Diphoton decay of the higgs from the Epstein–Glaser viewpoint
Abstract
We revisit a nearly 10-year old controversy on the diphoton decay of the Higgs particle. To a large extent, the controversy turned around the respective merits of the regularization techniques employed. The novel aspect of our approach is that no regularization techniques are brought to bear: we work within the Bogoliubov–Epstein–Glaser scheme of renormalization by extension of distributions. Solving the problem actually required an expansion of this method’s toolkit, furnished in the paper. Duch, P.; Dütsch, M.; Gracia-Bondía, J.M.
Full text
Eu . Phys. J. C (2021) 81:131
h ps://doi.o g/10.1140/epjc/s10052-021-08898-z
Regula A icle - Theo e ical Physics
Dipho on decay o he higgs om he Eps ein–Glase iewpoin
Paweł Duch1,2, Michael Dü sch3, José M. G acia-Bondía5,4,a
1Ins i u ü Theo e ische Physik, Uni e si ä Leipzig, 04103 Leipzig, Ge many
2Max-Planck Ins i u e o Ma hema ics in he Sciences, 04103 Leipzig, Ge many
3Ins i u ü Theo e ische Physik, Uni e si ä Gö ingen, 37077 Gö ingen, Ge many
4CAPA and Depa amen o de Física Teó ica, Uni e sidad de Za agoza, 50009 Za agoza, Spain
5Labo a o io de Física Teó ica y Compu acional, Uni e sidad de Cos a Rica, San Ped o 11501, Cos a Rica
Recei ed: 9 Augus 2020 / Accep ed: 21 Janua y 2021
© The Au ho (s) 2021
Abs ac We e isi a nea ly 10-yea old con o e sy on
he dipho on decay o he Higgs pa icle. To a la ge ex en ,
he con o e sy u ned a ound he espec i e me i s o he
egula iza ion echniques employed. The no el aspec o ou
app oach is ha no egula iza ion echniques a e b ough
o bea : we wo k wi hin he Bogoliubo –Eps ein–Glase
scheme o eno maliza ion by ex ension o dis ibu ions.
Sol ing he p oblem ac ually equi ed an expansion o his
me hod’s oolki , u nished in he pape .
Die Eule de Mine a beginn e s mi de einb echenden Dämme ung
ih en Flug
– Geo g Wilhelm F ied ich Hegel
1 In oduc ion: he con o e sy
Due o i s cleanness, i is ha d o o e s a e he expe imen-
al impo ance o he decay o he Higgs pa icle in o wo
pho ons. I goes mainly ia i ual W-bosons, he hea ie
cha ged pa icles o fla ou dynamics. The ampli ude o his
con ibu ion was calcula ed o he fi s non- anishing o de
(one-loop, cubic in he couplings) long ago in he ligh -higgs
limi [1] – and hen “exac ly” in [2]. The accep ed esul was
confi med many imes – see [3] o a pa icula ly cle e cal-
cula ion. I does no anish in he hea y-higgs limi – which
seems o fly in he ace o he “decoupling heo em” (DT) in
[4], as o en unde s ood.
Much mo e ecen ly, hose calcula ions we e ques ioned
in [5,6]. The ensuing deba e highligh s he heo e ical el-
e ance o his decay. The au ho s o hese pape s made he
poin ha , since he higgs canno couple di ec ly o he pho-
To he memo y o Gün e Scha and Raymond S o a.
ae-mail: jmgb@uniza .es (co esponding au ho )
ons, he one-loop con ibu ion mus be fini e: he e a e no
couplings equi ing “ eno maliza ion”. The oundabou p o-
cedu es h ough “ eno malizable gauges”, hey concluded,
we e unnecessa y. Eschewing dimensional egula iza ion,
hey ecompu ed he ampli ude in he uni a y gauge o elec-
oweak (EW) heo y. They did ob ain a esul di e ing om
he s anda d one by an addi i e cons an , which shows up
o ins ance in he hea y-higgs limi – whe eby hei esul
is equal o ze o.
The e was no sho age o ejoinde s [7–14] o[5,6]. The
au ho s o [9] a e he ones o he o iginal calcula ion [2].
Those pape s made se e al poin s, some a he implausibly
a guing ha a a gi en poin in he calcula ion in [6] elec o-
magne ic gauge in a iance is los , and c i icizing he in e -
p e a ion o he DT made in [5,6]. The e was in some o
he he ejoinde s an explana o y eliance on he heu is ics
o he B ou –Engle –Higgs mechanism, h owing back he
so-called “equi alence heo em” (GBET).
The c i icisms ecei ed a ejoinde in u n in [15]. This
la e pape a gues by he example ha wo compu a ions
o he same p ocess in di e en gauges (Rξ e sus uni a y
gauge) may yield di e en esul s. This goes agains he
g ain, al hough o cou se no heo em con adic s such an
asse ion. Meanwhile, a dispe sion ela ion calcula ion ca -
ied ou in [16] appea ed o suppo he con en ions o [5,6],
and go in u n a – qui e hough ul – ejoinde in [17]. Mo e
ecen pape s dealing wi h he same o ela ed issues a e
[18,19].
By and la ge, he majo i y’s opinion and he expe imen al
esul s [20] suppo he fi s ally. On he o he hand, om
he heo e ical poin o iew he si ua ion is s ill obscu e: i
had o be so, since bo h pa ies d aw s eng h om di e en
casuis ics o he calcula ions in pe u ba i e quan um field
heo y.
The deba e abou he uses and abuses o he uni a y gauge
and he ole o he decoupling and equi alence “ heo ems”
0123456789().: V,- ol 123
131 Page 2 o 25 Eu . Phys. J. C (2021) 81:131
is o be salu ed as salu a y. And i is sa e o admi ha up o
now we lack a ull concep ual unde s anding o he p oblem.
The cleanes way o add ess his lack is su ely o enounce
all he heu is ics o ma hema ically ill-defined quan i ies, in
a ou o a me hod in which he e can be no a gumen on he
meaning o infini e e ms. Such is he uly (pe u ba i ely)
s ingen scheme by Bogoliubo , Eps ein and Glase (BEG)
o “ eno maliza ion” wi hou egula iza ion, by ex ension o
dis ibu ions.
In he BEG cons uc ion, go e ned by causali y, he e is
no such hing as a “di e gen diag am”: one ne e encoun e s
infini ies. The e may, howe e , emain in he ex ension p o-
cedu es some addi i e ambigui y, ha can be es ic ed (bu
no always comple ely emo ed) by physical p inciples. This
is a he o be ega ded as a s eng h o he BEG pa adigm,
because hose ambigui ies exp ess p ecisely how, and o wha
ex en , he heo y is de e mined by he undamen al p inci-
ples o pe u ba i e QFT.
A pa icula ad an age o he induc i e BEG cons uc ion
[21] o he ( unc ional) S-ma ix is ha in p inciple one is
allowed o s ay on configu a ion space, which makes mo e
anspa en he physics unde examina ion. Fo examples o
calcula ions wi hin he BEG scheme explici ly ca ied ou in
configu a ion space, see [22]o [23, Sec . 3.5]. I is only o
compu a ional con enience ha we swi ch a some momen
o momen um space.
Since we do no deal in infini ies, we e e as no mal-
iza ion o he p ocesses aking he place o egula iza ion
and eno maliza ion in he BEG amewo k. Fo i s ela i e
pauci y o diag ams, in ou con ex he unde lying a gumen
is made clea e by wo king mos ly in he uni a y gauge –
whe eupon only he physical pa icles’ da a a e b ough o
bea .1
To summa ize, so a : we we e mo i a ed o ackle his
subjec by wonde ing why mos knowledgeable people, bo -
owing di e en (bu all appa en ly sound) me hods o wo k
on such a basic p ocess, we e di ided on he ou come. I all
u ns a ound a sub le y unco e ed by use o he BEG no -
maliza ion. Tha condenses he pu pose o he p esen pape .
1.1 Main esul s and plan o he a icle
In Appendix A we in oduce ou con en ions and no a ions,
ecalling a ew well-known o mulae o QFT needed in he
body o he pape , in pa icula he p opaga o s o he EW
heo y in he uni a y gauge. Le mhdeno e he mass o he
higgs h. The ampli ude coming om he one-loop calcula-
ions may be quo ed as [25–27]:
1The pape [24] dwells use ully on he subjec o he Rξ- e sus-uni a y
gauges, leaning o demons a e he alidi y o he la e a he quan um
le el.
A=gα
2πMF1(ρ)Pμν,
wi h α he fine s uc u e cons an , g he EW coupling con-
s an , M he mass o he in e media e W-boson and ρ:=
m2
h/4M2. The pola iza ion ac o Pμν, eflec ing elec omag-
ne ic gauge in a iance (EGI) o A,2is w i en in his pape
as
Pμν := (k1k2)gμν −k1νk2μ;(P•νk1)=(Pμ•k2)=0,
(1.1)
wi h k1,k2 he ou going pho ons’ momen a. Finally, o he
dimensionless ac o :
F1(ρ) := 2+3
ρ+3
ρ2−1
ρ (ρ). (1.2)
Now ha we a e a ha , we quo e as well he compa able
esul o a cha ged scala pa icle o mass Ma he place o
he W-boson:
F0(ρ) =1
ρ1− (ρ)
ρ;so ha
F1(ρ) =3F0(ρ) +6 (ρ)
ρ+2.(1.3)
Fo he benefi o he eade coming o he subjec o his pape
o he fi s ime, Appendix B in oduces he dis ibu ion (ρ)
appea ing in bo h F1(1.2) and F0(1.3) – as well as in he
ampli ude o dipho on decay o h ia i ual e mions.
The bone o con en ion is ha he fi s summand 2 in (1.2)
should no be he e, acco ding o [5,6,16]. Rela ions (B.3)
and (B.6) ell us ha , as ρ↓0:
F1=2+3
ρ+6
ρ−3
ρ2ρ+ρ2
3+8ρ3
45 +···
=7+22
15 ρ+O(ρ2);
so F1(0)=7 and F1(∞)=2 om(1.2). P ecisely he
o me figu e is wha was calcula ed in he pape [1]. The
esul a gued by he “he e ics” in he con o e sy is F1−
2, so hei espec i e asse ions a e ins ead F1(0)=5 and
F1(∞)=0. Also, om (1.3): F0(0)=−1/3 and F0(∞)=
0.
Appendices A and B o his pape deal wi h con en-
ions and ma hema ical p e equisi es. The basics o he BEG
scheme a e ecalled in Appendix C. Unde s anding o he
BEG me hod is indispensable in wha ollows, and e en ead-
e s amilia wi h i a e ad ised no o miss ou e iew. The
ela ion be ween he no maliza ion p oblem by ex ension o
dis ibu ions (o by “dis ibu ion spli ing”) and dispe sion
in eg als is ea ed in i s Sec . 1. New esul s in his espec
a e equi ed, announced in he sho Sec . 2and p o ed in
Sec s. 3.2 and 3.3 o his pape . So o aficionados o BEG
2Tha is, ans e sali y o he ou going pho ons.
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Eu . Phys. J. C (2021) 81:131 Page 3 o 25 131
no maliza ion he e is no el y he e – whose in e es goes
beyond he pa icula p oblem ha mo i a ed i .
Sec ions 3and 4cons i u e he hea o he pape . The
scala model leading o F0is wo ked ou in Sec . 3.Oneis
able o pe o m he “adiaba ic limi ” o Eps ein and Glase
a an in e media e s ep, which simplifies compu a ions – his
is igo ously jus ified. This “ oy model” allows he eade
o amilia ize wi h he BEG cons uc ion o ime-o de ed
p oduc s in a ela i ely simple case. Fo i , he ambigui y in
he Eps ein–Glase esul can be disposed o , and he unique
ou come happens o coincide wi h he esul o a “nai e”
on-shell calcula ion, o he kind pe o med in [16].
