Dear Celine,
our definition of the operators is identical what you use. For
reference, I have also attached a plot that is almost equivalent to the
one you've shown on p.16. The operator is now O_WWW as well, and the
coupling is almost the same: C_WWW/Lambda^2 = 6/TeV^2. The second page
shows the cross section for the 2->2 process W^+W^- -> W^+W^- (see below
why this is interesting).
The plot in my talk uses the dim-8 operator T1 with f_T1/Lambda^4 =
25/TeV^4, formfactor settings for the curve with formfactor are
Lambda_FF=1295 GeV, n_FF=2.
The bound is derived in the following way (I'll abbreviate lambda_1 as
l1, etc.):
We also start from the partial wave expansion and use orthogonality of
the D-functions to rewrite it as
a^j_(l1-l2,l3-l4) = 1/(32 pi) int_{-1}^{1} d cos(Theta) T D^j_(l1-l2,l3-l4)
Unitarity of the S-matrix tells you Re(a^j_(lm)) < 0.5 for any amplitude
T of a 2->2 scattering process. We choose vector boson scattering VV->VV
with the combinations W^+W^- -> W^+W^-, W^+Z -> W^+Z and additionally ZZ
-> ZZ for the operators contributing to the neutral vertices. These
amplitudes depend only on the center-of-mass energy sqrtS, which will
later be plotted on the x-axis, and the scattering angle Theta which is
integrated over.
Note that this is a theory calculation to check a theory value, so
experimental issues (like how to build a W+W- collider or how to exactly
measure the final-state momenta) do not play any role.
We find that the most stringent bounds typically come from the j=0
modes, so we check only those. In this case l1-l2=0=l3-l4, which leaves
us with 9 different helicity combinations to check for each process. We
finally require that none of them should violate unitarity individually,
so we take the maximum of them to determine the value where the
unitarity bound is crossed.
Best regards,
Michael
On 13.12.2012 18:35, Celine Degrande wrote:
> Dear Shih-Chieh,
>
> It seems indeed that the scale at which unitarity is violated is
> different for Michael and for me. However, there are differences : We do
> not use the same process, the same operator and maybe also not the same
> assumptions for the derivation of the unitarity bound. I will look at it
> for WW scattering with dimension-six to see where the difference mainly
> come from.
>
> Michael, could you give your operator with all the convention and some
> details how you derive this bound? You can find the details about our
> result in our paper (1205.4231).
>
> Cheers,
>
> Celine
>
>
>
> On Wed, 2012-12-12 at 13:48 -0800, Shih-Chieh Hsu wrote:
>> Sorry for that I didn't complete my email. Please see a small
>> modifications below.
>>
>> On 12/12/12 1:41 PM, Shih-Chieh Hsu wrote:
>>> Dear Oscar, Celine and Michael,
>>>
>>> Thank you very much for your comprehensive introductions of your
>>> electroweak effective field theories.
>>> I learned more depth about different aspects of your EFT today.
>>>
>>> I have a couple of questions looking forward to your further comments.
>>>
>>> Q1. What is the appropriate approach to probe gauge couplings
>>> in a consistent way in inclusive diboson, VBS diboson+jj and triboton
>>> production at hadron colliders?
>>>
>>> Celine commented that we don't need energy dependence form factor
>>> to preserve unitary for WW production by considering all dimension 6
>>> operators.
>>> However, Michael use VBS WW+jj production as an example to show that
>>> we need to
>>> introduce s^hat dependent form factor in order to preserve unitarity.
>>> The TGC limit derived in inclusive WW might not be compared to the TGC
>>> limit
>>> derived in VBS WW+jet neither Triboson production.
>>>
>>> Q2. Could Oscar give us a summary of the coupling constants with
>>> necessary form factor for the Dimension 6 and Dimension8 operators?
>>> To my understanding, we only introduce Dimension 8 operators for
>>> neutral gauge couplings.
>> For practical approaches, we agreed to only introduce Dimension 8
>> operators for neutral gauge couplings.
>>
>> Cheers,
>>
>> Shih-Chieh
>>
>>>
>>> Q3. It looks to me VBFNLO, MadGraph5 and Oscar's package all implement
>>> the same effective field theory.
>>> Do I understand it correctly that we should be able to reproduce
>>> the same calculations
>>> from each generator?
>>>
>>> Q4. In the end, the TGC/QGC operators are linear expansions of the
>>> effective Lagrangian.
>>> Do I expect the production cross-section can be represented as
>>> a quadratic dependent function
>>> of each gauge coupling coefficients?
>>>
>>> Sincerely yours,
>>>
>>> Shih-Chieh
>>>
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>>
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--
Michael Rauch Phone: +49 721 608 47028
Institut fuer Theoretische Physik Fax: +49 721 608 43582
Karlsruher Institut f. Technologie (KIT)
76128 Karlsruhe
Germany
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