# Electromagnetic properties of the $D\bar{D}^{*}K$ molecular hexaquark state

Ulaş Özdem<EMAIL_ADDRESS>Health Services Vocational School of Higher
Education, Istanbul Aydin University, Sefakoy-Kucukcekmece, 34295 Istanbul,
Turkey

###### Abstract

We systematically study the electromagnetic properties of multiquark states.
In this study, inspired by the recent series of studies that showed the likely
existence of a $D\bar{D}^{*}K$ state, we examine the magnetic moment of
$D\bar{D}^{*}K$ hexaquark state in three-meson molecular structure, as well as
having isospin and spin-parity quantum numbers $I(J^{P})=3/2(1^{-})$ via
light-cone sum rules. The magnetic moment obtained for the $D\bar{D}^{*}K$
molecular hexaquark state is quite large due to the double electric charge,
and its magnitude indicates that it is accessible in future experiments. As a
byproduct, the quadrupole moment of the $D\bar{D}^{*}K$ molecular hexaquark
state is also extracted. This value indicates a non-spherical charge
distribution. The magnetic moments of hadrons contains valuable knowledge on
the distributions of charge and magnetization their inside, which can be used
to better understand their geometric shape and quark-gluon organizations. The
results given in this study constitute an estimate of the magnetic moment of
this $D\bar{D}^{*}K$ state and should serve as an inspiration to conduct
experimental examinations of this state.

Magnetic moment, $D\bar{D}^{*}K$ molecular hexaquark state, Light-cone sum
rules

##  I Introduction

Since the discovery of the $X(3872)$ by Belle in 2003 [1], there have been
numerous candidates of multiquark states observed in particle experiments,
which cannot be well explained in the conventional quark model. A lot of
experimental and theoretical research has been done in the last few decades,
but their nature is still elusive. We refer to the reviews [2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14] and references therein for detailed discussions.

Theoretically, the attraction in the $DK$ and $D\bar{D}^{*}$ subsystems
constitutes an argument in favor of the existence of $D\bar{D}^{*}K$ state.
The observation of such $D\bar{D}^{*}K$ state might be possible in the current
experimental facilities and it would constitute an exciting improvement in the
multiquark states. This state, if found experimentally, definitely cannot be
accommodated in a conventional meson picture, and hence presents a clear case
of an multiquark state. In Ref. [15], based on the attractive force of the
isosinglet $D^{*}K$ system the mass of the $D\bar{D}^{*}K$ system has been
studied using Born-Oppenheimer approximation via delocalized $\pi$ bond with
isospin and spin-parity quantum numbers $I(J^{P})=1/2(1^{-})$. In Ref. [16],
the authors solved the Faddev equation to obtain the mass and decay width of
$D\bar{D}^{*}K$ state. The obtained mass and decay width are given as:
$m=(4307\pm 2)$ MeV and $\Gamma=(9\pm 2)$ MeV with $I(J^{P})=1/2(1^{-})$,
respectively. In Ref. [17], the possible two-body decay channels of
$D\bar{D}^{*}K$ state have been investigated via triangle diagrams with
$I(J^{P})=1/2(1^{-})$. In Ref. [18], the spectroscopic parameters of
$D\bar{D}^{*}K$ system have been obtained in the framework of the QCD sum
rules with quantum numbers $I(J^{P})=3/2(1^{-})$. In Ref. [19], the mass of
the $D\bar{D}^{*}K$ state have been investigated by solving the Schrödinger
equation with quantum numbers $I(J^{P})=1/2(1^{-})$.

There are different approaches in investigation of the structure of multiquark
states, especially, the most promising one in this direction is the study of
the electromagnetic properties of the multiquark states. Electromagnetic
properties of the multiquark states play a vital role in understanding their
internal structure and shape deformations. Besides, examining the magnetic
dipole and higher multipole moments of multiquark states is a crucial element
of understanding the heavy quark dynamics; however, the studies on the
magnetic moments of multiquark states are scarce. Motivated by these facts,
the magnetic moment of the $D\bar{D}^{*}K$ vector hexaquark state has been
obtained by using the light-cone sum rules [20, 21, 22]. While obtaining this
magnetic moment, it is taken into account that these state is in the three-
meson molecular structure, as well as having $I(J^{P})=3/2(1^{-})$ quantum
numbers. Light-cone sum rules have been proven to be a powerful and successful
nonperturbative method over the past few decades. Its starting point is to
construct the corresponding interpolating currents appropriate to the hadron
of concern, which have the prominent knowledge about the related hadron, like
quantum numbers and the constituent particles. With these interpolating
currents, the correlation function in the presence of the external
electromagnetic field, which has two representations, the QCD representation
and the hadronic representation, can be constructed. Matching these two
representations via quark-hadron duality, the light-cone sum rules will be
established, from which the magnetic moments, form factors, etc. may be
deduced.