Finally, in Sec . 4, we compu e he EW ampli ude, wo k-
ing fi s in he uni a y gauge. We s a in ea nes by illus a -
ing in his ele an ins ance he machine y o he BEG o -
malism in cons uc ing ime-o de ed p oduc s, a he lowes
non- i ial o de : om cubic in e ac ion e ices, iden ified
o ime-o de ed p oduc s a fi s o de in he couplings, we
de i e he qua ic, second-o de AAWW†- e ex.
I is ime o a e why he “no- eno maliza ion” a gumen
in [6] is no wa e igh . A di ec hγγ coupling in fla ou dy-
namics is o bidden also because o EGI. Thus o ob ain he
gene al ampli ude, which li es o -shell, one mus add o he
nai e calcula ions a polynomial in he ex e nal momen a,
o deg ee gi en by he singula o de o ha ampli ude.
Compu ing he 1-loop con ibu ion in he uni a y gauge by
he Eps ein–Glase me hod, we a i y his ac . To find he
coe ficien s o ha polynomial, beyond EGI he e we call
upon gauge-fixing independence o he on-shell ampli ude.
This locks in he inde e mina ion; and in he end we do
ob ain F1(ρ). Wi hin he uni a y gauge, a di e en a gumen
o he same pu pose is discussed a he end o his Sec . 4.
Sec ion 5is he conclusion.
2 The obs uc ion o dis ibu ion spli ing o null
momen a
Fo mula (C.16) in Appendix C is ou main wo kho se: in
momen um space he Eps ein–Glase dis ibu ion spli ing
amoun s o a dispe sion in eg al. Bu i pe ains o ema k
ha , by cons uc ion, p esc ip ions (C.14) and (C.16)a ein
p inciple alid only o imelike k. Thus, in o de o sol e
he p oblem in his pape , one has o un an ex a mile.
The explici spli ing p ocedu e in oduced he e exhibi s el-
e an no el ea u es: we ha e o compu e he cen al solu-
ion ac(k1,k2) o null momen a. Hence, one canno immedi-
a ely use he dispe sion in eg als (C.14)o (C.16). On ying
o wo k ins ead wi h he con olu ion in eg al (C.13), he e
appea s he p oblem ha , in spi e o k2
j=0, i gene ally holds
ha (kj− j)2= 0 because j∈V+;i does no su fice o
know he causal dis ibu ion d(k1,k2)only o k2
1=0=k2
2.
The nex sec ion sol es his p oblem o models such ha
0<(k1+k2)2<4M2and k0
1k0
2>0. The p oo ’s s a egy
is as ollows: s a ing om he dispe sion in eg al (C.14)
o k2
1>0, k2
2>0 and k0
1k0
2>0, we in end o show
ha d(k1,k2)is egula enough ha his in eg al commu es
wi h he limi (k2
1↓0∧k2
2↓0). The e o e he dispe -
sion in eg als (C.14) and (C.16) keep hei use ulness o
k2
1=0=k2
2: indeed, o compu ing ac(k1,k2)|k2
1=0=k2
2i
su fices o know d(k1,k2)only o k2
1=k2
2=0, because
k2
1=k2
2=0 implies ( k1)2=( k2)2=0 o all .
C ucially, in he esul ing dispe sion in eg als (C.14) and
(C.16) o k2
1=0=k2
2, he pa ame e ωis he singula o de
o he o -shell d(k1,k2). As a consequence, he gene al solu-
ion (p io o imposi ion o o he in a iance ules) o he dis-
ibu ion spli ing is ob ained by adding o ac(k1,k2)|k2
1=0=k2
2
a polynomial in k1,k2, in p inciple a bi a y, whose deg ee
is gi en by he singula o de o he o -shell ampli ude
d(k1,k2). Now, i equen ly happens ha he singula o de
o d(k1,k2)|k2
1=0=k2
2has a smalle alue. Consequen ly, i
may happen ha he equi ed dispe sion in eg al appea s o
be “o e sub ac ed” – i.e., i would be con e gen also o a
smalle alue o ω. Examples o his a e he “ oy model” in
he nex sec ion and he EW dipho on decay o he higgs in
he uni a y gauge (Sec s. 3.3 and 4.3, espec i ely).
These issues we e ealized by Raymond S o a, who, e e -
ing o he e y subjec p ocess o his pape , poin ed ou o
one o us ha he good beha iou o he abso p i e pa o he
o m ac o in ol ing Comp on sca e ing o he W-bosons
should no make one o ge ha BEG-gene a ed dispe sion
in eg als, jus as pe u ba i e eno maliza ion heo y in gen-
e al, applies o -shell. 3
3 Higgs o dipho on decay ia a cha ged scala field
The scala elec odynamics compu a ion leading o F0wo ks
like a kind o oy model, allowing he eade o amilia ize
wi h ou me hods in a less complica ed, al hough non- i ial
case. We de elop i in he p esen sec ion. No ice he ollow-
ing: in he Eps ein–Glase scheme he “seagull” e2AAϕϕ†-
e ex is de i ed by implemen ing EGI wi hin he cons uc-
ion ules o he me hod – as any o he pa o T2[28]. We
gi e ull de ails on how his comes abou o he qua ic e -
ex in he EW heo y in Sec . 4.1. The game he e would be
simila , only simple . The eade is ad ised o keep in mind
he me hods and s anda d no a ions ecalled in Sec . C.2.
3P i a e communica ion, ea ly 2013.
123
131 Page 4 o 25 Eu . Phys. J. C (2021) 81:131
3.1 A causal dis ibu ion on-shell
The s a ing poin is gi en by he lowe o de ime-o de ed
p oduc s (TOPs):
T1(x3)=gM h(x3)ϕ(x3)ϕ†(x3);
T1(xj)=−ieAλ(xj)ϕ†(xj)←→
∂λϕ(xj), j=1,2;
T2(x1,x2)
=−e2Aμ(x1)Aν(x2)ϕ†(x1)∂
μF(x1−x2)∂
νϕ(x2)
−∂μϕ†(x1)F(x1−x2)∂
νϕ(x2)
+ϕ†(x1)∂ν∂μF(x1−x2)+igμν δ(x1−x2)ϕ(x2)
−∂μϕ†(x1)∂
νF(x1−x2)ϕ(x2)
+(x1↔x2)+T1(x1)T1(x2)
+[i ele an loop diag am e ms],
whe e Fdeno es he Feynman p opaga o (A.3).
F om ou o mulas (C.4) and (C.5):4
D3(x1,x2,x3)=−[T1(x1), T2(x2,x3)]
−[T1(x2), T2(x1,x3)]+[T2(x1,x2), T1(x3)].(3.1)
Because he pho ons emi ed a x1,x2a e on-shell, only
he hi d commu a o is ele an he e – in he language o
Cu kosky ules, one needs only he iangle cu sepa a ing
he higgs e ex om he p opaga o connec ing he pho ons.
We gi e he explana ion u he on. F om he gene al o mula
o he an ich onological p oduc (C.3), we pa icula ly know
ha
T1(x1)=T1(x1);T2(x1,x2)=−T2(x1,x2)
+T1(x1)T1(x2)+T1(x2)T1(x1). (3.2)
Fo he same easons jus a gued, only he connec ed ee
diag am pa o he T2(x1,x2)summand in T2(x1,x2)con-
ibu es.
A mos con enien pa allel o he coming calcula ion is
he ea men o he e ex unc ion in QED in he fi s edi ion
o he fini e QED book by Scha [29, Sec . 3.8]. Going o
he con ac ions, b inging in he e ices and he p opaga o s
(A.2), (A.4), apa om a ac o 4ge2Mwe ob ain:
Aμ(x1)Aν(x2)h(x3)−(1)∂μF(1−2)∂ν−(2)
−∂μ−(1)∂νF(1−2)
−(2)
−∂μ−(1)F(1−2)∂ν−(2)+−(1)∂μ∂νF(1−2)
+igμν δ(1−2)−(2)
−[ he same ou e ms wi h − eplaced by +]+···
=: Aμ(x1)Aν(x2)h(x3)dμν(1,2),
4The Dna e always linea combina ions o commu a o s.
whe e 1 ≡y1:= x1−x3,2≡y2:= x2−x3. He e and
u he down, he do s s and o he e ms coming om he
o he wo cu s and u he e ms no con ibu ing o he on-
shell ampli ude. No e he ad e ised addi ional +igμν δ o
∂μ∂νF, co esponding o he “closed seagull” o fish-like
diag am con ibu ion o he h→2γdecay in his model.
We now p oceed o momen um space, whe e compu-
a ions a e ca ied ou mo e simply. Fo Fou ie ans o -
ma ions, consul he con en ion (C.7). In his sec ion and
he nex , in keeping wi h physicis s’ no a ion, we indi-
ca e he ans o ms by jus exhibi ing he a iables, namely:
dμ(k1,k2)≡ˆ
dμ(k1,k2). We ob ain
dμν(k1,k2)=1
(2π)24(Iμν
+−Iμν
−)+2kν
2(Iμ
+−Iμ
−)
−2kμ
1(Iν
+−Iν
−)−kμ
1kν
2(I+−I−)
−i
(2π)2gμν (J+−J−)+··· (3.3)
wi h he in eg als
I{·|μ|μν}
±(k1,k2):= d4k{1|kμ|kμkν}±(k1−k)
F(k)
±(k+k2),
J±(k1,k2):= d4k±(k1−k)
±(k+k2), (3.4)
whe e he J±- e m is he con ibu ion o he fish-like diag am.
Keep in mind ha he e ms belonging o A
3:= A3−T3a e
hose coming om he in eg als I·|μ|μν
−and J−, whe eas he
con ibu ion o R
3:= R3−T3is gi en by he in eg als I·|μ|μν
+
and J+.
Fo ou pu poses one may pe o m he adiaba ic limi
al eady a his s age. Since all in e nal lines o he diag ams
co espond o massi e fields, his limi can be done he e in
he nai e way by jus se ing he swi ching unc ion g(x)
in (C.1) o1:
dx1dx2dx3Aμ(x1)Aν(x2)h(x3)dμν (x1−x3,x2−x3)
=(2π)2dk1dk2h(k1+k2)Aμ(−k1)Aν(−k2)dμν (k1,k2).
(3.5)
In his limi he momen a k1and k2become he momen a o
he ex e nal pho ons: k2
1=k2
2=0.
F om now on, we compu e dμν(k1,k2)|k2
1=0=k2
2.We ewe
o ha e included he o he cu s in (3.1)o T1T1T1- e ms, he e
would appea ±- ype p opaga o s a he place o he Feyn-
man p opaga o s abo e. The o me a e ∼δ(k2−M2), wi h
kdeno ing he in e nal momen um a iable in he loop: so o
speak, in con as wi h he Feynman p opaga o s, he ±a e
123
Eu . Phys. J. C (2021) 81:131 Page 5 o 25 131
“always on-shell”, e en wi hin loops.5Thus no u he in e -
nal momen a can be on-shell: assuming k2=M2one ob ains
(k1−k)2=M2−2(k1k)= M2; simila ly o (k+k2).6
Scala in eg als I±.
We ha e o compu e
I∓(k1,k2)
:= i
(2π)4d4kθ(∓(k0
1−k0)) δ((k1−k)2−M2)
×1
k2−M2+i0θ(∓(k0+k0
2)) δ((k+k2)2−M2).
Le us make a change o a iable q:= k+k2, and in oduce
P:= k1+k2, no ing o la e pu poses ha P2=2(k1k2).
One ob ains he in eg al:
d4qθ(∓(P0−q0)) δ((P−q)2−M2)
1
(q−k2)2−M2+i0θ(∓q0)δ(q2−M2). (3.6)
I ollows ha I∓(k1,k2)∝θ(∓P0)θ(P2−4M2), and ha
sgn k0
1=sgn k0
2 o P2≥4M2.
