[Psqateic] [EXTERNAL] Re: PSQ at EIC White Paper

Keh-Fei Liu liu at g.uky.edu
Wed Aug 17 15:35:21 EDT 2022 

Dear Oleg:

          Thank you for the comment. You are right, \bar{C} is import to 
allow one to obtain the same expression of T^{00} FF as from the
Hamilton and T_{\mu}^{\mu} with the anomaly in the operator form. In the 
rest frame, it is in fact the stability condition due to the 
cancellation between the traceless and trace part of the rest energy. 
The expression of \bar{C} in the rest frame has been worked out
by Cedric and me. If one wants to calculate T_{\mu\nu} separately for 
quark and glue, then one has the expression. However, the separation
is scale and scheme dependent. It is found (Liu:2021gco) that it is more 
physical to regroup them to the scale and scheme independent forms
which has physical implications.

Best,
Keh-Fei

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

          I have written a summary to Volker. Let me share with you in 
the following:

         Yes, Cedric first showed that \bar{C}_q and \bar{G}_g cancel. 
This is the consequence of energy-momentum conservation, i.e.
\partial_{\mu} T_{\mu\nu} = 0. Since both of them depend on \mu and 
scheme, it is not clear what they mean physically and how to
measure them experimentally. I noticed that they can be regrouped into 
terms that are scale and scheme independent and they have deep
physical meaning. These are written in my paper (Liu:2021gco) and I am 
working on a manuscript to explain it more transparently. In short,
\bar{C} = E_S - 1/3 E_T where E_S and E_T are those denoted as scalar 
and tensor energies for the nucleon where E_0 = E_T + E_S in
Xiangdong's notation. Note that T^{ii} = - 3\bar{C}  M in the rest frame 
and T^{ii} is the pressure-volume work. Therefore
PV = - V\partial E_0/\partial V = 1/3 E_T - E_S.   The volume dependence 
of PV infers an equation of state  E_0 = cV +d V^{-1/3}, where
c and d are constants. For example, c = - 1/4 <T_{\mu\mu}>/V, i.e. the 
negative of the glue and quark condensate density in the vacuum.
This is the basis for the MIT bag model formulation that E_0 = BV + 
\sigma/R.

Consequences:

1) \bar{C}= 0 is the pressure at equilibrium, a stability condition that 
reflects the cancellation of what can be termed `kinetic energy'
(i.e. E_T = (<x>_q + <x>_g) M) and the `potential energy' (i.e. E_S = 
1/4(Trace anomaly and chiral condensate from the vacuum condensate)). In 
Liu:2021gco, I have shown that the trace anomlay gives the potential 
energy in the charmonium whose string tension agrees with that of the 
Cornell potential very well. The coefficient of E_S and E_T in \bar{C} 
reflects their volume dependence.

2) The stability condition involves the cancellation of two energies 
with opposing volume (size) dependence. This fact implies that
these two energies should appear in the rest energy as denoted as <T> 
(for kinetic energy) and <V> (for potential energy). All bound states
have sizes and should have this energy-stability correspondence.

3) As an example, the binding energy of the Coulomb potential can be 
written as E_0 = - <T> or E_0 = 1/2 <V> from the Virial theorem.
However, we know these are not the correct physical picture. Solving the 
differential equation, we have E_0 = <T> +<V> for the Hamiltonian.
Since the stability condition is 2 <T> + <V> = 0, we can add it to E_0 = 
- <T> to obtain the correct answer E_0 = <T> + <V>. This is
what is done with the \bar{C} in Eq. (19). If one drops \bar{C}, one 
obtains the forward GFF T^00 as A_(0) M (equals 4/3 E_T) which is the 
kinetic energy. When the \bar{C} is added, one obtains E_0 = E_T + E_S 
which is the same expression as from the Hamiltonian as a cross check.

4) As a lattice practitioner, I care about expression that can be 
calculated on the lattice and measurable experimentally. The ongoing 
debate on the form of renormalized operators of the EMT confounds me. 
The dimensional regularization is a perturbative scheme, it does not
calculate the matrix elements. We have done the lattice calculations of 
the components in T^00. One only needs the BARE operators from
the continuum, discretize them on the lattice and consider 
renormalization and mixing under the lattice scheme. We have done all 
these. There is no controversy. The original Ji's traceless and trace 
separation of the Hamiltonian is applied. There is no problem. Most of 
the quantities
in E_T and E_S are experimentally measurable and they are all calculable 
on the lattice.

5) As you can see that even though \bar{C} is zero, it is due to the 
cancellation between E_T and E_S. Thus, both E_T and E_S should appear
in the expression for the rest energy. Only when \bar{C} is retained 
does one get the same T^00 as from the Hamiltonian. Also, as I explained 
in the previous emails, retaining \bar{C} will give the correct 
expression for T_{\mu\mu} which contains the anomaly. I should
point out that mass and rest energy are equal, but they are not the 
same. Mass is not additive in terms of its constituents (example given
by Okun as cited in my paper), but energy is.

