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Prolegomena MF Gauge Schemes THU Conc lusions
Heavy Higgs LineshapeMany Questions Few Ans wers
Giampiero PASSARINO
Dipartimento di Fisica Teorica, Universita di Torino, Italy
INFN, Sezione di Torino, Italy
Loops and Legs at Wernigerode, 16 April 2012
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Prolegomena MF Gauge Schemes THU Conc lusions
One might wonder why
considering an heavy SM Higgs boson. There are classicconstraints on the Higgs boson mass coming from
unitarity
triviality
vacuum stability
precision electroweak data
absence of fine-tuning
However, the search for a SM Higgs boson over a mass rangefrom 80 GeV to 1 TeV is clearly indicated as a priority in manyexperimental papers (confirmed by Higgs conveners).
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Prolegomena MF Gauge Schemes THU Conc lusions
Preludio
From matrix� elements to pseudo-obser vables
Most of the people just want to use some well-definedrecipe without having to dig any deeper;
however, there is no alternative to a complete descriptionof LHC processes which has to include the completematrix elements for all relevant processes;
splitting the whole S -matrix element into components isjust conventional wisdom.
However, the precise tone and degree of formality must bedictated by gaug e invariance .
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Prolegomena MF Gauge Schemes THU Conc lusions
Heavy or light, standar d or not
the Higgs boson is an unstable particle; as such it is describedby a comple� x pole on the second Riemann sheet
sH � M2H�
SHH sH � M2t � M2
H � M2W � M2
Z 0
To lowest order accuracy the Higgs propagator can be rewrittenas
�� 1H s � sH
The complex pole describing an unstable particle isconventionally parametrized as
si 2i � i i � i
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Prolegomena MF Gauge Schemes THU Conc lusions
Example� of a none xistent object
is it contradictory (Hume) or logically ill-formed (Kant) orwholeheartedly embraceable (Leibniz)?A (gauge) meaningful definition of a nonexistent object. Take
H � P ��� Z��� p1 � � Z��� p2 �Work in the R� -gauge; for any quantity f ����� write
f ����� f � 1 � � f ����� �
f � 1 � 0
and look for � -independence
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Prolegomena MF Gauge Schemes THU Conc lusions
Step 1
Given the� Higgs self-energy,
SHH � s � S 1 !HH�#"
g4 g2
16 $ 2 % HH � s � �&"g4
Let MH be the renormalized Higgs mass, we obtain
% 1 !HH ��� � s � M2H � s � M2
H ' 1 !HH ��� � s � M2H �
The main equation is the one for the Higgs complex pole,
sH � M2H�
S 1 !HH � � sH � M2H 0
from which we derive M2H sH
�#"g2
(
Prolegomena MF Gauge Schemes THU Conc lusions
Step 2
Easy to prove:)) � S 1 !HH � � sH � sH 0
Next consider the one-loop vertices contributing to H � ZZ
and obtain an S -matrix element
A 1 !V V 1 !d * �+� � V 1 !p p2� p1� e� � p1 �-, 1 � e
� � p2 �-, 2 �
.
Prolegomena MF Gauge Schemes THU Conc lusions
Step 3
Using the� decomposition
V 1 !d / p � � s � M2H V 1 !d / p 1 � s � M2
H�
V 1 !d / p � � s � M2H
we obtain the following results:
1 after reduction to scalar form-factors there are no scalarvertices remaining in
V 1 !d � � sH � sH .
2 Furthermore,
V 1 !p � � sH � sH 0.