Pe o ming he q0-in eg a ion and using he no a ion
Eq:= |q|2+M2, we ex ac
I∓(k1,k2)=i
(2π)4θ(∓P0)θ(P2−4M2)
×d3q
2Eq
θ(∓(P0−q0)) δ((P−q)2−M2)
1
(q−k2)2−M2+i0q0=∓Eq
.
Since P2>0, one may choose a pa icula Lo en z ame
such ha
P=(P0,0);hence
k1=−k2,k0
1=∓|k1|=∓|k2|=k0
2=1
2P0.(3.7)
Taking in o accoun q2=M2, we obse e ha (P−q)2−
M2=2P0(1
2P0−q0), which yields
δ((P−q)2−M2)=δ(q0−1
2P0)
2|P0|=δ(Eq−1
2|P0|)
2|P0|,
by using q0=∓Eq. Fo la e aims, we poin ou ha in he
chosen ame his dis ibu ion implies q0=k0
2; hence
kP =(q−k2)P=(q0−k0
2)P0=0.(3.8)
5This poin is made in [30,Sec .6.4].
6Compa e he discussion a e [29, Eq. (3.8.24)].
F om ∓q0=∓
1
2P0comes ∓(P0−q0)=∓
1
2P0>0.
The e o e he ac o θ(∓(P0−q0)) is edundan . Changing
he in eg a ion a iables,
d3q···=∞
M
dEqEqE2
q−M2dq···,
he Eq-in eg a ion can i ially be done, and we a e le wi h:
I∓(k1,k2)=iθ(∓P0)θ(P2−4M2)
(P0)2−4M2
(2π)48|P0|dq
(q−k2)2−M2+i0q0=P0/2
.(3.9)
Le αbe he angle be ween k2and q, and le z:=
cos α.Due oq2=M2,k2
2=0, |q|=E2
q−M2=
1
2P2
0−4M2and ela ions (3.7) and (3.8), we ob ain
(q−k2)2−M2=−2(k2q)=−2(k0
2q0−|q|·|k2|z)
=a
2(−a+bz), (3.10)
whe e
a:= |P0|>0,0≤b:= (P0)2−4M2<a.
We poin ou ha (−a+bz)<0 o all z∈[−1,1]: he e
is no in a ed p oblem in ou iangle g aph. The emaining
q-in eg al can be easily compu ed:
4π
a1
−1
dz
−a+bz
=4π
|P0|(P0)2−4M2log (P0)2−|P0|(P0)2−4M2−2M2
2M2.
(3.11)
To ob ain he esul in a gene ic Lo en z ame, eplace (P0)2
by s:= P2=2(k1k2),so
I∓(k1,k2)=iθ(∓P0)θ(s−4M2)
4(2π)3slogs−s(s−4M2)
2M2−1
=: θ(∓P0)θ(s−4M2)F(s). (3.12)
The esul o J±(k1,k2)can be ead o om (3.9)by
omi ing he Feynman p opaga o i(2π)−2((q−k2)2−M2+
i0)−1. One ob ains o he con ibu ion o he J-in eg als:
J±(k1,k2)=1
8πθ(±P0)θ(s−4M2)1−4M2/s.
Vec o in eg als Iμ
∓.
Fo he same easons as o he scala in eg al, i mus
hold ha Iμ
∓(k1,k2)∝θ(∓P0)θ(s−4M2). F om Lo en z
co a iance and Iμ
±(k1,k2)=−Iμ
±(k2,k1)i ollows
Iμ
∓(k1,k2)=θ(∓P0)θ(s−4M2)(kμ
1−kμ
2)G(s)
123
131 Page 6 o 25 Eu . Phys. J. C (2021) 81:131
o app op ia e G(s). An immedia e consequence is IμPμ=
0. To p ocu e G(s), compu e
k2,μ Iμ
∓(k1,k2)=1
2θ(∓P0)θ(s−4M2)sG(s)
=−i/8(2π)3θ(∓P0)θ(s−4M2)1−4M2/s
The second equali y is ob ained by compa ing wi h he scala
in eg al: he e is an ex a ac o (k2k)=(k2q)=−a(−a+
bz)/4, whe e (3.10) is used. Then he q-in eg al becomes
i ial. Thus we glean
G(s)=−i
32 π3s1−4M2/s.(3.13)
Tenso in eg als I μν
∓.
P oceeding analogously o he ec o in eg als, one a gues
ha
Iμν
∓(k1,k2)=θ(∓P0)θ(s−4M2)(kμ
1kν
1+kμ
2kν
2)A(s)
+(kμ
1kν
2+kμ
2kν
1)B(s)+gμν C(s).
We need h ee independen iden i ies o compu e A(s),B(s)
and C(s). A fi s one is:
Iμν
∓k2μk2ν=θ(∓P0)θ(s−4M2)A(s)s2/4
=θ(∓P0)θ(s−4M2)−i
25(2π)3s1−4M2/s.
(3.14)
The second equali y is ob ained by a modifica ion o he
compu a ion o he scala in eg al: he e is he ex a ac o
(k2k)2=a2(−a+bz)2/16. This yields A(s)=G(s)/2.
A second iden i y is gi en by he ace. The esul is again
ob ained by compa ing wi h he compu a ion o he scala
in eg al: he e is an addi ional ac o k2=(q−k2)2=
M2−2(k2q)=M2−2(kk2), hence
Iμ
∓,μ =θ(∓P0)θ(s−4M2)(sB +4C)=M2I∓−2k2,μ Iμ
∓.
A hi d iden i y ollowing om (3.8) eads:
Iμν
∓Pν=θ(∓P0)θ(s−4M2)Pμ(A+B)s/2+C=0.
Pulling oge he hese esul s, one a i es a
B(s)=−M2F(s)/sand C(s)=M2F(s)/2−sG(s)/4.
•A his poin we a e able o show ha he iangle plus
fish-like pa s cons i u e a gauge-in a ian quan i y. Fo ha ,
inse he esul s al eady known o he in eg als in o (3.3),
ob aining:
dμν(k1,k2)k2
1=0=k2
2=sgn(P0)θ(s−4M2)
(2π)2kμ
1kν
2[4G(s)
−(1+4M2/s)F(s)]
+2M2gμν F(s)−kν
1kμ
2
4M2
sF(s)
=sgn(P0)θ(s−4M2)4M2
(2π)2Pμν F(s)
s.(3.15)
The kμ
1kν
2- e ms ha e been d opped in he las iden i y, due
o kμAμ(−k)=0. The emainde is elec omagne ically
gauge-in a ian . In oducing he dimensionless a iable
˜ρ:= s
4M2=P2
4M2,
keeping in mind o mula (3.12), and on use o (B.5), equa ion
(3.15) can be ew i en as
dμν
gi (k1,k2)k2
1=0=k2
2:= isgn(P0)θ(˜ρ−1)
(2π)5Pμν b(˜ρ)
(3.16)
wi h
b(˜ρ) := 1
16 M2˜ρ2log2˜ρ−2˜ρ( ˜ρ−1)−1
=− 1
16 M2˜ρ2log 1+1−˜ρ−1
1−1−˜ρ−1,
whe e ‘gi’ s ands o he gauge in a ian pa . The singula
o de o dμν
gi k2
1=0=k2
2
is ω=−2 by powe coun ing; whe eas
o he o -shell dμν(k1,k2) he alue is ω=0.
3.2 Regula i y o abso p i e pa s in momen um space
This subsec ion is de o ed o p o e essen ial egula i y
p ope ies o he o -shell d-dis ibu ion, mo e p ecisely o
dμν(k1,k2), o (k1,k2)∈V:= V+ {0}×2∪V− {0}×2.
We look a he e ms coming om (3.3) by means o (3.4).
In oducing he new in eg a ion a iable q:= −k+1
2(k1−
k2), he in e nal lines’ momen a a e
q1=q+1
2P,q2=q−1
2P,q3=q−1
2(k1−k2),
(3.17)
and one sees ha he conside ed e ms a e all o he ype
Hμν(k1,k2)
:= d4qθ(q0
1)θ(−q0
2)
−θ(−q0
1)θ(q0
2)δ(q2
1−M2)δ(q2
2−M2)hμν(k1,k2,q)
M2−q2
3
(3.18)
123
Eu . Phys. J. C (2021) 81:131 Page 7 o 25 131
o (k1,k2)∈V1:= V {0}×2wi h V:= V+∪V−, and
whe e hμν :R4×3→Cis a polynomial o deg ee 2. We
ha e used ha o (k1,k2)∈V1i holds ue ha
d4qθ(q0
1)θ(−q0
2)
−θ(−q0
1)θ(q0
2)δ(q2
1−M2)δ(q2
2−M2)δ(q2
3−M2)=0.
(3.19)
This las ela ion can be a gued as ollows:7 he a ious θ-
and δ-dis ibu ions yield he es ic ions (q1,q2)∈(H+
M×
H−
M)∪(H−
M×H+
M)and q3∈H+
M∪H−
M; aking mo eo e
in o accoun ha q3=q2+k2and q3=q1−k1, i ensues
ha he a ious es ic ions on q3a e no compa ible.
The same iden i y implies ha e ms o he kind T1(xπ1)
T1(xπ2)T1(xπ3)do no con ibu e o he hi d commu a o
in o mula (3.1) o D3when (k1,k2)∈V1, o all pe mu-
a ions π: hewhole con ibu ion o dμν(k1,k2)|(k1,k2)∈V1
coming om his commu a o is o he kind (3.18).
The con ibu ions o dμν(k1,k2)|(k1,k2)∈Vcoming om
he o he wo commu a o s in (3.1) a e o he same o m up
o cyclic pe mu a ions k1→ k2→−(k1+k2)→ k1o
he ex e nal momen a. He e we use ha (k1,k2)∈Vimplies
(k2,−k1−k2)∈V1and (−k1−k2,k1)∈V1, hence we may
apply he iden i y (3.19) also o he pe mu ed momen a.
Howe e , no e ha he polynomials hμν
j,j=2,3, belong-
ing o hese o he wo cu s a e no ob ained by cyclic pe mu-
a ions o he ex e nal momen a in he o iginal polynomial
hμν
1, mean in (3.18). This is due o he di e ence be ween he
higgs e ex and he pho on e ices; in pa icula , hese o he
wo cu s con ain no e m gi ing ise o a fish-like diag am.
Summing up, i holds ha
dμν(k1,k2)(k1,k2)∈V
=Hμν
1(k1,k2)+Hμν
2(k2,−k1−k2)+Hμν
3(−k1−k2,k1),
(3.20)
o some Hμν
j(j=1,2,3)o he o m (3.18), he pe inen
polynomials hμν
jbeing o deg ee 2.
Lemma 1 Le q1,q2,q3and V1be defined as abo e in (3.17)
and a e (3.18), and le Hμν :R4×2→Cbe gi en in e ms
o a gene ic polynomial hμν :R4×3→Co deg ee ζ∈N0,
as in (3.18). Then o all (k1,k2)∈V1and o some C >0
he unc ion Hμν is con inuous in he egion V1, and can be
bounded as ollows:
|Hμν (k1,k2)|
≤C(1+|(k1,k2)|)ζ
|(k1k2)|θ((k1+k2)2−4M2)log((k1+k2)2/M2).
(3.21)
7We bo ow he s anda d no a ion o he mass shell: H±
M:= { p∈
R4:p2=M2,±p0>0}.
No e ha |(k1k2)|>0i (k1,k2)∈V1and (k1+k2)2≥4M2.
P oo Le P:= k1+k2and k:= k1−k2. We fi s obse e,
on he s eng h o
q2
1−q2
2=2(Pq), q2
1+q2
2−2M2=2(q2+1
4P2−M2)
and o M2−q3=(M2−q2−1
4P2)+1
4P2−1
4k2+(kq) ha
Hμν (k1,k2)∼sgn(P0)d4qδ(q2+1
4P2−M2)
δ((Pq)) hμν (k1,k2,q)
P2/4−k2/4+(kq),
omi ing i ele an p e ac o s. Since q1−q2=Pand
(q1,q2)∈(H+
M×H−
M)∪(H−
M×H+
M), we know ha
Hμν(k1,k2) anishes o P2<4M2. Hence, o pe o m
he in eg als in q0and |q|using he Di ac del as, we may
wo k in he ame in which P=0. The e he δ-dis ibu ions
yield q0=0 and |q|=P2
0/4−M2.
