracted
γγ φ†φ0 as follows relationfor Φ(s) in Eq.(14), write Φ=F /s,and obtain
h | | i φ
α
hγγ|φ†φ|0i=−Φ(s)· 4π f1µνf2,µν, (9) F (s)= s ∞ ds1ImFφ(s1) , (16)
φ
where fµν = kµǫν kνǫµ. The physical form factor is π Z4m2φ s1(s1−s−i0)
then F (is)=mi2 iΦ(−s)=i isΦ(s).
φ H which is a once-subtracted dispersion relation for the
The form factor Φ(s) at large s = (k +k )2 m2
1 2 ≫ φ formfactor Fφ(s). Therefore,the subtractionof the dis-
equals to [3, 18, 19]
persion relation for F (s) at s =0, which enforces the
φ
2 condition Fφ(s = 0)=0, appears automatically provided
Φ(s)= . (10) that we use the unsubtracted dispersion relations only
s
for quantities (e.g. Φ(s)) that are computed in a theory
AftermultiplyingEq.(9)bythe“couplingconstant”m2H, whereallinteractionsarerenormalizablebynaivepower-
identifying m2H with s and taking the s → ∞ limit, we counting. This is not the case for both, the toy model
obtain lims→∞Fφ(s) = 2, which reproduces the non- with the interactionterm as in Eq.(7) and the Standard
decoupling constant in Eq.(2). Modelintheunitarygauge,sothattheuseoftheunsub-
It is straightforward to reproduce this result in the tracted dispersion relations in both of these cases leads
dispersive approach. Indeed, by unitarity the imaginary to incorrect results.
part of the γγ φ†φ 0 amplitude is We elaborate on the last statement. Suppose that we
h | | i
do not perform the integration by parts in the interac-
2Im γγ φ†φ0 = dLips(p ,p ,K )Mγγ, (11) tiontermEq.(7)anduseitdirectly tocompute H γγ
h | | i Z 1 2 12 φφ →
amplitude. Roughly speaking, this is a situation that
where dLips denotes the elementof standardLorentzin- corresponds to calculations in the unitary gauge in the
variant phase space of two φ particles with momenta p full Standard Model. The imaginary part of this ampli-
1
and p and Mγγ is the amplitude of φ(p )+ φ¯(p ) tude is given by the imaginary part of the physical form
2 φφ 1 2 →
γ(k1)+γ(k2) annihilation, factor Fφ(s). If we now use this imaginary part in the
unsubtracteddispersionrelation, we obtaina resultthat
Mγγ =2e2 (ǫ ǫ ) (p1ǫ1)(p2ǫ2) (p1ǫ2)(p2ǫ1) . (12) differs from Fφ(s) in Eq.(16) by a subtraction constant
φφ n 1 2 − (p1k1) − (p1k2) o
ds
1
After integrationoverthe phasespaceoftwoφparticles, Im[F (s )]=2. (17)
φ 1
−Z s
we obtain 1
β 1+ 1 β We will now check that we can get the correct result
ImΦ(s)=−πθ(1−βφ) sφ ln1 p1−βφ . (13) for the form factor using the unsubtracted dispersionre-
− − φ lations even if we work with the non-renormalizable in-
p
Weusethisresultintheunsubtracteddispersionrelation teraction in Eq.(7) but regulate the theory in the ultra-
and find violet, in spite of the fact that the final result turns out
to be finite.
∞
1 ds ImΦ(s ) 2 AsimpleformoftheUVregularizationisanintroduc-
1 1
Φ(s)= = 1 β f(β ) , (14)
π Z s s i0 s − φ φ tion of Pauli-Villars fields. In our case it means that a
4m2 1− − (cid:0) (cid:1) contribution of the loop of charged scalar particles with
φ
4
the mass m should be subtracted from the loop of If we consider value of √s that are much smaller than
PV
φ-fields. The introduction of the Pauli-Villars regulator thescaleµexp(1/(2ǫ)),whichplaysaroleoftheUVcut-
leadstoachangeintheimaginarypartoftheformfactor off, ∆ F (s) adds the required constant 2 to F (s), that
ǫ φ φ
ImF (s) at s 4m2 , is reconstructed from the unsubtracted and unregulated
φ ≥ PV
dispersion relation.
1+√1 β
∆ [ImF ]=πθ(1 β )β ln − R , (18) Byconsideringthetoymodelforthe interactionofthe
PV φ R R
− 1 √1 βR longitudinally-polarizedgaugebosonswiththeHiggsbo-
− −
where β =4m2 /s. We find son,weshowedthatthereasonfortheappearanceofthe
R PV constant contribution to the H γγ form factor is the
∞ fact that interactions between t→he Higgs boson and the
1 ds ∆ [Im[F (s )]
1 PV φ 1
∆ F (s)= =2β f(β ). electroweak bosons in the unitary gauge are not renor-
PV φ R R
π Z s s i0
4m2 1− − malizable by power counting. The correct result can be
R obtainedbyeitherintroducingexplicitultravioletregula-
(19)
We are interested in the limit β =4m2 /s ; in tor,in spite ofthe factthat the computationofthe form
R PV → ∞ factorleadstoafinite result,orbyswitchingtoaformu-
that limit
lationofthetheorywhereinteractionsarerenormalizable
∆PVFφ(s)=2βRf(βR) 2, (20) by power counting. We also note that the UV regular-
→
ization leads automatically to a result that is consistent
which is the same constant that appears in Eq.(17).