0
Prolegomena MF Gauge Schemes THU Conc lusions
Step 4
Compute renormalization Z -factors for the external legs
s � M2H�
S 1 !HH � � s � M2H
s � sH 1� S 1 !HH � � s � sH � S 1 !HH � � sH � sH
s � sH
1�
ZH s � sH�#"
s � sH2
1
Prolegomena MF Gauge Schemes THU Conc lusions
Step 5
The main result follows:
V 1 !d � � sH � sH � 1
2
ZH ����� � ZZ ����� A 0 ! 0
which gives to key to deal with processes where unstableparticles play a role:
Define them at the comple x pole
2
Prolegomena MF Gauge Schemes THU Conc lusions
The Unbearab le Heaviness of Higgsing
3
Prolegomena MF Gauge Schemes THU Conc lusions
Table: The Higgs boson complex pole at fixed values of the W 4 tcomplex poles5 compared with the complete solution for sH 4 sW and st
6H [GeV] 7 W [GeV] f 7 t [GeV] f 7 H [GeV] d
200 2 8 088 1 8 481 1 8 355250 3 8 865300 8 8 137350 14 8 886400 26 8 5986
H [GeV] 7 W [GeV] d 7 t [GeV] d 7 H [GeV] derived200 2 8 130 1 8 085 1 8 356250 2 8 119 0 8 962 3 8 823300 2 8 193 0 8 836 8 8 139350 2 8 607 0 8 711 14 8 653400 3 8 922 0 8 566 25 8 498
9
Prolegomena MF Gauge Schemes THU Conc lusions
: ;<;>=�?A@ BCBEDGF<HAIKJML>N O
P+Q-=RF SAHTL>N O
:UQWV�XZY[=\S]@ ^_S`DGF<H IKJ L>N O
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Prolegomena MF Gauge Schemes THU Conc lusions
Lineshape Master Form ulas
The final result is written in terms of pseudo-obser vables
' ij b H b F � s � 1$c d ' ij b H
c d s2
s � sH2
c d e H b Ffs
' ij b H b F � s � 8-8-8c d e tot
Hfs
BR � H � F �
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Prolegomena MF Gauge Schemes THU Conc lusions
h
i
j
k�lnm�o pqlnm�or
sEtvu wKtMx kElRyEpzl{y+|}�~ s��]�_�Ux pql ~
�
� x pql ~
|}��
�\���\� �U���������A�������� �����`� �+���¡ ���£¢¤�¡�
¥ ���£¦§�A���`��A���`�©¨Zª ¬«®£¯
� �\� �U���������A�������� ���¡�{� ��� ��� ¨Zª ����¢°���
¥ ����¦+�©�¡�{� «±£¯�²
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Prolegomena MF Gauge Schemes THU Conc lusions
It is wor th noting that
the introduction´ of complex poles does not imply complexkinematics. Only the residue of the propagator at the complexpole becomes complex, not any element of the phase-spaceintegral (details are non trivial aspects of L � C � and analyticcontinuation).
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Prolegomena MF Gauge Schemes THU Conc lusions
About on-off gaug e variance
If the Higgs¶ boson is off shell,
in LO and NLO QCD in most cases the matrix element stillrespects gauge invariance,
but in NLO EW gauge invariance is lost, unless the rightscheme is used.
Technically speaking, we have a matrix element
e � H � F � f s · 2H
where s is the virtuality of the external Higgs boson, 6 H is themass of internal Higgs lines and Higgs wave-functionrenormalization has been included.
¸
Prolegomena MF Gauge Schemes THU Conc lusions
The follo wing happens:
f � sH � sH � is gauge-parameter independent to all orderswhile
f � 6 2H � 6 2
H � is gauge-parameter independent at one-loopbut not beyond,
f � s � 6 2H � is not.
In order to account for the off-shellness of the Higgs boson wecan use (at one loop level) f � s � s � , i.e.,
we intuitively replace the on-shell decay of the Higgsboson of mass 6 H with the on-shell decay of an Higgsboson of mass
fs and
not with the off-shell decay of an Higgs boson of mass 6 H.
¹
Prolegomena MF Gauge Schemes THU Conc lusions
º
Prolegomena MF Gauge Schemes THU Conc lusions
Simulating interf erence?
Of course, the behavior for s �¼» is known and any correcttreatment½ of PT (no mixing of different orders) will respectunitarity cancellations. The Higgs decays almost completelyinto longitudinals Zs, thus for s �¼»
AH ¾ sm2q
2M2Z
H ln2 s
m2q
AB ¾ � m2q
2M2Z
ln2 sm2
q
but the behavior for s �¼» (unitarity) should not/cannotbe used to sim ulate the interference for s ¿ M2
H.
The only relevant message is: unitarity requires theinterference to be destructive at large s.
À
Prolegomena MF Gauge Schemes THU Conc lusions
Schemes beyond ZWA
OFFBW
S �TÁ � 8-8-8A� Vprod ��Á � 8-8-8A� BW �TÁG� Vdec �TÁG�
OFFP
S �TÁ � 8-8-8A� Vprod �ÂÁ � 8-8-8K� prop �TÁq� Vdec �TÁG�
CPP
S �TÁ � 8-8-8A� Vprod sH � 8-8-8 prop �TÁG� Vdec � sH �
Ã
Prolegomena MF Gauge Schemes THU Conc lusions
The proofÄ that CPP-scheme
satisfies gaug e-parameter independence can be sketched asfollows:
S � s � Vprod � sH � Vdec � sH �1 � S ÅHH � sH � � s � sH � �)
) � Vprod / dec � sH � 1 � S ÅHH � sH � � 1Æ 2 0
where � is an arbitrary gauge parameter and S ÅHH � sH � is thederivative of SHH � s � computed at s sH. Note that thisequation follows from the use of Nielsen identities.