Wi h he no a ion ˆp:= p/|p| o p∈{q,k}, i ollows
ha
1
4P2−1
4k2+(kq)=(k1k2)1−(ˆqˆ
k)1−4M2/P2|P0||k|
2(k1k2),
and one e ifies ha
0≤P2
0|k/2|2=(k1k2)2−k2
1k2
2,(3.22)
wi h k1=(k0
1,k1)and k2=(P0−k0
1,−k1). Wi h he help
o hese esul s we ob ain
(k1k2)Hμν(k1,k2)∼sgn(P0)θ(P2−4M2)1−4M2/P2
×S2d(ˆq)hμνk1,k2,(0,P2/4−M2ˆq)
1−(ˆqˆ
k)(1−4M2/P2)(1−k2
1k2
2/(k1k2)2)
,
(3.23)
alid in he ame in which P=0. Le mo eo e VM
1:=
{(k1,k2)∈V1:(k1+k2)2≥4M2}. We know ha
4M2/P2∈(0,1]and k2
1k2
2/(k1k2)2∈[0,1] o (k1,k2)∈VM
1;
hence a:= (1−4M2/P2)(1−k2
1k2
2/(k1k2)2)∈[0,1).
In pa icula , he denomina o in he in eg and o (3.23)
does no anish o (k1,k2)∈VM
1. Since θ(P2−4M2)
1−4M2/P2is con inuous, Hμν is con inuous on V1.
Obse e now ha o all ˆq∈S2 he inequali y
hμνk1,k2,(0,P2/4−M2ˆq)≤cons (1+|(k1,k2)|)ζ
holds, wi h |(k1,k2)|2:= 3
j=0(k2
1j+k2
2j).
Se ing z:= ˆqˆ
k, he emaining in eg al is o he ype
1
−1
dz
1−az =1
alog1+a
1−a≤2(1−log(1−a)),
123
131 Page 8 o 25 Eu . Phys. J. C (2021) 81:131
alid o a∈[0,1). Using ha a≤1−4M2/P2≤(1−
2M2/P2)and mono onici y o he loga i hm, we see ha
−log(1−a)=log 1
1−a≤log P2
2M2.
Pu ing oge he he es ima es, we end up wi h
|(k1k2)Hμν(k1,k2)|
≤cons ·θ(P2−4M2)(1+|(k1,k2)|)ζ1+log(P2/2M2),
(3.24)
impliying (3.21), since 1 +log(P2/2M2)<2log(P2/M2)
o P2≥4M2.
The eade should keep in mind ha dμν(k1,k2)is sup-
po ed ou side a ce ain neighbou hood o he o igin on
momen um space – ha e a look back a Eq. (3.23).
Co olla y 2 The o -shell d-dis ibu ion dμν (k1,k2)gi en
in (3.20) is con inuous on Vand ulfills he bound:
|dμν(k1,k2)|≤cons (1+|(k1,k2)|)ω+2
|(k1k2)|
log(2+|(k1,k2)|/M) o all (k1,k2)∈V.(3.25)
P oo Con inui y ollows immedia ely om Lemma 1.Fo
he bound (3.25) we ha e subs i u ed ω+2≡ω(d)+2 o ζ
o he Lemma, since he singula o de o Hμν
j(j=1,2,3)
is ζ−2 by powe coun ing in (3.18). In addi ion, o
Hμν
1(k1,k2)we ha e used ha (k1+k2)2≤4|(k1,k2)|2,
and in o de o omi he θ-dis ibu ion we ha e eplaced
log(2|(k1,k2)|/M)by 2 log(2+|(k1,k2)|/M). One deals
analogously wi h Hμν
2(k2,−k1−k2)and Hμν
3(−k1−k2,k1).
3.3 Dis ibu ion spli ing by he dispe sion in eg al o null
momen a
Recall ha o (k1,k2)∈Vη×Vη he ad anced pa aμν
o dμν can be compu ed by he dispe sion in eg al (C.14).
Using he egula i y p ope ies o dμν gi en in Co olla y 2,
we finally aim o show ha he limi k2
1↓0, k2
2↓0in(C.14)
commu es wi h in eg a ion; ha is, he dispe sion in eg al is
also alid o k2
1=0=k2
2. To o mula e he asse ion, le
K:= {(k1,k2)∈(R4)×2:k2
1,k2
2<4M2,
(k1+k2)2<4M2,(k1k2)= 0}.(3.26)
Bea ing in mind he ac o s θ(q2−4M2) o q∈
{k1,k2,k1+k2}appea ing in each e m o dμν(k1,k2),we
see ha o (k1,k2)∈(Vη×Vη)∩K, o mula(C.14) can be
ew i en as:
aμν(k1,k2)=iη
2π| |≥ min
d dμν( k1, k2)
ω+1(1− ),(3.27)
o some min >1 depending on k1,k2. Now, as discussed
in Sec . 1, one knows aμν(k1,k2) o be analy ic on he
egion K. The Lebesgue domina ed con e gence heo em
[31, Th. 4.6.3] wi h he bound (3.25) allows us conclude ha
(3.27) is a alid iden i y o (k1,k2)∈V∩K. Indeed, in o-
ducing he se o limi poin s
M:= V∩K∩{(k1,k2)∈R8:k2
i=0}
={(k1,k2)∈R8:k2
i=0,0<(k1+k2)2<4M2},
i is enough o obse e ha o any (˜
k1,˜
k2)∈M– implying
(˜
k1˜
k2)>0 and ˜
k0
1˜
k0
2>0 – he e is a neighbou hood U(˜
k1,˜
k2)
such ha
θ(| |− min)d( k1, k2)
ω+1(1− )
≤cons ·θ(| |− min)
| (1− )|1+|(k1,k2)|ω+2
|(k1k2)|
log(2+| ||(k1,k2)|/M)
≤Cθ(| |− 1)
| (1− )|log(2+C1| |),
o all (k1,k2)∈(Vη×Vη)∩K∩U(˜
k1,˜
k2), o someC,C1>0
and some 1>1 independen o (k1,k2). The unc ion on
he igh hand side is absolu ely in eg able in – he e we see
he eason o he condi ion (k1k2)= 0in(3.26).
3.4 No maliza ion o he scala model by dis ibu ion
spli ing
We mus finally compu e he gauge in a ian pa μν
gi (k1,k2)
o momen a lying on he se M. Conside ing he o mula
T3=A3−A
3and eckoning ha aμν(k1,k2)|k2
1=0=k2
2con-
ains he ac o θ(P2−4M2)whe e P:= k1+k2,wesee
ha on Mi s con ibu ion anishes, ha is μν =aμν he e.
The upsho o he p eceding wo subsec ions is ha we
may compu e alid e ms o he cen al solu ion aμν|M≡
acμν|Mby inse ing he on-shell ampli ude (3.15) in o he
dispe sion in eg al, wi h ω he singula o de o he o -shell
dμν, equal o 0 in he p esen case.
Looking a (3.15), obse e ha a kμ
kν
s-o gμν- e m o dμν
goes o e o a kμ
kν
s-o gμν- e m o aμν , espec i ely. The e-
o e, such ac o s may be aken ou o he dispe sion in eg al.
Since mo eo e Pμν ( k)= 2Pμν(k), we see ha he gauge
in a ian pa aμν
gi can be ob ained by inse ing jus he gauge
in a ian pa dμν
gi in (3.16) in o he dispe sion in eg al. The
la e iso he o m(C.15). So we may use he e sion (C.16)
o he dispe sion in eg al.
Las ly, μν
gi |M=aμν
gi |Mis ob ained om (C.16) by se ing
ω=0 and subs i u ing he e b(u˜ρ) as gi en in (3.16) o
u(k2
1,k2
2,(k1+k2)2)– in ou case only (k1+k2)2=s
is p esen . Allowing o he dila ion ac o in Pμν his leads,
123
Eu . Phys. J. C (2021) 81:131 Page 9 o 25 131
o (k1,k2)∈M, o
μν
gi (k1,k2)
=−Pμν
(2π)6∞
˜ρ−1
du ub(u˜ρ)
u(1−u)
=Pμν
16M2(2π)6∞
˜ρ−1
du
˜ρ2u2(1−u)log 1+1−u−1˜ρ−1
1−1−u−1˜ρ−1
=− Pμν
8(2π)6
J2(˜ρ)
M2,(3.28)
whe e
2J2(˜ρ) := ∞
1
d 1
( −˜ρ) 2log 1+√1− −1
1−√1− −1,
a e he change o in eg a ion a iable := u˜ρ.In eg als
like J2ha e been compu ed in [19]. F om Appendix C o
ha e e ence:
J1(˜ρ,a):= 1
2∞
1
d 1
( −˜ρ)( −a)log 1+√1− −1
1−√1− −1
= (˜ρ) − (a)
˜ρ−a(3.29)
o 0 ≤˜ρ≤1, 0 ≤a≤1, whe e is he dis ibu ion (B.3).
We in e ha
J2(˜ρ) =∂
∂aa=0
J1(˜ρ,a)
= (˜ρ)
˜ρ2−1
˜ρ o 0 ≤˜ρ≤1,(3.30)
by b inging in he alues (0)=0 and (0)=1, which
can be ead o om (B.6).
Summing up, he final esul eads, as expec ed:
μν
gi (k1,k2)=Pμν
8(2π)6
1
M21
˜ρ− (˜ρ)
˜ρ2
=Pμν
8(2π)6
F0(˜ρ)
M2 o (k1,k2)∈M,(3.31)
whe e F0was gi en in (1.3). We conjec u e ha his o mula
holds ue o all (k1,k2)sa is ying k2
1=0=k2
2and (k1+
k2)2>0.
The eade should emembe ha (3.31) s ands in p inciple
o jus a membe o a solu ion se . Since ω=0, he gen-
e al Lo en z-in a ian Eps ein–Glase solu ion is ob ained
by adding o exp ession (3.31) a e m o he ype Cgμν wi h
C∈Ca bi a y. Bu such a e m wi h C= 0 would iola e
EGI. The e o e we ega d he abo e esul as unique.
Reco e ing o mula (3.5) and he ac o 4ge2M, one ends
up wi h
dx1dx2dx3T3(x1,x2,x3)
=gα
(2π)3Mdk1dk2h(k1+k2)Aμ(−k1)Aρ(−k2)Pμν F0(˜ρ),
which, on subs i u ing ρ o ˜ρ, ha is, m2
h o s≡(k1+k2)2,
ag ees wi h he li e a u e [25].
Rema k 1 In he occasion an (unsub ac ed) dispe sion in e-
g al applied o b(u), pe o med in [16, Eq. (3.2)], leads o
he same in eg al (3.28) and so he same co ec esul . As
he nex sec ion shows, his does no hold o he higgs o
dipho on decay ia EW ec o bosons.
4 Higgs o dipho on decay ia EW ec o bosons
4.1 De i a ion o he qua ic AAWW†- e ex in he uni a y
gauge
The ampli ude in ques ion in his pape desc ibes an EW
decay p ocess a hi d o de in he coupling cons an . I s
s uc u e is gi en by he cubic e ices in he fi s TOP T1
– ha is he sole “empi ical” inpu . He e in going om T1
o T2we de i e he AAWW†- e ex which con ibu es by
a “fish-like” diag am o he ampli ude o be compu ed, see
Fig. 1.
