heavy higgs lineshape many questions few answerspersonalpages.to.infn.it/~giampier/ll12.pdf · 0...

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

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).

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 .

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

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

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

Step 1

Given the� Higgs self-energy,

SHH � s � S 1 !HH�#"

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

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 �

Step 3

Using the� decomposition

V 1 !d / p � � s � M2H V 1 !d / p 1 � s � M2

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.

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

ZH s � sH�#"

s � sH2

Step 5

The main result follows:

V 1 !d � � sH � sH � 1

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

The Unbearab le Heaviness of Higgsing

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

: ;<;>=�?A@ BCBEDGF<HAIKJML>N O

P+Q-=RF SAHTL>N O

:UQWV�XZY[=\S]@ ^_S`DGF<H IKJ L>N O

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

BR � H � F �

k�lnm�o pqlnm�or

sEtvu wKtMx kElRyEpzl{y+|}�~ s��]�_�Ux pql ~

� x pql ~

|}��

�\��\� �U��A�� `� �+��¡ ��£¢¤�¡�

¥ ��£¦§�A��`��A��`�©¨Zª ¬«®£¯

� �\� �U��A�� ¡�{� �� ¨Zª ��¢°��

¥ ��¦+�©�¡�{� «±£¯�²

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).

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.

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.

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

H ln2 s

AB ¾ � m2q

ln2 sm2

but the behavior for s �¼» (unitarity) should not/cannotbe used to sim ulate the interference for s ¿ M2

The only relevant message is: unitarity requires theinterference to be destructive at large s.

Schemes beyond ZWA

S �TÁ � 8-8-8A� Vprod ��Á � 8-8-8A� BW �TÁG� Vdec �TÁG�

S �TÁ � 8-8-8A� Vprod �ÂÁ � 8-8-8K� prop �TÁq� Vdec �TÁG�

S �TÁ � 8-8-8A� Vprod sH � 8-8-8 prop �TÁG� Vdec � sH �

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.

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).

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 .

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 .

��

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 .

�!�! "�!�! # !�!�!$#�# !�!%#�&�!�!$# '�!�!%# (!�!)# *�!�!!�+ !�!�!

!�+ !�!�#

!�+ !�!&

,�- .�/ 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

WYX[Z.

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

� s � M2H � 2 � 8-8-8

anGF M2

Derive

7 H GF M3

2$ 2a1 1

� GF M2H

� 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

No large variations up to 1TeV with a breakdo wn of theperturbative expansion around 1 8 74 TeV .

in the Higgs-Goldstone model one has

bH�H 1 8 1781 gH

�0 8 4125 g2

1 8 1445 g3H

gH GF� 2

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 .

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.

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

��

�� ¡ ¢ £¤

¥¦ ¤

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.

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

an , n XHG � , GF Á2f

Let e p Xpf Á the width computed by PROPHECY4F, we

redefine the total width as

e totf Á � Xp � XHG � � XHG 3

an , n

where now a0 Xp � XHG.

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

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

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 �

º�»�» ¼»�» ½�»�»¿¾�»�»ÁÀ�»�» Â »�»�» Â�º�»�»»

»�Ã Ä

Â�Ã 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 .

Û�Ü�Ü Ý�Ü�Ü Þ�Ü�Ü ß Ü�Ü�Ü ß à�Ü�ÜÜ

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 .

Off-shell Pandora box

ö e Hö MH ö � ö ' prod

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

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 �

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

s � sH2

c ds / c

AH b F2

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

s Aij � s �T8

Step 4

e H b F � s � 12f

sd û H b F

s / cAH b F

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.

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

' ij b H b F � s � 8-8-8c d e tot

BR � H � F �

The comple x-mass scheme

can be translated into a more familiar language by introducingthe Bar – scheme.

M2H 6 2

H� 7 2

H e H MH 7 H

It follows a remarkable identity:

1s � sH

MHs � M

� 1 �

showing that the Bar-scheme is equivalent to introducing arunning width in the propagator with parameters that are notthe on-shell ones.

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! "# $ % & ' ( ) * + , - . / 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 jk l m n o p q r s t u v w x y z { | } ~ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼½ ¾ ¿ ÀÁ Â Ã Ä Å Æ Ç È É Ê

ËÌ ÍÎ Ï Ð Ñ Ò

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µ¶ · ¸ ¹º »

¼ ½ ¾ ¿ À Á Â Ã Ä Å ÆÇ È É ÊË Ì Í Î Ï Ð Ñ Ò Ó

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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 Â 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

From Seymour paperit’s Bar - scheme !!!

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

Modified Seymour scheme?

ss � m2

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.

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

� H ln2 sm2

AB � � m2q

ln2 sm2

but the behavior for s �� (unitarity) should not/cannotbe used to sim ulate the interference for s � M2

The only relevant message is: unitarity requires theinterference to be destructive at large s.

Leading K -factor for the decay width H VV:

OS K � 1�

a1GFM2

16 � 2� 2

CPP K � 1�

a1GFsH

16 � 2� 2

� � a2�

3i � a1 � GFsH

16 � 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.

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 ��

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 " .

heavy higgs lineshape many questions few answerspersonalpages.to.infn.it/~giampier/ll12.pdf · 0...

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