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Prolegomena MF Gauge Schemes THU Conc lusions
È_ÉCÉ ÊAÉCÉ ËCÉCÉ ÌCÉCÉ Í<ÉCÉCÉ ÍZÈ_ÉCÉ Í ÊAÉCÉÉWÎ ÉCÉ
ÉWÎ ÉWÍ
ÉWÎ ÉAÈ
ÉWÎ ÉCÏ
ÉWÎ É_Ê
ÉWÎ ÉCÐ
Ñ`Ò ÓKÔ ÕAÖ
×Ø ÙÚÙ Ú ×Ø
Figure: The normalized invariant mass distribution in theOFFP-scheme with running QCD scales for 600 GeV (black),700 GeV (blue), 800 GeV (red).
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Prolegomena MF Gauge Schemes THU Conc lusions
Ü_ÝCÝ ÞAÝCÝ ßCÝCÝ àCÝCÝ á<ÝCÝCÝ áZÜ_ÝCÝ á ÞAÝCÝÝWâ ÝCÝ
ÝWâ ÝWá
ÝWâ ÝAÜ
ÝWâ ÝCã
ÝWâ Ý_Þ
ÝWâ ÝCä
å`æ çKè éAê
ëì íîí î ëì
Figure: The normalized invariant mass distribution in theOFFP-scheme (blue) and OFFBW-scheme (red) with running QCDscales at 800 GeV .
ï
Prolegomena MF Gauge Schemes THU Conc lusions
ðCñCñ òCñCñ ó_ñCñ ôCñCññWõ ñCñ
ñWõ ñWö
ñWõ ñA÷
ñWõ ñCð
ñWõ ñ_ø
ñWõ ñCò
ù`ú ûKü ýAþ
ÿ� �� � � ÿ�
Figure: The normalized invariant mass distribution in theOFFP-scheme with running QCD scales for 600 GeV . The blue linerefers to 8 TeV , the red one to 7 TeV .
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Prolegomena MF Gauge Schemes THU Conc lusions
����� ���� ���� ����� � ����� ������� � ������� ������ ������ ����� ������ ������ ���
��� ��� ��
�� �� � � ��
Figure: The normalized invariant mass distribution in theOFFP-scheme with running QCD scales for 800 GeV . The blue linerefers to 8 TeV , the red one to 7 TeV .
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Prolegomena MF Gauge Schemes THU Conc lusions
�!�! "�!�! # !�!�!$#�# !�!%#�&�!�!$# '�!�!%# (!�!)# *�!�!!�+ !�!�!
!�+ !�!�#
!�+ !�!&
,�- .�/ 01
234 5 4 678 9:;
< => ? @A =BCA =
DFE�GIH J / K L M N O P Q R /D / S S N S H J / K L M N O P Q R /
Figure: The invariant mass distribution in the OFFP-scheme forTH U 800 GeV with THU introduced by V tot
H
WYX[Z.
\
Prolegomena MF Gauge Schemes THU Conc lusions
Estimate� of THU is also neededIt would we desirable to include two- and three-loopcontributions as well in 7 H and for some of these contributionsonly on-shell results have been computed so far.
Use the Higgs-Goldstone Lagrangian of the SM
SHH � s � A M2H�
B � s � M2H � � C
M2H
� s � M2H � 2 � 8-8-8
A ]
n ^ 0
anGF M2
H
2f
2$ 2
n
_
Prolegomena MF Gauge Schemes THU Conc lusions
Derive
7 H GF M3
H
2f
2$ 2a1 1
� GF M2H
2f
2$ 2
a2
a1
� 8-8-8
The ratio a2 ` a1 can be used to estimate that the fir stcorrection to 7 H is roughly given by
* H 0 8 350119GF
6 2H
2f
2$ 2
No large variations up to 1TeV with a breakdo wn of theperturbative expansion around 1 8 74 TeV .
a
Prolegomena MF Gauge Schemes THU Conc lusions
in the Higgs-Goldstone model one has
bH�H 1 8 1781 gH
�0 8 4125 g2
H�
1 8 1445 g3H
gH GF� 2
H
2 c 2d 2
7 H 168 8 84 GeV (LO), 180 8 94 GeV (NLO),186 8 59 GeV (NNLO) for 6 H 700 GeV .
Using the three known terms in the series we estimate a68 e credib le inter val of 7 H 186 8 59 f 1 8 93 GeV .