The gene al idea is o examine he p opaga o which is
o become he in e nal line linking he di-pho on in he
one-loop, h ee- e ex g aph, and o ob ain he one-loop,
wo- e ex g aph om a modifica ion o ha p opaga o ,
demanded by EGI – by which he e we p ecisely unde s and
in a iance o he S-ma ix unde he a ia ions Aμ(x)→
Aμ(x)+∂μ(x): in e ac ion dic a es symme y. The me hod
is simila o he de i a ion o he AAϕϕ†“seagull” e ex
om he cubic coupling in scala QED, fi s pe o med in
his way in [28].
The concep wo ks on configu a ion space, as ollows.
Recall he pe inen He mi ian e ex – see o ins ance [32,
Sec . 7.2.2], explici ly e e ing o he uni a y gauge. Wi h
Gμν := ∂μWν−∂νWμ, one has:
T1(x1)=ie[(WμG†
μν −W†μGμν)Aν−WμW†
νFν
μ](x1).
(4.1)
All indica ed ope a o p oduc s a e Wick p oduc s. We copy
a second e ex simila o (4.1):
T1(x2)=ie[(WρG†
ρλ −W†ρGρλ)Aλ−WρW†
λFλ
ρ](x2),
(4.2)
123
131 Page 16 o 25 Eu . Phys. J. C (2021) 81:131
Le us now o come back o e e ence [16]. I is a gued
he e ha he con e gen in eg al
0(˜ρ) := ∞
˜ρ−1
du b1(u˜ρ)
1−u
leads o he co ec esul . F om he s andpoin o his e e -
ence, o mula (4.27) is “o e sub ac ed”. One ob ains he e,
ye again
Pμν 0(˜ρ) =Pμν∞
˜ρ−1
du b1(u˜ρ)
1−u
=− 3Pμν
8M2(2π)6(2J1(˜ρ,0)−J2(˜ρ))
=− Pμν
8M2(2π)63
˜ρ+6 (˜ρ)
˜ρ−3 (˜ρ)
˜ρ2
=− Pμν
8M2(2π)6(F1(˜ρ) −2). (4.31)
So he nai e on-shell compu a ion yields a pa icula
Eps ein–Glase solu ion. In he p esen case, howe e , equa-
ion (4.30) ells us ha we a e o ced o add (a leas ) a poly-
nomial o deg ee wo espec ing EGI, ha is, a e m CPμν
o Can inde e mina e cons an – wi h which ou esul o
he ampli ude is compa ible wi h he gene ally accep ed one.
Rema k 2 F om ou iewpoin , he exp ession in (4.31)is he
unique Eps ein–Glase solu ion espec ing EGI, co espond-
ing o he ollowing causal d-dis ibu ion: le he esul (4.26)
o dμν
gi (k1,k2)(ob ained by ligh -cone es ic ion o he pho-
on momen a) be in e p e ed as an un es ic ed elemen o
S(R8), ha is, all alues (k1,k2)∈R8a e admi ed. One
easily e ifies ha his d-dis ibu ion has causal suppo , so
he spli ing p oblem is well defined, and since i s singula
o de is ze o, he EGI equi emen selec s a unique spli ing
solu ion. W i ing he la e sui ably as a dispe sion in eg al
in momen um space, one e ifies he claim. This p ocedu e
s ongly simplifies explici compu a ions, bu i is no con-
cep ually co ec .11
4.4 Fixing he no maliza ion polynomial by ag eemen
wi h he Feynman gauge
In o de o de e mine he no maliza ion polynomial we may
as well in oke he compu a ion o he h→γγ decay in
he Feynman gauge and gauge-fixing independence, namely,
he equi emen ha obse able quan i ies should no depend
11 Ac ually, in he fi s edi ion o he book by Scha on quan um elec-
odynamics (i.e., [29] a he han [39]), he e ex unc ion in QED a
hi d o de was compu ed by such a me hod.
on he choice o gauge.12 Mo i a ed by esul s o [43],13 we
con end ha he “en i ely on-shell” ampli ude coming ou
o ou p e ious compu a ion should coincide wi h ha o
an Eps ein–Glase compu a ion in he Feynman gauge. By
“en i ely on-shell” we mean ha no only he pho ons, bu
also he higgs is on-shell, ha is, ˜ρ=ρ:= m2
h/4M2.
Deno e he Eps ein–Glase esul o he d-dis ibu ion
in he Feynman gauge by d1
μν. In con as wi h he uni a y
gauge, he e addi ionally con ibu e diag ams wi h S ück-
elbe g fields and Faddee –Popo ghos s (as inne lines) o
d1
μν, see e.g. [15]. We spa e he eade he de ails o he con-
s uc ion o he TOPs, and in pa icula he de i a ion o he
AAWW†- e ex in his con ex . Fo pho ons on-shell wi h
physical pola iza ions (se ing k2
1=0=k2
2and omi ing
pu e gauge e ms ∼k1μo ∼k2ν), ou esul eads:
d1
μν(k1,k2)=− 1
23(2π)6M2
×(k1k2)gμν −3
˜ρ2+7
˜ρ−ρ
˜ρ2
−k2μk1ν−3
˜ρ2+8
˜ρ−2ρ
˜ρ2 (˜ρ). (4.32)
The edious compu a ion o he abo e abso p i e pa was
done wi h he aid o he Ma hema ica package FeynCalc [44].
The compu a ion p oceeds along he lines o he compu a-
ions o ela ed abso p i e pa s in scala elec odynamics
and elec oweak heo y in he uni a y gauge p esen ed in
ull de ail in Sec s. 3.1 and 4.2 , espec i ely. As be o e,
all e ms con ibu ing o he dis ibu ion d1
μν can be ep e-
sen ed by Feynman diag ams wi h cu s – o he comple e
lis see e.g. [15]. Jus like in Sec s. 3.1 and 4.2 , because o
he kinema ic cons ain s one needs o conside only he cu
sepa a ing he higgs e ex om he pho on e ices. All he
appea ing exp essions ha e a e y simila s uc u e o hose
ha ha e been al eady conside ed in he abo e-men ioned
pa s. Thanks o he p esence o he cu , each in eg al o e
he ou -momen um flowing in he loop can be con e ed in o
an in eg al o e a sphe e, which can be e alua ed explici ly.
We s ess he ac ha , due o compac ness o he egion o
in eg a ion, he compu a ion o he abso p i e pa does no
in ol e any egula iza ion.
12 The equi alence o inequi alence o calcula ions pe o med in di -
e en gauges was a nagging wo y o Raymond S o a in his las yea s.
The classic pape [42] illus a es he di ficul ies lu king he e.
13 This e e ence wo ks wi h a o mula ion o gauge in a iance sui able
o he BEG scheme. In ha amewo k i was shown o he a ious
Rξ-gauges ha he T-p oduc s can be no malized in such a way ha he
physical S-ma ix (i.e., o in- and ou -s a es being on-shell) does no
depend on he gauge-fixing pa ame e ξin he o mal adiaba ic limi ;
and ha his no maliza ion is compa ible wi h gauge in a iance in he
men ioned sense.
123
Eu . Phys. J. C (2021) 81:131 Page 17 o 25 131
An impo an ea u e o elec oweak heo y in he Rξ-
gauges is he ac ha all in e ac ion e ices ha e dimensions
lowe o equal o ou (because dim Wμ=1, in con as o
he alue dim Wμ=2 o he uni a y gauge). In pa icula ,
a s aigh o wa d powe coun ing a gumen gi es he uppe
bound ωd1
μν≤0 o he singula o de o he o -shell
dis ibu ion d1
μν. No ing ha ˜ρ=(k1k2)/2M2and (˜ρ) =
O(log ˜ρ) we see ha he on-shell es ic ion o d1
μν(k1,k2)
in Eq. (4.32) g ows loga i hmically o big alues o ˜ρ.Fo
he o -shell d1
μν, his implies he equali y ωd1
μν=0. This
should be con as ed wi h he bounds 6 ≥ωdμν≥2in he
case o he abso p i e pa compu ed in he uni a y gauge.
The o -shell dis ibu ion d1
μν is again o he ype con-
side ed in Sec s. 3.2. In pa icula , he me hod o dis ibu ion
spli ing de eloped in Sec . 3.3 is applicable. Fo pho ons on-
shell wi h physical pola iza ions, he cen al solu ion eads
1c
μν (k1,k2)=− 1
23(2π)6M2
×gμν(k1k2)−3
˜ρ2+7
˜ρ−ρ
˜ρ2 (˜ρ) +3
˜ρ+2ρ
˜ρ
−k1νk2μ−3
˜ρ2+8
˜ρ−2ρ
˜ρ2 (˜ρ) +3
˜ρ+2ρ
˜ρ.
(4.33)
Acco ding o he pos ula e ‘Di e gence deg ee’ (in Sec .
C.1), we ha e o demand o he o -shell 1
μν ha
ω 1
μν=ωd1
μν=0.
This implies ha he pe aining no maliza ion eedom con-
sis s o a cons an e m which is a enso wi h wo indices. By
he Lo en z in a iance such a e m has o be p opo ional o
he me ic enso . Consequen ly, he gene al o -shell solu ion
o he spli ing p oblem is o he o m
1
μν(k1,k2)= 1c
μν (k1,k2)+gμν D,(4.34)
whe e Dis an a bi a y cons an ; no e ha his ela ion holds
also a e es ic ion o on-shell pho ons wi h physical pola -
iza ions.
Obse e ha , in con as o he uni a y gauge, as long
as he higgs is o -shell, he dis ibu ions (4.32) and (4.33)
a e no elec omagne ically gauge-in a ian . This was o be
expec ed and is ela ed o he p esence o unphysical deg ees
o eedom in elec oweak heo y in he Rξ-gauges. How-
e e , en i ely on-shell EGI can be sa isfied: se ing ˜ρ:= ρ
in (4.33), we plainly ge
1
μν(k1,k2)˜ρ=ρ
=− 1
23(2π)6M2Pμν(k1,k2)F1(ρ) +Dg
μν,(4.35)
and one sees ha Dmus be pu equal o ze o. This fixes
comple ely he no maliza ion eedom in he cons uc ion
o 1
μν in he Feynman gauge. A his le el he e is o cou se
coincidence wi h he esul in [45], despi e di e en game
ules.
Recall ha in he uni a y gauge, o on-shell pho ons
wi h physical pola iza ions, he gene al no maliza ion ee-
dom ulfilling elec omagne ic gauge in a iance and Lo en z
co a iance is gi en by he las e m in (4.30), whe e ω≡
ω(dμν). We s ess ha he cons an s Ckappea ing in ha
e m canno be fixed wi hou imposing some u he no mal-
iza ion condi ions. To add ess his p oblem, obse e ha i
is possible o adjus he coe ficien s Cko he polynomial in
he exp ession (4.30) o gi,μν in he uni a y gauge in such a
way ha he ollowing equali y
gi,μν(k1,k2)˜ρ=ρ= 1
μν(k1,k2)˜ρ=ρ.(4.36)
holds en i ely on-shell, i.e. o ˜ρ=ρ. In ac , we mus se
C0:= 2 and Ck:= 0 o all k≥1in(4.30), which fixes
comple ely he no maliza ion eedom o gi,μν .Eq.(4.36)
exp esses he independence o he physical ampli ude o he
dipho on decay o he higgs o he choice o he gauge. We
ega d (4.36) as a no maliza ion condi ion o ime-o de ed
p oduc s. We ha e shown ha his condi ion can be sa isfied
in he case a hand and de e mines uniquely he inde e mina e
no maliza ion polynomial o gi,μν in he exp ession (4.30).
In summa y, ou final esul o he en i ely on-shell EW
h→γγ decay eads:
μν(k1,k2)˜ρ=ρ=− 1
23(2π)6M2Pμν(k1,k2)F1(ρ),
in ag eemen wi h he majo i y o he li e a u e.
4.5 On se ling he con o e sy
Should one in e ha by compu ing in he “physical” uni a y
gauge he e is no way o en i ely se le he con o e sy ha
mo i a es his wo k, by emo ing he emaining ambigui y in
de e mining he ampli ude in ques ion? No wi hou a leas
ponde ing c edible “hea y-higgs” (o M→0) and “ligh -
higgs” (o M→∞) a gumen s o bols e he case o F1(ρ)
e sus F1(ρ) −2, ha ha e been made in he li e a u e.