withthe low-energyconstraintwhichis F (s=0)=0 in
We will now demonstrate that the same result is ob- φ
our toy model . As we saw, imposing this condition was
tained if dimensional regularization is used for the UV
sufficient for the dispersive reconstruction. All of these
cut-off. It is convenient to choose the photon polariza-
approaches can be used to compute the complete form
tion vectors as
factorF (s)inthe dispersiveapproach;inthenextSec-
W
ǫ =(0,1, i,0)/√2, ǫ ǫ = 1, (21) tion, we will do that by performing the dispersive com-
1,2 1 2
± · − putation in a renormalizable R gauge and studying if
ξ
in the reference frame where the photon momenta are the unitary gauge result is recoverdin the ξ limit.
→∞
along z axis. Then from the unitarity relation
2Im γγ H = dLips(p ,p ,K ) φφ¯ H Mγγ (22) III. IMAGINARY PART AND THE
h | i Z 1 2 12 h | i φφ RENORMALIZABLE GAUGE
we obtain
ImFφ=(4π)2µ2ǫZdLips(p1,p2,K12)n−1+ pp2x20−+pp2z2yo, uwneOsuhubarvtregaocsaetleednis,attnohdiscourmnerqpeuugitureelastthaeedfofdorimrsmpuelrafasticiootnonrroeFflWatth(ieosn)tshu.esoiAnrygs
(23)
whererenormalizabilityisapparent. Hence,weareforced
where µ is the normalization point and the factor µ2ǫ
to consider the R gauges.
restores a correct dimension. ξ
At d = 4 this expression shows that ImF m2 Similar to what has been done before, we will calcu-
φ ∝ φ latetheformfactorusingdispersionrelations;forthiswe
and leads to ImF given in Eq.(15). At d = 4 2ǫ we
φ − willneedtocomputeitsimaginarypartfors=m2 . Itis
should split the φ particle momentum pµ into the four- importanttorecognizethattheamplitudetha6 tdeHscribes
dimensionalpartandthepartp livinginremaining 2ǫ
ǫ − the transition of the off-shell Higgs to two photons be-
dimension. To determine anadditionalpart∆ ImF we
ǫ φ comes gauge-dependent; this applies to the dependence
put m =0. Then,
φ of the imaginary part on the electroweak gauge param-
n2 eter ξ as well as to the loss of the transversality of the
∆ ImF =(4π)2µ2ǫ dLips(p ,p ,K ) ǫ
ǫ φ Z 1 2 12 sin2θ electromagnetic current.
The second problem is easy to avoid by choosing the
=ǫ(4π)2µ2ǫ dLips(p1,p2,K12) (24) non-linearRξ gaugewheretheelectromagneticgaugein-
Z
variance is explicitly maintained. To this end, we can
Ω(d−1) s −ǫ s −ǫ use
= ǫ 2πǫ .
2d−3(2π)d−4 (cid:16)µ2(cid:17) ≈ (cid:16)µ2(cid:17) 1
= D Wµ iξm φ2, (26)
Ipnatnhdeske,eaqnudatiΩo(nds−1n)ǫis=thpeǫ/s|opl|i,dthanegalengilnedθ is1bestpwaetieanl Lgauge −ξ | µ − W |
−
dimensions. Thecorrectiontotheimaginarypartinduces as the gauge fixing term with Dµ = ∂µ ieAµ. If we
−
the following change in Fφ choose this gauge, some Feynman rules of a linear Rξ
gaugeget modified but this is not important for us. The
∞
∆ F (s)=1 ds1∆ǫIm[Fφ(s1)]=2 s −ǫ. (25) importantpointisthatthegauge-fixingtermLgaugeelim-
ǫ φ π Z s1 s i0 (cid:16)− µ2(cid:17) inates the φWγ vertex. In addition, it is important for
4m2R − − what follows that in the Rξ gauge Lagrangian, linear or
5
not, the only interaction vertex that explicitly contains To compute this integral, we use the expression for the
m2 is the interaction vertex involving the Higgs boson imaginarypartasinEq.(32)andrealizethatthe disper-
H
and the two Goldstone φ-fields sion integral of Im[Fc (β)] reconstructs Fc , see Eq.(2).
W W
We also substitute m2 s, to conform with the pre-
m2 H →
= HHφ†φ. (27) vious notations, and write the final result for the form
LHφ†φ − v factor as
The interactionof the φ-fields with the photons are that
∞
of the scalar QED and follow from the Lagrangian 1 ds
F (s)=Fc (s) 1 Im[F (ξβ )]
= D φ2. (28) W W − π Z s1 φ 1 (34)
Lφ | µ | 4ξm2
W
The final remark that we need to make is that the mass =Fc (s)+2.