The difference NNLO � LO is 17 8 8 GeV and in the full SMour estimate is 7 H 163 8 26 f 11 8 75 GeV .
g
Prolegomena MF Gauge Schemes THU Conc lusions
It is better to quantify the uncer tainty
at the level of those quantities that characterize the resonance.
' prod total production cross-section
% d ' prod
d Á differential distributionh Á max � % max i the maximum of the lineshapeh Ákj � 12 % max i the half-maxima of the lineshape
A l�mlon d Á % the area between half-maxima
where Á is the Higgs virtuality.
p
Prolegomena MF Gauge Schemes THU Conc lusions
Table: Theoretical uncer tainty on the production cross-section, theheight of the maximum, the position of the half-maxima and the areaof the resonance.
qH [GeV] r H s tFu rwv prod s tFu rwx max s tFu y{z�|~}�y{z�� [GeV] r A s tFu600 5 � 3 n
6 � 0 m 6 � 3 n10 � 8 m 11 � 4 � n 2 � 5 } m 2 � 5 ��� m 2 � 5 } n 2 � 5 � n
4 � 8 m 6 � 0700 7 � 2 n
8 � 0 m 8 � 6 n14 � 8 m 16 � 0 � n 9 � 0 } m 8 � 0 ��� m 4 � 0 } n 4 � 0 � n
7 � 0 m 11 � 8800 9 � 4 n
9 � 7 m 10 � 6 n19 � 3 m 21 � 5 � n 18 � 2 } m 18 � 2 ��� m 6 � 1 } n 6 � 1 � n
8 � 7 m 9 � 5
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Prolegomena MF Gauge Schemes THU Conc lusions
���� ����� ����� ����� � �������� �����
��� �����
��� ����
��� ��� ��
��� � � ��� ��� � ¡ ¢ £¤
¥¦ ¤
Figure: The invariant mass distribution in the OFFP-scheme withrunning QCD scales for T H U 700 GeV . The red lines give theassociated theoretical uncertainty.
§
Prolegomena MF Gauge Schemes THU Conc lusions
The right factor e totH �TÁq�
in the master¨ formulas represents the “on-shell” decay of anHiggs boson of mass
f Á and we have to quantify thecorresponding uncertainty.
e Hf Á HG
3
n ^ 1
an , n XHG � , GF Á2f
2$ 2
Let e p Xpf Á the width computed by PROPHECY4F, we
redefine the total width as
e totf Á � Xp � XHG � � XHG 3
n ^ 0
an , n
where now a0 Xp � XHG.
©
Prolegomena MF Gauge Schemes THU Conc lusions
As long as , is not too large
we can define a p e ¿ 80 e credib le inter val as
e tot �TÁG� e p �TÁG�ªf e e p �TÁG�{� 1 f * e � e 54
maxh{«
a0« � a1 i p e , 4 Á
It is easily seen that
forf Á 929 GeV the two-loop corrections are of the
same size of the one-loop corrections
forf Á 2 8 6 TeV one-loop and Born become of the
same size
¬
Prolegomena MF Gauge Schemes THU Conc lusions
Table: Theoretical uncertainty on the total decay width, V totH . V p is the
total width computed by PROPHECY4F and ®¯V gives the credibleintervals.
f Á [GeV] e p[GeV] * e±° 68 e¯² * e±° 95 e¯²600 123 0 8 25 ° e¯² 0 8 42 ° e¯²700 199 0 8 62 ° e¯² 1 8 03 ° e¯²800 304 1 8 35 ° e¯² 2 8 24 ° e¯²900 449 2 8 63 ° e¯² 4 8 38 ° e¯²
1000 647 4 8 72 ° e¯² 7 8 85 ° e¯²1200 1205 13 8 1 ° e¯² 21 8 7 ° e¯²1500 3380 34 8 7 ° e¯² 57 8 8 ° e¯²2000 15800 98 8 9 ° e¯² 165 ° e¯²
³
Prolegomena MF Gauge Schemes THU Conc lusions
Table: Total theoretical uncertainty on the production cross-section,the height of the maximum, the position of the half-maxima and thearea of the resonance. The total is obtained by considering the THUon ´ H and on V tot with a cut µ X~¶ 1 · 5 TeV .