Now, o he p esen au ho s he ques ion is no whe he
ei he class o a gumen s is compelling enough. Ins ead, he
ques ion is whe he hey can be made wi hin he BEG p e-
sc ip ions, and a he le el o igou o his pape . The a gu-
men s in he fi s -named class in ol e plays wi h field ans-
o ma ions, powe coun ing ules and he adiaba ic limi ha
we find ha d o coun enance in he BEG o malism.
Howe e , hose o he second class a e pe suasi e wi hin
ou pu iew. No e ha F(0), o bo h scala and ec o boson
cha ged fields, as well as o Di ac e mions, mus coincide
wi h ( he fi s coe ficien o ) he β- unc ion se ies associa ed
123
131 Page 18 o 25 Eu . Phys. J. C (2021) 81:131
o elec ic cha ge eno maliza ion.14 I was a o una e his-
o ical ac ha a calcula ion o he e ec i e Lag angian o
cha ged P oca pa icles [46] was al eady a ailable when he
fi s “exac ” compu a ion o he higgs o digamma p ocess
ha we a e awa e o was pe o med [2] – hus making possi-
ble a dependable “ligh -higgs” a gumen . A compu a ion o
he eno maliza ion o he elec ic cha ge o massi e ec o
bosons in he uni a y gauge by means o BEG echnology
is in p inciple easible – c . in his espec [41, Sec . 7] and
[47] – and expec ed o yield he equi ed alue F1(0)=7.
Tha would comple e he analysis o his pape , wi hou going
beyond he uni a y gauge amewo k.
5 Conclusion
Con a y o cus om, we begin his sec ion by decla ing wha
we ha e no done in he pape . Fini e QFT àlaBogoliubo –
Eps ein–Glase is ma hema ically a igo ous me hod. So,
e e ing o wha is ound in he li e a u e – like ha ci ed in
he In oduc ion – we ha e no employed dimensional egu-
la iza ion, deemed an “a i ac ” by some. No do we bo ow
Pauli–Villa s’, no cu o egula iza ions, o ha ma e . We
did no ha e o p ac ice “judicious ou ings o he ex e nal
momen a” [6], no adop he “loop egula iza ion me hod”
[10], o any o he echniques o handle di e gen in eg als,
esul ing om he blind applica ion o Feynman g aph ech-
nology on momen um space. We do no po e o e di e gen
in eg als, a all. Each and e e y one o he in eg als appea ing
in his pape p oduces an unambiguous esul ; each ampli-
ude is fini e.
We expec ed he BEG p ocedu e o yield a concep ually
clea unde s anding o he EW h→γγ decay in he uni a y
gauge. We ha e succeeded in his – a a p ice. Acco ding o
Eps ein and Glase , he adiaba ic limi is o be pe o med a e
dis ibu ion spli ing. Such an o -shell p ocedu e o he h→
γγ decay in he uni a y gauge demands compu a ions mo e
han one o de o magni ude g ea e han he ones pe o med
in his pape – compa e he compu a ion o he QED e ex
unc ion in [39, Chap. 3.8] and in [48].
We we e no disposed o inflic his on ou sel es, no ou
su i ing eade s. Thus we we e o ced o inno a e on he
me hod, gene alizing he spli ing dispe sion in eg al o p o-
duc ion o massless pa icles, and showing ha in he p esen
si ua ion he adiaba ic limi may be pe o med be o e dis i-
bu ion spli ing. Only, hen one may ha e o add o he esul
so ob ained an a p io i inde e mina e polynomial in he ex e -
nal momen a, o a deg ee gi en by he singula o de o he
ampli ude o -shell. I is p ecisely he addi ion o his poly-
nomial ha is missing in e e ences [5,6] and [16]. We ha e
14 F0(0)=−1/3, which has been calcula ed in his pape , means
p ecisely his.
esol ed he ambigui y by ecou se o gauge-fixing indepen-
dence o he en i ely on-shell ampli ude. Al e na i ely, he
ambigui y could be esol ed wi hin he uni a y gauge in he
BEG scheme, by in oking he low-ene gy a gumen .15 We
ha e no a emp ed he e a igo ous p oo o his a gumen ,
no compu ed he ele an coe ficien o he be a unc ion,
lea ing he ask o a sepa a e analysis in u u e wo k.
Acknowledgemen s We a e g a e ul o E. Al a ez, L. Al a ez-Gaumé,
M. He e o, C. P. Ma ín, J. C. Vá illy and T. T. Wu o commen s, dis-
cussions and help ul ema ks. We pa icula ly hank I. T. Todo o o
keen help in he beginning, and his con inued and hough -p o oking,
i con a ian, in e es in his wo k. As well we hank an anonymous
e e ee o knowledgeable epo ing, defini ely con ibu ing o imp o e
he pape . Du ing he incep ion and w i ing o his a icle, PD ecei ed
unding om he Na ional Science Cen e , Poland, unde he G an
UMO-2017/25/N/ST2/01012. He also g a e ully acknowledges he hos-
pi ali y o he Uni e si y o Za agoza. JMG-B ecei ed unding om
he Eu opean Union’s Ho izon 2020 esea ch p og amme unde he
Ma ie Skłodowska-Cu ie G an ag eemen RISE 690575; om P ojec
FPA2015–65745–P o MINECO/Fede ; om CERN; om he COST
ac ions MP1405 and CA18108. Hospi ali y o CERN, IFT-Mad id, ITP-
Gö ingen and ZiF-Biele eld is g a e ully acknowledged.
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Appendix A: No a ions and p e equisi es
Ou Minkowski me ic is mos ly-nega i e. The Minkowski
inne p oduc o wo ec o s x≡xμ,p≡pνis deno ed
wi h pa en heses: (xp)=xμpμ. When (we hope) i does no
cause con usion, we o en deno e p2=(pp).
We signal he s anda d o mula o ime-o de ed 2-poin
unc ion:
Tϕ(x)χ(x) := i
(2π)4d4pe−i(p(x−x))
p2−M2+i0Mϕχ(p),
(A.1)
15 Va ian s o he “ligh -higgs” o “low ene gy” a gumen besides [2,
9,17] a e ound o ins ance in [27, Ch. 24.8], in [49]andin[50].
123
Eu . Phys. J. C (2021) 81:131 Page 19 o 25 131
whe e Mϕχ is he mul iplie appea ing in he co esponding
2-poin unc ion o he fields ϕ,χ wi h he same mass M.
P opaga o s o a (complex) scala field. Clea ly, o
(say, complex) scala fields he Feynman p opaga o
F(x−x):= Tϕ(x)ϕ†(x) (A.2)
ulfils
F(x−x)=−M2F(x−x)−iδ(x−x), (A.3)
whe e Mis he mass o he ϕ-field. Also, wi h θdeno ing he
Hea iside unc ion, he Wigh man unc ions
+(x−x):= ϕ(x)ϕ†(x)
=ϕ†(x)ϕ(x) = 1
(2π)3d4pθ(p0)δ(p2−M2)e−i(p(x−x))
so ha (+M2)+(x)=0,
−(x):= −+(−x),
(A.4)
a e used in ou calcula ions.
Massi e ec o fields. A d eibein e (p)on Minkowski
momen um space, wi h he p ope ies:
e (p)es(p)=−δ s o ,s=1,2,3;pe
(p)=0,
desc ibes pola iza ion s a es o pa icles wi h squa ed mass
M2=p2>0 and spin j=1. F om he abo e iden i ies,
one de i es he p ojec o o mula:
3
=1
eμ
(p)eν
(p)=−gμν +pμpν
M2.(A.5)
The se eis ega ded as an in e wine ma ix mapping he
na u al ep esen a ion space o he Lo en z g oup on o he
ep esen a ion space C3 o spin 1 objec s. Le a†
(p)and
a (p)be espec i ely he c ea ion and annihila ion ope a-
o s on he boson Fock space o such pa icles – whose 1-
pa icle subspace is he co esponding Wigne uni ep space;
and b†
(p)and b (p) o hei an ipa icles.
The e is a quan um ec o field ac ing on ha space gi en
by he o mula
Wμ(x):=
dμ(p)ei(px)eμ
(p)b†
(p)
+e−i(px)eμ∗
(p)a (p);(A.6)
In (A.6) and in o he o mulas dμ(p)deno es he usual in a i-
an measu e d3p/2E(p)=d3p/m2+|p|2o e he mass
hype boloid H±
M:= {p∈M|p2=M2∧±pn>0}.By
i s defini ion, he cha ged P oca field Wis di e genceless:
(∂W)=0. I s equa ions o mo ion can be a iously w i en
as
(+M2)Wμ=(+M2)Wμ−∂μ(∂W)
=∂νGνμ(x)+M2Wμ=0,(A.7)
whe e Gμν := ∂μWν−∂νWμ.
The heo y o massi e ec o fields is a gauge heo y
[51,52], i s P oca e sion being a “uni a y gauge” o i . I
has been analyzed, in e ms pa allel o Maxwell field heo y,
in [53]; whe ein he associa ed BRST machine y is “decon-
s uc ed” in e ms o Koszul cohomology.
The high-ene gy limi o (−gμν +pμpν/M2)/(p2−
M2)appa en ly signals quad a ic di e gences and ouble
wi h uni a i y o he sca e ing ma ix: c oss-sec ions would
appea o g ow wi hou bound due o he longi udinal momen-
um s a es. The di ficul y lies wi h he closu e ela ion (A.5)
o he in e wine s e , whose dimension does no allow he
s anda d su ficiency c i e ion o eno malizabili y. This is
usually “cu ed” nowadays by he cohomological ex ension
o he Wigne ep esen a ion space o massi e spin-1 pa -
icles in o spaces popula ed by Faddee -Popo ghos s and
an i-ghos s and S ückelbe g fields.
In his pape we wo k mainly wi h he P oca field (i.e., we
use he uni a y gauge), whe e hese addi ional unphysical
fields do no appea ; he appa en ly bad UV-beha iou o he
p opaga o s is unde con ol, as we e i y, hanks o amazing
cancella ions in he ampli udes.
P opaga o s o he EW heo y in he uni a y gauge.
We will make equen use o
α
β(x−x):= TWα(x)W†
β(x) = −(gα
β+∂α∂β/M2)
×F(x−x), (A.8)
whe e Mis he mass o he W-field, and i s p ope ies:
α
β=−M2α
β+i(gα
β+∂α∂β/M2)δ;∂μμ
ν=i∂νδ/M2.
The co esponding o mulas o he Wigh man unc ions
espec i ely ead:
α+
β(x−x):= Wα(x)W†
β(x) = Wα†(x)Wβ(x)
=−(gα
β+∂α∂β/M2)+(x−x)
and α+
β=−M2α+
β,∂
μμ+
ν=0.
We will in oke also he Maxwell-like fields, whe e # =†o
naugh ,
Fμν := ∂μAν−∂νAμ;G#
μν := ∂μW#
ν−∂νW#
μ,
and in oduce he p opaga o
Dαμ
βρ (x−x):= TGαμ(x)G†
βρ (x) = TGαμ †(x)Gβρ (x)
=T(∂αWμ(x)−∂μWα(x))(∂βW†
ρ(x)−∂ρW†
β(x))
=−∂μ∂ρα
β(x−x)−∂βα
ρ(x−x)
+∂α∂ρμ
β(x−x)−∂βμ
ρ(x−x)
=(gα
β∂μ
ρ−gα
ρ∂μ
β−gμ
β∂α
ρ+gμ
ρ∂α
β)F(x−x),
∂α
ρ:= ∂α∂ρ,(A.9)
123
131 Page 20 o 25 Eu . Phys. J. C (2021) 81:131
since in
∂ρα
β−∂βα
ρ=(−gα
β∂ρ+gα
ρ∂β)F(A.10)
he e ms wi h h ee de i a i es cancel ou , due o he an i-
symme y o G#
μν. Tha ac is ele an in his pape . Analo-
gously we ob ain
Dαμ+
βρ (x−x):= Gαμ(x)G†
βρ (x) = Gαμ †(x)Gβρ (x)
=(gα
β∂μ
ρ−gα
ρ∂μ
β−gμ
β∂α
ρ+gμ
ρ∂α
β)+(x−x),
wi hou hi d-o de de i a i es. We also no e ha
∂μDαμ
βρ =(gα
β∂ρ−gα
ρ∂β)F=(−gα
β∂ρ+gα
ρ∂β)(M2F+iδ),
since hi d-o de de i a i es appea only in he o m ∂F,
emo able wi h he help o (A.3).