6) It would be great if one can measure the trace anomaly form factor 
from the threshold J/\Psi production. But if it proves to be harder to
do than GPD, then one can obtain it from the combination of A, B, and C 
alternatively as in illustrated in the whitepaper. I suppose this is the 
motivation to derive Eq. (20) which is correct mathematically. Since 
\bar{C} is zero for all t, one can drop it to obtain Eq. (20) without 
having to
involve the explicit expression of \bar{C} (t) which will involve trace 
anomaly and may end up not having a
new expression in Eq. (20). In this case, it is a good motivation to 
drop \bar{C} in Eq. (20) only when one wants an alternative way to
obtain the trace anomaly form factor.

         We can retain \bar{C} in Eq. (19) as in the literature. For 
those who want to obtain the trace FF, they can drop
\bar{C} to obtain Eq. (20). For those who want to have a unique 
expression for the rest energy from T^00 FF and the Hamiltonian and the 
mass from T_{\mu\mu}, one can use \bar{C} to secure such a goal. People 
can choose what they want to do with the \bar{C}. On the other hand,
if one drops it from Eq. (19), one is shut out of the option of having 
unique expressions for T^00 and T_{\mu\mu}.





On 8/17/22 1:16 AM, teryaev wrote:
> Dear Keh-Fei,
>
>
> You are completely right, there are terms cancelling when the when 
> quark and gluon contributions are summed (due to equivalence 
> principle, whose extension may be used to explain their smallness even 
> separately), including C\bar ("proton cosmological constant")  
> together with
> anomalous gravitomagnetic moment, and often forgotten gravitoelectric 
> dipole moment. violating also CP. Indeed, C \bar is crucially 
> important when (separate) traces are calculated.
>
> Maybe in the text one can add after (6) something like "There are also 
> terms cancelling
> in the sum of quark and gluons due to conservation laws and 
> equivalence principle"?
>
> best regards,
> Oleg
>
> Keh-Fei Liu писал(а) 2022-08-13 08:56:
>> Dear Volker:
>>
>>        Now that we agree that there are \bar{C} terms in Eq. (6), take
>> the trace of the GFF in the rest frame, one
>> has the expression <T_{\mu\mu}> = [(A_q(0) + A_g(0)]M + Tr-Anomaly +
>> chiral condensate - [(A_q(0) + A_g(0)]M = Tr-Anomaly
>> +chiral condensate which is the correct expression. The last three
>> terms come from \bar{C} which is zero (for a good reason because
>> it is the stability condition). If one drops them first, one ends up
>> with the expression of [(A_q(0) + A_g(0)]M. Please explain why
>> it is acceptable and not wrong with an expression where the trace of
>> EMT does not contain trace anomaly. \bar{C} exists in many
>> publications in the literature, I don't understand the rationale for
>> dropping it in this document. It leads to (unintended) consequences.
>>
>> Best,
>> Keh-Fei
>>
>> On 8/12/22 4:07 PM, Volker Burkert wrote:
>>
>>> Dear Keh-Fei,
>>>
>>> * There is agreement for eq.(6) which deals with quark and gluon
>>> EMT separately, and the \bar{C} is needed (with opposite sign for q
>>> and g).
>>> * For eq.(19), which deals with the full EMT (quark + gluons} the
>>> \bar{C} terms should cancel out to zero, and the term \bar{C} should
>>> not appear.
>>>
>>> *
>>>
>>> *
>>> * Volker
>>>
>>> *
>>>
>>> -------------------------
>>>
>>> From: Keh-Fei Liu <liu at g.uky.edu>
>>> Sent: Friday, August 5, 2022 8:35 AM
>>> To: Volker Burkert <burkert at jlab.org>; psqateic-request at jlab.org
>>> <psqateic-request at jlab.org>
>>> Cc: Lorcé Cédric (M.) <cedric.lorce at polytechnique.edu>
>>> Subject: [EXTERNAL] Re: [Psqateic] PSQ at EIC White Paper
>>>
>>> Dear Volker and Latifa:
>>>
>>> Thank you for your tremendous effort in organizing this
>>> valuable whitepaper for the community.
>>>
>>> I now have a firm argument to show that one cannot drop