6H[GeV] * ' prod ° e¸²600 � 5 8 5 �
5 8 9700 � 7 8 0 �
7 8 5800 � 7 8 7 �
8 8 8900 � 7 8 0 �
8 8 9
¹
Prolegomena MF Gauge Schemes THU Conc lusions
º�»�» ¼»�» ½�»�»¿¾�»�»ÁÀ�»�»  »�»�» Â�º�»�»»
»�à Ä
Â�à ÀÆÅ Â »Ç�È
É�Ê Ë�Ì ÍÎ
ÏÐÑ Ò Ñ ÓÔÕ Ö×Ø
Figure: The invariant mass distribution in the OFFP-scheme (black)and in the CPP-scheme (red) for T H U 700 GeV for the processgg Ù H Ù ZcZc .
Ú
Prolegomena MF Gauge Schemes THU Conc lusions
Û�Ü�Ü Ý�Ü�Ü Þ�Ü�Ü ß Ü�Ü�Ü ß à�Ü�ÜÜ
á
Þ â ß Üãä
å�æ ç�è éê
ëìí î í ïðñ òóô
Figure: The invariant mass distribution in the OFFP-scheme (black)and in the CPP-scheme (red) for T H U 800 GeV for the processgg Ù H Ù ZcZc .
õ
Prolegomena MF Gauge Schemes THU Conc lusions
Off-shell Pandora box
ö e Hö MH ö � ö ' prod
÷
Prolegomena MF Gauge Schemes THU Conc lusions
Conc lusions
A conclusion is the place where you get tired of thinking(A. Bloch)
Many questions, few answers 8-8-8 but
aking the right questions takes as much skill as giving theright answers
Higgs signal is in a good shape
Interference is not in a good shape
ø
Prolegomena MF Gauge Schemes THU Conc lusions
Define the Signal: Step 1
General structure of any process containing a Higgs bosonintermediate state:
A � s � f � s �s � sH
�N � s � �
Signal (S) and background (B) are defined as follows:
A � s � S � s � � B � s �S � s � f � sH �
s � sH
B � s � f � s � � f � sH �s � sH
�N � s �
ù
Prolegomena MF Gauge Schemes THU Conc lusions
Step 2
Consider the process ij � H � F where i � j ú partons and F isa generic final state; the complete cross-section will be writtenas follows:
' ij b H b F � s � c 1
2 sd û ij b F
c ds / c
Aij b H
2
c d 1
s � sH2
c ds / c
AH b F2
ü
Prolegomena MF Gauge Schemes THU Conc lusions
Step 3
Strictly speaking and for reasons of gauge invariance, oneshould consider only
the residue of the Higgs-resonant amplitude at thecomplex pole
If we decide to keep the Higgs boson off-shell also in theresonant part of the amplitude (interference signal/backgroundremains unaddressed) then we can write
d û ij b Hs / c
Aij b H
2
s Aij � s �T8
ý
Prolegomena MF Gauge Schemes THU Conc lusions
Step 4
e H b F � s � 12f
sd û H b F
s / cAH b F
2 �
which gives the partial decay width of a Higgs boson ofvirtuality s into a final state F.
' ij b H Aij � s �s �
which gives the production cross-section of a Higgs boson ofvirtuality s.
þ
Prolegomena MF Gauge Schemes THU Conc lusions
Step 5
We can write the final result in terms of pseudo-obser vables
' ij b H b F � s � 1$c d ' ij b H
c d s2
s � sH2
c d e H b Ffs
8
' ij b H b F � s � 8-8-8c d e tot
Hfs
BR � H � F �
ÿ
Prolegomena MF Gauge Schemes THU Conc lusions
The comple x-mass scheme
can be translated into a more familiar language by introducingthe Bar – scheme.
M2H 6 2
H� 7 2
H6
H e H MH 7 H
It follows a remarkable identity:
1s � sH
1�
i e H
MHs � M
2H�
i e H
MHs
� 1 �
showing that the Bar-scheme is equivalent to introducing arunning width in the propagator with parameters that are notthe on-shell ones.