Fo he 2-poin unc ions wi h one G#plus one W#,we
use (A.10) o ge id o he e ms wi h h ee de i a i es. Fo
ime-o de ed ones we ob ain
TWμ(x)G†
αν(x) = TWμ†(x)Gαν(x)
=−(∂αμ
ν(x−x)−∂νμ
α(x−x)) =(gμ
ν∂α−gμ
α∂ν)
×F(x−x),
TG†
αν(x)Wμ(x)
=TGαν(x)Wμ†(x)
=(−gμ
ν∂α+gμ
α∂ν)F(x−x). (A.11)
Wi h he pa allel Wigh man unc ions we p oceed simila ly:
Wμ(x)G†
αν(x) = Wμ†(x)Gαν(x)
=(gμ
ν∂α−gμ
α∂ν)+(x−x),
G†
αν(x)Wμ(x) = Gαν(x)Wμ†(x)
=(−gμ
ν∂α+gμ
α∂ν)+(x−x).
(A.12)
Compa ing wi h he Feynman gauge, in which he W# wo-
poin unc ions α
βand α+
βa e eplaced by −gα
βF(A.13)
and −gα
β+(A.14), espec i ely, we find he W#G#,G#W#
and G#G# wo-poin unc ions o be he same, hanks o he
cancella ions in (A.10).
P opaga o s o he EW heo y in he Feynman gauge.
The Feynman p opaga o and he Wigh man wo-poin unc-
ion o he W-field in he Feynman gauge ead
TWα(x)W†
β(x) = −gα
βF(x−x), (A.13)
Wα(x)W†
β(x) = Wα†(x)Wβ(x) = −gα
β+(x−x).
(A.14)
Besides he W-field he compu a ion om Sec s. 4.4 in ol es
he S ückelbe g fields ϕ±and he ghos and an i-ghos fields
C±,¯
C±– whe e φ±:= 1
√2(φ1±iφ2 o φ=ϕ, C,¯
C.
Below we lis he non- anishing Feynman p opaga o s and
wo-poin unc ions o hese fields:
Tϕ+(x)ϕ−(x) = F(x−x),
(A.15)
ϕ+(x)ϕ−(x) = ϕ−(x)ϕ+(x) = +(x−x),
(A.16)
TC+(x)¯
C−(x) = TC−(x)¯
C+(x) = F(x−x),
(A.17)
C+(x)¯
C−(x) = − ¯
C−(x)C+(x) = +(x−x),
(A.18)
C−(x)¯
C+(x) = − ¯
C+(x)C−(x) = +(x−x).
(A.19)
Appendix B: An in e es ing dis ibu ion
In his appendix we s udy he dis ibu ion (ρ) appea ing in
he ampli ude o he h→γγ decay ia bo h scala QED and
fla ou dynamics.
To define √·:C→Cand log:C→Cone uses a cu on
he nega i e eal axis:
e
iϕ=√ e
iϕ/2,log eiϕ=log +iϕ,
bo h wi h ϕ∈(−π,π].
The complex unc ion
˜
:C (−∞,0)∪(1,∞)−→ C
z−→ −log(√1−z+i√z)2(B.1)
is analy ic, in iew o he wo cu s on he eal axis. The
dis ibu ion (ρ) is defined by
:[0,∞)−→ C:ρ−→ (ρ) := ˜
(ρ +i0). (B.2)
We claim ha
(ρ) =(a csin √ρ2=a c an ρ
1−ρ22
o 0 ≤ρ≤1,
(B.3)
(ρ) =−1
4log 1+1−ρ−1
1−1−ρ−1−iπ2
o ρ≥1,(B.4)
om which one easily ob ains he ollowing o mula o he
imagina y pa :
(ρ) =θ(ρ −1)π
2log √ρ+√ρ−1
√ρ−√ρ−1
=−θ(ρ −1)π
2log2ρ−2ρ(ρ −1)−1.(B.5)
123
Eu . Phys. J. C (2021) 81:131 Page 21 o 25 131
The fi s claim (B.3) ollows immedia ely om he iden i y
a csin √ρ=−ilog1−ρ+i√ρ o ρ∈[0,1],
which is ob ious om exp(ia csin x)=√1−x2+
ix,|x|≤1.
To p o e he second claim (B.4), fi s no e ha one has
√1−(ρ +i0)=−i√ρ−1 o ρ≥1. Hence, he e holds:
log1−(ρ +i0)+i√ρ=log√ρ−ρ−1+iπ/2
=1
2log(√ρ−ρ−1)2+iπ
=1
2log √ρ−√ρ−1
√ρ+√ρ−1+iπ−1
2log 1+1−ρ−1
1−1−ρ−1−iπ,
om which asse ion (B.4) ollows.
We poin ou ha , o ρ∈[0,1], in he dis ibu ion
F0(ρ) =ρ−11−ρ−1 (ρ)in Eq. (1.3) he e ms∼ρ−1
cancel. We b ing in he powe se ies expansion
a csin x=x+x3
2·3+3x5
2·4·5
+3·5x7
2·4·6·7+··· o |x|≤1,yielding
(ρ) =(a csin √ρ)
2=ρ+ρ2
3
+8ρ3
45 +··· so ha F0(ρ) =−1
3−8
45 ρ+···.
(B.6)
Appendix C: Bogoliubo –Eps ein–Glase no maliza ion
Eps ein and Glase [21,54] s a ed om Bogoliubo ’s unc-
ional S[g]-ma ix [55, Sec . 21], based on [56] and on p e i-
ous wo k by S ückelbe g and Ri ie [57]. Tha is an expan-
sion o ope a o - alued dis ibu ions (OVD) on configu a ion
space, o he o m
S[g]=1
+∞
n=1
in
n!d4x1···d4xnTn(x1,...,xn)g(x1)···g(xn),
g∈S(R4,R). (C.1)
We ha e aken ¯
h=1. The g’s a e mul iple s o coupling
unc ions which wo k as adiaba ic cu o s. The Tn,symme -
ic in hei a gumen s, a e iden ified wi h ch onological o
ime-o de ed n-p oduc s. This is Bogoliubo ’s e sion o he
summands in he o mal Dyson expansion o he sca e -
ing ma ix in he in e ac ion pic u e. One ies o ecu si ely
build he Tn om na u al pos ula es: he ul a iole p oblem
is sol ed in ha cons uc ion. In he “adiaba ic limi ” g↑1
he unc ional sca e ing ma ix (C.1) is expec ed o con e ge
o he physical Sin sui able senses [58].
C.1: The Eps ein–Glase pos ula es
Beginning o induc ion: The p ocedu e is pe u ba i e,
he basic building blocks being fini e se s o quan um
ee fields on hei co esponding Fock spaces. P ecisely,
T1(x)is a Wick polynomial in hose and hei de i a i es
– a well-defined OVD.16 The coupling cons an s o he
model a e included in he Tn, he expansion being a powe
se ies on hem. The o he pos ula es shall enable us o
cons uc he Tn om T1by induc ion on n.
Causali y: This is he key equi emen , o which he
Eps ein–Glase manu ac u ing o TOPs is also called
“causal pe u ba ion heo y”. Le V±and V± espec i ely
deno e he open o wa d and backwa d ligh cones and
hei closu es. I g1,g2a e such ha
supp g2∩supp g1+V−=∅, hen
S[g1+g2]=S[g2]S[g1];
equi alen ly, Tn(x1,...,xn)
=T (x1,...,x )Tn− (x +1,...,xn)whene e
{x1,...,x }∩{x +1,...,xn}+V−=∅,
o all and nwi h1 ≤ ≤n−1.
This is a powe ul pos ula e, called causal ac o iza ion.
I means ha on la ge open se s o he n-poin Minkowski
space (M4)×n≡Mn he TOP Tncan be buil up om
i s lowe -o de coun e pa s. In he induc i e s ep o he
Eps ein–Glase me hod, his equi emen uniquely de e -
mines Tnon he se o Schwa z unc ions S(Mn n),
in e ms o he gi en Tka lowe o de s k≤n−1,
whe e nis he “ hin” diagonal n:= {(x1,...,xn):
x1=x2= ··· = xn}. Pe u ba i e no maliza ion is
he ex ension o he ope a o - alued dis ibu ion Tn om
S(Mn n) o S(Mn). The gis o BEG no maliza ion
is ha in local quan um field heo y his p oblem finds a
solu ion, he induc ion p ocess going h ough. So he e is
no need o deal wi h infini ies. The solu ion o he ex en-
sion p oblem is non-unique: in p inciple one may add any
OVD which is suppo ed on n. All u he pos ula es o
Eps ein–Glase ha e he pu pose o gi ing guidance o
his p oblem; hence hey may be called “no maliza ion
condi ions”.
Causal Wick expansion: The TOPs a e equi ed o sa -
is y he Wick expansion o mula. We display he la e
in e ms o he in e ac ion T1(x)=ϕk(x), o ϕa eal
16 One can hink o T1as an “in e ac ion Lag angian”. Howe e , he
Lag angian mindse is inessen ial he e.
123
131 Page 22 o 25 Eu . Phys. J. C (2021) 81:131
scala field:
Tnϕk(x1),...,ϕk(xn)
=
k
l1,...,ln=0k
l1···k
lnTn(ϕk−l1(x1), . . . ,
×ϕk−ln(xn)) ϕl1(x1)···ϕln(xn)
wi h · · · deno ing acuum expec a ion alue. This pos-
ula e educes he ex ension p oblem o he OVD Tn(···)
o one o nume ical dis ibu ions – a simple ask.
Poinca é Co a iance: Le he e be gi en he s anda d
li ing U(a,) o Fock space o he Poinca é uni a y
i educible ep esen a ions (uni eps) on 1-pa icle sub-
spaces. Then
U(a,)S[g]U†(a,)=S(a,)·g,
whe e ((a,) ·g)(x)=g(−1(x−a)). In pa icu-
la , ansla ion in a iance implies ha he coe ficien s
in he causal Wick expansion depend only on he el-
a i e coo dina es. The e o e, he ex ension p oblem o
he nume ical dis ibu ions is s ep by s ep simplified o
an ex ension o one poin , namely om S(R4(n−1) {0})
o S(R4(n−1)).
Uni a i y (conse a ion o p obabili y):
S[g]S†[g]=S†[g]S[g]=1;he e we deno e:
S−1[g]
=: 1+∞
n=1
(−i)n
n!d4x1···d4xnTn(x1,...,xn)
×g(x1)···g(xn).
Di e gence deg ee: Heu is ically, his is he equi emen
ha no maliza ion does no make he T-p oduc “mo e
singula ” (in he UV- egion). This is exp essed in e ms
o he scaling deg ee o he coe ficien s (i.e., he nume -
ical dis ibu ions) in he causal Wick expansion o he
T-p oduc : ha deg ee may no be inc eased by he ex en-
sion. The s anda d defini ions o he scaling deg ee sd( )
and he singula o de ω( )o a dis ibu ion ∈S(Rk)
o ∈S(Rk {0})– see, e.g., [23, Sec . 3.2.2] – a e as
ollows:
sd( ):= in { ∈R:lim
λ↓0λ (λx)=0},ω( ):= sd( )−k,
(C.2)
whe e in ∅:=∞and in R:= −∞. Fo ins ance, o
a ansla ion-in a ian dis ibu ion d(x1−x3,x2−x3)∈
S(R8) ulfilling sd(d)=8, equi alen ly ω(d)=0, we
say ha he ampli ude supe ficially is “loga i hmically
di e gen ”.