>>> the \bar{C} term in Eqs. (6) and (19). Take the
>>> forward FF of the trace T_{\mu\mu}, they give (A_q(0) + A_g(0))M.
>>> This is a trivial and meaningless result. It relates
>>> the matrix element of the trace to the mass via a sum rule of
>>> traceless matrix elements. It does not have the anomaly.
>>> Only when the \bar{C} is retained, do you get the correct expression
>>> with the anomaly. We should put the \bar{C} term back
>>> in these equations.
>>>
>>> Best,
>>> Keh-Fei
>>>
>>> On 8/3/22 4:33 PM, Volker Burkert wrote:
>>>
>>>> Dear Colleagues,
>>>>
>>>> We are sending you this email because you have registered to
>>>> participate in at least one of the workshops presented below,
>>>> which were held on December 15-16, 2020, March 17-19, 2021, and
>>>> July 19-23, 2021.
>>>>
>>>> One of the main goals of these workshops was to develop a white
>>>> paper focused on high-impact science requiring high integrated
>>>> luminosity at the low-to-medium center-of-mass energy of the U.S.
>>>> Electron-Ion Collider project.
>>>>
>>>> This white paper is ready to be submitted for publication. As a
>>>> participant in the workshop series, we invite you to become a
>>>> co-author of this white paper, which can be found at this link:
>>>>
>>>
>> https://www.femtocenter.org/sites/default/files/docs/PSQ_EIC_White_Paper.pdf 
>>
>>>> [1]
>>>>
>>>> Please visit this link and review the scope and content of the
>>>> white paper. If you decide to co-author the WP, please send an
>>>> email to burkert at jlab.org no later than August 12.  You may also
>>>> suggest minor changes to the text, acknowledgements, and
>>>> references, or make comments that would improve the WP.
>>>>
>>>> We have an agreement with a journal to publish the paper and we
>>>> need to submit the final version by August 20.
>>>>
>>>> With best regards,
>>>>
>>>> Volker & Latifa
>>>>
>>>> https://urldefense.proofpoint.com/v2/url?u=https-3A__indico.bnl.gov_event_9794&d=DwIDaQ&c=CJqEzB1piLOyyvZjb8YUQw&r=lRQm7JQTTNgh9H7STdZUgbq2inxaGxMaxrjYUFO8eiw&m=l2lzfiSorzZaJmWy6sbaGANWQcOigdLw-j8u1bRQXcd1m4NuZ6iZWC-PGLzXjUb6&s=V9gt5WONfE16eMHu286r_LnJYas1GgGEO9UF6VeTN_M&e=   [2]
>>>>
>>>> https://urldefense.proofpoint.com/v2/url?u=https-3A__indico.bnl.gov_event_10677&d=DwIDaQ&c=CJqEzB1piLOyyvZjb8YUQw&r=lRQm7JQTTNgh9H7STdZUgbq2inxaGxMaxrjYUFO8eiw&m=l2lzfiSorzZaJmWy6sbaGANWQcOigdLw-j8u1bRQXcd1m4NuZ6iZWC-PGLzXjUb6&s=_jiIzHApRK6pazlUA_zjbgRtdHUdkmWFZSzk7hTyJO4&e=   [3]
>>>>
>>>> https://urldefense.proofpoint.com/v2/url?u=https-3A__indico.bnl.gov_event_11669&d=DwIDaQ&c=CJqEzB1piLOyyvZjb8YUQw&r=lRQm7JQTTNgh9H7STdZUgbq2inxaGxMaxrjYUFO8eiw&m=l2lzfiSorzZaJmWy6sbaGANWQcOigdLw-j8u1bRQXcd1m4NuZ6iZWC-PGLzXjUb6&s=Tm4e8NSSGO76TJ0VG4GLrcicPYBNXeJitQxUdn1hIbM&e=   [4]
>>>>
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>>
>>
>> Links:
>> ------
>> [1] 
>> https://www.femtocenter.org/sites/default/files/docs/PSQ_EIC_White_Paper.pdf
>> [2] 
>> https://urldefense.proofpoint.com/v2/url?u=https-3A__indico.bnl.gov_event_9794&d=DwMDaQ&c=CJqEzB1piLOyyvZjb8YUQw&r=vVqGQabj76hiyFv-SdFNhS9JnbDxELkn2szNoXMYQyo&m=5AYW9vwxxp8DN9axYcgThaO2tDVhNhrtj-XHvMm22kfS_MbCzGceMTTjt26SJakY&s=4daDQQx1-C0DPsG0iFTkbWjsPATQb5ZddY_N9OGquj0&e=
>> [3] 
>> https://urldefense.proofpoint.com/v2/url?u=https-3A__indico.bnl.gov_event_10677&d=DwMDaQ&c=CJqEzB1piLOyyvZjb8YUQw&r=vVqGQabj76hiyFv-SdFNhS9JnbDxELkn2szNoXMYQyo&m=5AYW9vwxxp8DN9axYcgThaO2tDVhNhrtj-XHvMm22kfS_MbCzGceMTTjt26SJakY&s=vIOR4EUCQxbO8iay3K1a8zheeXLc26lrKytTPA4AF_U&e=
>> [4] 
>> https://urldefense.proofpoint.com/v2/url?u=https-3A__indico.bnl.gov_event_11669&d=DwMDaQ&c=CJqEzB1piLOyyvZjb8YUQw&r=vVqGQabj76hiyFv-SdFNhS9JnbDxELkn2szNoXMYQyo&m=5AYW9vwxxp8DN9axYcgThaO2tDVhNhrtj-XHvMm22kfS_MbCzGceMTTjt26SJakY&s=ULJyvLIlwTDZWjvQsqoes6fwCcIVQE67gO8XWJup3aI&e=
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