�
Prolegomena MF Gauge Schemes THU Conc lusions
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Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ � � � � � � � � � � � � � � � � � � � � � � � �� � � � � ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^_ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Ã Ä Å Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø
Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ � � � � � � � � � � � � � � � � � � � � �� � � � � � � � ! " # $ % & ' (
) * + , - . / 01 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t uv w x y z { | } ~ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹º » ¼ ½ ¾ ¿ À Á Â Ã Ä Å Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û üý þ ÿ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = >? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b
c de f gh i j k lm n o pq r s tu v w x yz { | } ~ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹º »¼ ½ ¾ ¿ ÀÁ  ÃÄ Å Æ ÇÈ É Ê Ë Ì Í Î Ï Ð Ñ ÒÓ Ô ÕÖ × Ø Ù Ú Û ÜÝÞ ß à á â ã ä åæ çè é ê ëì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " # $ % & ' ( ) * +
, - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n op q r s t u v w x y z { | } ~ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ®¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â
Ã
From Seymour paperit’s Bar - scheme !!!
Ä
Prolegomena MF Gauge Schemes THU Conc lusions
From Seymour paperno need to appr oximate
nothing to do with interf erence!!!
Å Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6
7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p qr s t u v w x y z { | } ~ � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Ã Ä Å Æ Ç È É Ê Ë Ì Í ÎÏ Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ � � � � � � � � � � � � � � � �� � � � � � � � � � � � � ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R ST U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~ � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Ã Ä Å Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ ÒÓ Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U VW X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~ � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Ã Ä Å Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à áâ ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " # $ % & '( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c
de f g hi j
k l m n o p q r s t uv w x yz { | } ~ � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Ã Ä Å ÆÇ È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ � � � � � � � �� � � � � � � � � � � � � � � � � � � � � ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K LM N O P Q R S T U V W X Y Z [ \ ] ^ _
`a b c de f
g h ij k lm n o pq r s t u v w x y z
{ | } ~ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸¹ º » ¼ ½ ¾ ¿ À Á Â Ã Ä Å Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý
þ
ÿ
Prolegomena MF Gauge Schemes THU Conc lusions
Modified Seymour scheme?
m2H
ss � m2
H�
i�
H
mHs� 1
not normalizable in � 0 ����not derivable from first principles, not justifiable, notsimulating interference
The Higgs propagator doesn’t know about real and imaginaryparts of boxes, e.g., gg ZZ.
I � 2 Re � H Re AH A B�
2 Im � H Im AH A BAH ��� s � sH � AH.
�
Prolegomena MF Gauge Schemes THU Conc lusions
Modified Seymour scheme?
Of course, the behavior for s �� is known and any correcttreatment� of PT (no mixing of different orders) will respectunitarity cancellations. The Higgs decays almost completelyinto longitudinals Zs, thus for s ��
AH � sm2q
2M2Z
� H ln2 sm2
q
AB � � m2q
2M2Z
ln2 sm2
q
but the behavior for s �� (unitarity) should not/cannotbe used to sim ulate the interference for s � M2
H.
The only relevant message is: unitarity requires theinterference to be destructive at large s.
�
Prolegomena MF Gauge Schemes THU Conc lusions
Leading K -factor for the decay width H VV:
OS K � 1�
a1GFM2
H
16 � 2� 2
�a2
GFM2H
16 � 2� 2
2
CPP K � 1�
a1GFsH
16 � 2� 2
� � a2�
3i � a1 � GFsH
16 � 2� 2
2 �
a1 � 1 � 40 � 11 � 35 i a2 ��� 34 � 41 � 21 � 00 i
Above 1 TeV the NNLO term dominates theK -factor � 0 � 17 � MH � 1 TeV � 4.
�
Prolegomena MF Gauge Schemes THU Conc lusions
More on interf erenceConsider gg 4 f, one would like to have the bestprediction for signal, i.e.,
� � H 4 f � at NLO+NNLO (NNLOdominates for large masses).
Therefore the Signal is (at least) at two-loop level and isnot gauge invariant for off-shell Higgs.
One-loop (complete) Background(+ Interference) is underconstruction, two-loop Background seems out of reach forthe foreseeable future.
Dura lex sed lex �����
�
Prolegomena MF Gauge Schemes THU Conc lusions
Warning
It is clear� that it does not make much sense to have anerror estimate beyond 1 � 3 TeV and, therefore, all resultsfor the Higgs lineshape that have a sizable fraction ofevents in this high-mass region should not be taken tooseriously. Here, once again, the only viable alternative todefine the Higgs signal is the CPP-scheme.
above 0 � 93 TeV perturbation theory becomesquestionable since the two-loop corrections start bebecome larger than the one-loop ones
above 1 � 3 TeV the error estimate also becomesquestionable since the expansion parameter is !� 0 � 7 andthe 95 " credible interval (after inclusion of the leadingtwo-loop effects) is 32 � 2 " .
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