The rest of article is arranged as follows: After the introduction, in Sect.
II we derive the light-cone sum rules for the magnetic moment of the
$D\bar{D}^{*}K$ hexaquark molecular state. In Sect. III, we present the
numerical results and discussions. The summary of the paper is given in the
Sect. IV.

##  II Formalism

The starting point of the light-cone sum rules is the correlation function
constructed from two hadronic currents in the presence of the external
electromagnetic field with the following form:

$\Pi_{\mu\nu}(p,q)=i\int d^{4}xe^{ip\cdot x}\langle
0|\mathcal{T}\\{J_{\mu}(x)J_{\nu}^{\dagger}(0)\\}|0\rangle_{\gamma},$ (1)

where $\mathcal{T}$, $J_{\mu}(x)$ and $\gamma$ represent the time-ordered
product of two currents, the interpolating current of $D\bar{D}^{*}K$ state
and the external electromagnetic field, respectively. We need explicit
expressions for $J_{\mu}(x)$ to make progress in the calculations. In the
three-meson state, $J_{\mu}(x)$ with quantum numbers $I(J^{P})=3/2(1^{-})$ can
be written in the following form

$\displaystyle J_{\mu}(x)$ $\displaystyle=$
$\displaystyle\Big{\\{}\Big{[}\bar{u}^{a}(x)i\gamma_{5}c^{a}(x)\Big{]}\Big{[}\bar{c}^{b}(x)\gamma_{\mu}d^{b}(x)\Big{]}\Big{[}\bar{u}^{c}(x)i\gamma_{5}s^{c}(x)\Big{]}\Big{\\}},$
(2)

where the $a$, $b$, $c$ denote color indices and the C is the charge
conjugation matrix.

As the hadronic representation of the correlation function is concerned, we
insert a complete set of intermediate hadronic state with same quantum numbers
as the interpolating currents into the correlation function and we acquire

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)$ $\displaystyle={\frac{\langle 0\mid
J_{\mu}(x)\mid
D\bar{D}^{*}K(p,\varepsilon^{\theta})\rangle}{p^{2}-m_{D\bar{D}^{*}K}^{2}}}\langle
D\bar{D}^{*}K(p,\varepsilon^{\theta})\mid
D\bar{D}^{*}K(p+q,\varepsilon^{\delta})\rangle_{\gamma}\frac{\langle
D\bar{D}^{*}K(p+q,\varepsilon^{\delta})\mid{J_{\nu}^{\dagger}}(0)\mid
0\rangle}{(p+q)^{2}-m_{D\bar{D}^{*}K}^{2}}$ $\displaystyle+\mbox{higher
states}.$ (3)

The amplitude $\langle 0\mid J_{\mu}(x)\mid
D\bar{D}^{*}K(p,\varepsilon^{\theta})\rangle$ can be parameterized in terms of
residue $\lambda_{D\bar{D}^{*}K}$ and polarization vector
$\varepsilon_{\mu}^{\theta}$ of $D\bar{D}^{*}K$ state as

$\displaystyle\langle 0\mid J_{\mu}(x)\mid
D\bar{D}^{*}K(p,\varepsilon^{\theta})\rangle$
$\displaystyle=\lambda_{D\bar{D}^{*}K}\varepsilon_{\mu}^{\theta}\,,$ (4)

while the matrix element $\langle D\bar{D}^{*}K(p,\varepsilon^{\theta})\mid
D\bar{D}^{*}K(p+q,\varepsilon^{\delta})\rangle_{\gamma}$ is given by

$\displaystyle\langle D\bar{D}^{*}K(p,\varepsilon^{\theta})\mid
D\bar{D}^{*}K(p+q,\varepsilon^{\delta})\rangle_{\gamma}$
$\displaystyle=-\varepsilon^{\tau}(\varepsilon^{\theta})^{\alpha}(\varepsilon^{\delta})^{\beta}\bigg{\\{}G_{1}(Q^{2})~{}(2p+q)_{\tau}~{}g_{\alpha\beta}+G_{2}(Q^{2})~{}(g_{\tau\beta}~{}q_{\alpha}-g_{\tau\alpha}~{}q_{\beta})$
$\displaystyle-\frac{1}{2m_{D\bar{D}^{*}K}^{2}}G_{3}(Q^{2})~{}(2p+q)_{\tau}~{}q_{\alpha}q_{\beta}\bigg{\\}},$
(5)

where $\varepsilon^{\tau}$ is polarization of the photon and $G_{i}(Q^{2})$’s
are electromagnetic form factors, with $Q^{2}=-q^{2}$.