W
squared of the Goldstone boson φ is m2 =ξm2 .
φ W
Aswewillnowshowthisinformationisallthatweneed We note that the integral over ImF in the above equa-
φ
to perform the computation of F (s), given the results tion is ξ independent and coincides with a similar inte-
W
that we already presented in Section II. To facilitate the gralin the toy model, see Eq.(17). Ingeneral,the above
computation of the imaginary part of the form factor computation shows that the form factors calculated in
F (s), we use the already-mentioned fact that among the R gaugeandthe unitary gaugediffer by a constant,
W ξ
many diagrams that contribute to the form factor, the related to the contribution of Goldstone bosons to the
onlyinteractionvertexthatisproportionaltom2 comes imaginary part of the H γγ amplitude. The mass of
from the Hφ+φ interaction term in Eq.(27). MHotivated theGoldstonebosonm2 =→ξm2 remainsarbitraryinthe
φ W
by this observation, we write the imaginary part as the calculation, so that the limit ξ can be studied. It
→ ∞
sum of two terms follows from Eq.(34) that the Goldstone boson contribu-
tionto F (s) does not decouple inthe limit ξ ; this
Im[FRξ(r ,β,ξ)]=r G (β,ξ)+G (β,ξ), (29) W →∞
W H H 1 2 feature leads to a difference between the results of the
where r =m2 /s and β =4m2 /s. The functions G calculations in the unitary and the Rξ gauges. Finally,
H H W 1,2
the Goldstone boson contribution does not have a pole
canbe computed directly fromFeynmandiagrams,how-
ats=m2 and,therefore,doesnotcontributetothedis-
everthis isnotnecessary. Indeed, there isone constraint H
on the two functions that is available to us since if we continuity of the form factor; for all practical purposes,
compute the imaginary part for s = m2 , we should re- it is a subtraction term.
H
cover the ξ-independent result for the imaginary part in
the unitary gauge. This implies
IV. CONCLUSIONS
G (β,ξ)+G (β,ξ)=Im[Fc (β)], (30)
1 2 W
where Im[Fc (β)] is the imaginary part of the form Inthispaper,wediscussedhowthedispersionrelation
W
computation of the H γγ decay amplitude through
factor in the unitary gauge defined in Eq.(4). Next,
→
since the only m2 -dependent term in the calculation of the W-boson loop can be reconciled with the results of
H
Im[FRξ(r ,β,ξ)] comes from the diagrams with φ†φ- thediagrammaticcomputationsthatemploydimensional
W H regularization. As was pointed in Ref. [13], if one com-
intermediatestate,wecanreadoffG fromtheimaginary
1 putestheimaginarypartoftheformfactorF (s)inthe
part of the form factor F in Eq.(15). We obtain W
φ four-dimensional space-time and then uses it in an un-
G (β,ξ)=Im[F (β =ξβ)]. (31) subtracted dispersion integral to calculate the full form
1 φ φ
factor,oneobtainstheresultthatdiffersfromthecorrect
We can use the above constraints to rewrite the imag- onebyaconstantterm. Theappearanceofthisconstant
inary part of the form factor in a general Rξ gauge in a can be interpreted as the need to perform a subtraction
useful way. By adding and subtracting G1, we find in a finite dispersion integral which is quite unusual.
Im[FRξ(r ,β,ξ)]=Im[Fc (β)]+(r 1)Im[F (ξβ)]. (32) We have shown that the need to perform the subtrac-
W H W H− φ tion in the dispersion integralfor form factors computed
Thesecondtermhereshowsthatitistheoff-shellbehav- intheunitarygaugeisaconsequenceofthefactthatthe
iorthatdifferentiatesthesingularunitarygaugefromthe SMinthe unitarygaugeis notexplicitly renormalizable.
renormalizable R gauge. If one regularizes the (apparently finite) calculation by
ξ
We can now restore the real part of the form factor either introducing explicit UV regulator or starts from
fromitsimaginarypartusingtheunsubtracteddispersion the formulation of the theory where the renormalizabil-
relationfors=m2 . Theresultofthecalculationshould ityis,infact,apparent,onealwaysobtainsanadditional
H
be correctsincethe theoryinR gaugeisrenormalizable contribution to the real part of the form factor. For val-
ξ
by power-counting. To this end, we need to compute uesofsbelowtheultravioletcut-off,thiscontributionis,
essentially,aconstantandcanbe interpretedasthesub-
F (m2 )= 1 ds1 Im[FWRξ(rH,β1,ξ)]. (33) traction term in the dispersion relation. Unfortunately,
W H π Z s m2 unregulated and unsubtracted dispersion relation calcu-
1− H
6
lations, that employ unitary gauge, do not seem to be and L. Tancredi for useful conversations.
sufficient even if they lead to finite results.
Acknowledgments K.M. would like to thank F. Caola
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