O he in a iance ules and physical equi emen s: Dis-
c e e symme ies can be accomoda ed in he Eps ein–
Glase cons uc ion [59]. A Wa d iden i y playing a
pa amoun ole in his pape co esponds o EGI – see
Sec s. 3.1 and 4.2 o his. Fo di e en ypes o equi e-
men s, consul Sec s. 4.4 and 4.5.
C.2: I e a i e building o he ime-o de ed p oduc s
To assemble he Tnou side o he hin diagonal n om
he induc i ely known (Tk)1≤k≤n−1di ec ly by causal ac-
o iza ion, one would need a pa i ion o uni y subo dina e
o an open co e o Mn n–see[60] and [23, Sec . 3.3].
This is p oblema ic o p ac ical compu a ions. Fo his ea-
son he o iginal Eps ein–Glase cons uc ion [21,39]isless
di ec : i in oduces an in e media e Dn-dis ibu ion ha ing
causal suppo ; and he c ucial s ep is he spli ing o Dnin o
i s ad anced and e a ded pa s. This spli ing co esponds
p ecisely o he abo e-men ioned ex ension p oblem, ha is,
o pe u ba i e no maliza ion. A decisi e ad an age o he
me hod is ha he p oblem is sol ed in momen um space by
a dispe sion in eg al.
To explain he cons uc ion, we fi s exp ess he an ich ono-
logical p oduc Tnin e ms o he TOPs (Tk)1≤k≤n.Le
N={x1,...,xn}and I⊆Nwi h |I| = 0 elemen s. Define
T|I|(I)=T|I|(xi:xi∈I). By he s anda d in e sion o a
o mal powe se ies wi h noncommu ing e ms in e ms o
se composi ions, we ob ain
T|N|(N)=
n
k=1
(−)n+k
I1$···$Ik=N
T|I1|(I1)···T|Ik|(Ik),
(C.3)
whe e he disjoin union is o e nonemp y blocks I j.The e -
minology o an ich onological p oduc s is app op ia e, since
i I∩(J+V−)=∅, hen T(I∪J)=T(J)T(I).
Re a ded and ad anced p oduc s, deno ed by Rnand An
espec i ely, a e he coe ficien s in he pe u ba i e expansion
o he espec i e e a ded and ad anced in e ac ing fields. Fo
hem we ollow he con en ion in he book [23], iden ical o
ha o [21] excep ha Rnand Anha e an ex a ac o in−1.
In gene al, Bogoliubo ’s defini ions ead:
Rn+1(x1,...,xn+1):= in
I⊂{1,...,n}
(−1)|I|T|I|(I)T|Ic|+1(Ic,xn+1),
(C.4)
An+1(x1,...,xn+1):= in
I⊂{1,...,n}
(−1)|I|T|Ic|+1(Ic,xn+1)T|I|(I),
(C.5)
123
Eu . Phys. J. C (2021) 81:131 Page 23 o 25 131
whe e Ic:= {1,...,n} I. Eps ein and Glase [21] p o e ha
An+1,Rn+1ha e ad anced o e a ded suppo , espec i ely:
supp An+1⊆{x∈Mn+1:xj−xn+1∈V+∀j};
supp Rn+1⊆{x∈Mn+1:xj−xn+1∈V−∀j}.
In he induc ion s ep n→n+1 nei he he Tn+1no he Rn+1
no he An+1a e known. Bu by he induc ion hypo hesis
he di e ence Dn+1, defined by Dn+1:= An+1−Rn+1,
only depends on known quan i ies. Fo ins ance, in D3 he
unknown T3has d opped ou – and T1,T2a e uniquely gi en
in e ms o T1and T2.I ollows ha Dn+1has causal suppo :
supp Dn+1⊆{x∈Mn+1:xj−xn+1∈V+∀j}
∪{x∈Mn+1:xj−xn+1∈V−∀j}.
I one finds a way o ex ac he ad anced pa An+1o Dn+1,
ha is, o spli he OVD Dn+1in o An+1and −Rn+1in
such a way ha he la e wo sa is y he jus gi en suppo
p ope ies, hen one can cons uc a candida e o Tn+1.17
Fo he sake o no maliza ion condi ions, a his s age we
may add o Tn+1any OVD suppo ed on n+1which is sym-
me ic in x1,...,xn+1.TheDn+1 ulfils all he no maliza-
ion condi ions, in pa icula he ‘Causal Wick expansion’
and ‘T ansla ion in a iance’, because o he alidi y o hose
o he induc i ely gi en (Tk)1≤k≤n. The e o e, he spli ing
p oblem o Dn+1 ansla es in o a consonan p oblem o he
coe ficien s d(x1−xn+1,...,xn−xn+1)∈S(R4n,C)in he
Wick expansion o Dn+1, yielding a, (x1−xn+1,...,xn−
xn+1)∈S(R4n,C), which a e he coe ficien s in he Wick
expansion o An+1and Rn+1, espec i ely.
In fine, by he induc ion p ocess, one specifies he ambigu-
i y in he acuum expec a ion alue o each Tn+1by adding
o i a con ac e m, ha is,
(x1−xn+1,...,xn−xn+1)
+
|a|≤ω
ca∂aδ(x1−xn+1,...,xn−xn+1), (C.6)
whe e ωis he singula o de o he pe inen d(x1−
xn+1,...) and he coe ficien s ca∈Cdepending on he
mul i-index aa e a bi a y, up o es ic ions coming
om he ‘Poinca é co a iance’ and ‘O he in a iance ules’
equi emen s.
C.3: Dispe sion in eg als om spli ing in BEG
no maliza ion: he cen al solu ion
Fo simplici y, he e we es ic ou sel es o he case o wo
ou - a iables, ele an o his pape . Fo he Fou ie ans-
17 Tha some imes needs o be symme ized, by adding a sui able OVD
suppo ed on n+1.
o m o ∈SR8we employ he ollowing con en ion:
(y1,y2)=(2π)−4dk1dk2e−i(k1y1+k2y2)ˆ
(k1,k2).
(C.7)
Le ±:= V±×V±hence o h. Gi en a “causal dis ibu-
ion”, ha is, d∈S(R8)wi h
supp d⊆+∪−and sd(d)<∞,(C.8)
byaspli ing solu ion o dwe mean a dis ibu ion a∈S(R8)
wi h
(a−d)S(R8 −)=0,supp a⊆+and sd(a)≤sd(d).
(C.9)
In wha ollows we assume ha he Fou ie ans o m ˆ
do
he causal d-dis ibu ion we wish o spli anishes in an open
ball R⊂R8cen e ed a k=0. This holds i all p opaga o s
con ibu ing o da e massi e, as i is he case in his pape –
see [21, Sec . 5.2]. Also in [21] i is shown o any spli ing
solu ion a ha ˆ
d|R=0 en ails analy ici y o ˆa(k)on R.
In his case he e exis s a dis inguished spli ing solu ion, he
so-called cen al solu ion ac, cha ac e ized by he condi ions
∂aˆac(0)=0, o all |a|≤ω(d). (C.10)
As indica ed in Eq. (C.6), o sd(d)≥8 – i.e., o ω(d)≥0,
as defined in Eq. (C.2) – he spli ing solu ion o dis no
uniquely de e mined. Any wo solu ions a1and a2di e by
a1(y)−a2(y)=
ω(d)
|a|=0
Ca∂aδ(y)o equi alen ly,
ˆa1(k)−ˆa2(k)=1
(2π)4
ω(d)
|a|=0
Ca(−ik)a,
wi h a bi a y cons an s Ca∈C.
Essen ial o dealing wi h ou si ua ion is ha he cen al
solu ion o he spli ing p oblem in momen um space can
be compu ed by a dispe sion in eg al. Now we ske ch he
de i a ion o a ew e sions o his dis inguished spli ing
in eg al. 18 The nai e way o ex ac he ad anced pa ao
dis o mul iply he la e by a θ- unc ion:
anai e(y1,y2):= d(y1,y2)χ(y1,y2)wi h
χ(y1,y2):= θ(y1 1)+(y2 2),
whe e := ( 1,
2)∈V+×V+is a bi a y. Bu o
sd(d)≥8 he poin wise p oduc dχexis s only as an ele-
men o S(R8 {0}). The e o e, he spli ing p oblem is an
18 Fo u he de ail we e e o [39, Sec . 3.2], which elies on [21,
Sec . 6.5].
123
131 Page 24 o 25 Eu . Phys. J. C (2021) 81:131
ex ension p oblem: we ha e o ex end dχ∈S(R8 {0}) o
an a∈S(R8)such ha sd(a)=sd(dχ) =sd(d).19
The p oblem is s udied in i s pa icula s in [23, Sec . 3.2.2].
Gi en awi h singula o de ω, he e exis s an ob ious ex en-
sion aω, belonging in he dual space S
ω(R8)o
Sω(R8):= { ∈S(R8):∂b (0)=0 o all |b|≤ω},
uniquely de e mined by he equi emen ha sd(aω)=
sd(d). Nex , a p ojec ion is in oduced:
Wω:S(R8)−→ Sω(R8);
Wω (y):= (y)−w(y)
ω
|b|=0
yb
b!∂b (0), (C.11)
whe e he sui ably decaying unc ion wmus ulfil w(0)=1
and ∂bw(0)=0 o 1≤|b|≤ω. One e ifies ha a solu ion
aw(depending on he choice o he unc ion w) o he spli ing
p oblem (C.9) is ob ained by se ing
aw| :=aω|Wω .(C.12)
The aωin ol ed he e is dχwi h enla ged domain. I u he -
mo e assump ion d|R=0 is sa isfied, he in a ed beha iou
o d(y)is ha mless. Hence, one may simply choose w(y)=1
o all y∈R4. Then he co esponden spli ing solu ion
aw=1is ac ually he cen al solu ion ac(C.10). Subs i u ing
dχ o aωand u he using he con olu ion o mula
ˆ
ˆχ(k)=(2π)4i
2πR
d
+i0ˆ
(k− )
and
g =(2π)−4ˆ
ˆg, we see ha
ˆac(k)=i
2πR
d
+i0ˆ
d(k− ) −
ω
|b|=0
kb
b!∂bˆ
d(− ).
(C.13)
This spli ing in eg al does no depend on he choice o ∈
V+×V+. Mo eo e , o k∈Vη×Vη, whe e η∈{+,−},
we may choose := ηk– ha a y wi h kis admissible.
Wi h some ex a wo k [39, P op. 3.4], his o mula is hen
simplified in o a con e gen dispe sion in eg al:
ˆac(k)
=iη
2πR
d ˆ
d( k)
( −ηi0)max{ω+1,0}(1− +iη0) o k∈Vη×Vη.
(C.14)
In he applica ions ea ed in his pape , d( k)is o he o m
ˆ
d( k)
=ηsgn( )θ( 2− 2
min) 2k2
1, 2k2
2, 2(k1+k2)2 o k∈Vη×Vη,
(C.15)
19 A p io i, i migh happen ha sd(dχ)<sd(d); bu in he applica-
ions o Eps ein–Glase no maliza ion known o us one always finds
sd(dχ) =sd(d). Hence we assume he la e ela ion o hold ue.
o some ∈S(R3), whe e min >0 depends on he squa es
o he momen a. So finally, in oducing he new in eg a ion
a iable u:= 2, he in eg al (C.14) goes o e in o
ˆac(k)
=i
2π∞
2
min
du (uk2
1,uk2
2,u(k1+k2)2)
umax{ω/2+1,0}(1−u+iη0) o k∈Vη×Vη,
(C.16)
whe e ·deno es he in ege pa .
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