Using Eqs. (II)-(5) and after doing some necessary calculations the final form
of the correlation function is obtained as

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)$
$\displaystyle=\frac{\varepsilon_{\rho}\,\lambda_{D\bar{D}^{*}K}^{2}}{[m_{D\bar{D}^{*}K}^{2}-(p+q)^{2}][m_{D\bar{D}^{*}K}^{2}-p^{2}]}\bigg{\\{}G_{1}(Q^{2})(2p+q)_{\rho}\bigg{(}g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m_{D\bar{D}^{*}K}^{2}}-\frac{(p+q)_{\mu}(p+q)_{\nu}}{m_{D\bar{D}^{*}K}^{2}}$
$\displaystyle+\frac{(p+q)_{\mu}p_{\nu}}{2m_{D\bar{D}^{*}K}^{4}}(Q^{2}+2m_{D\bar{D}^{*}K}^{2})\bigg{)}+G_{2}(Q^{2})\bigg{(}q_{\mu}g_{\rho\nu}-q_{\nu}g_{\rho\mu}-\frac{p_{\nu}}{m_{D\bar{D}^{*}K}^{2}}\big{(}q_{\mu}p_{\rho}-\frac{1}{2}Q^{2}g_{\mu\rho}\big{)}$
$\displaystyle+\frac{(p+q)_{\mu}}{m_{D\bar{D}^{*}K}^{2}}\big{(}q_{\nu}(p+q)_{\rho}+\frac{1}{2}Q^{2}g_{\nu\rho}\big{)}-\frac{(p+q)_{\mu}p_{\nu}p_{\rho}}{m_{D\bar{D}^{*}K}^{4}}\,Q^{2}\bigg{)}$
$\displaystyle-\frac{G_{3}(Q^{2})}{m_{D\bar{D}^{*}K}^{2}}(2p+q)_{\rho}\bigg{(}q_{\mu}q_{\nu}-\frac{p_{\mu}q_{\nu}}{2m_{D\bar{D}^{*}K}^{2}}Q^{2}+\frac{(p+q)_{\mu}q_{\nu}}{2m_{D\bar{D}^{*}K}^{2}}Q^{2}-\frac{(p+q)_{\mu}q_{\nu}}{4m_{D\bar{D}^{*}K}^{4}}Q^{4}\bigg{)}\bigg{\\}}\,.$
(6)

To calculate the magnetic moment, we need only $G_{2}(Q^{2})$ of the form
factors described above. The magnetic form factor, $F_{M}(Q^{2})$, is written
as follows

$\displaystyle F_{M}(Q^{2})=G_{2}(Q^{2})\,.$ (7)

The $F_{M}(Q^{2}=0)$ is proportional to the magnetic moment
$\mu_{D\bar{D}^{*}K}$:

$\displaystyle\mu_{D\bar{D}^{*}K}=\frac{e}{2\,m_{D\bar{D}^{*}K}}\,F_{M}(0).$
(8)

To evaluate the QCD representation of the light-cone sum rules, we insert the
interpolating currents to the correlation function, contract relevant quark
fields and obtain QCD representation of the correlation function in terms of
the light and heavy quark propagators

$\displaystyle\Pi^{QCD}_{\mu\nu}(p,q)$ $\displaystyle=$ $\displaystyle-i\int
d^{4}xe^{ip\cdot x}\langle 0\mid\Bigg{\\{}{\rm
Tr}[\gamma_{\mu}S_{d}^{bb^{\prime}}(x)\gamma_{\nu}S_{c}^{b^{\prime}b}(-x)]{\rm
Tr}[\gamma_{5}S_{c}^{aa^{\prime}}(x)\gamma_{5}S_{u}^{a^{\prime}a}(-x)]$ (9)
$\displaystyle\times{\rm
Tr}[\gamma_{5}S_{s}^{cc^{\prime}}(x)\gamma_{5}S_{u}^{c^{\prime}c}(-x)]-{\rm
Tr}[\gamma_{\mu}S_{d}^{bb^{\prime}}(x)\gamma_{\nu}S_{c}^{b^{\prime}b}(-x)]{\rm
Tr}[\gamma_{5}S_{c}^{aa^{\prime}}(x)i\gamma_{5}$ $\displaystyle\times
S_{u}^{a^{\prime}c}(-x)\gamma_{5}S_{s}^{cc^{\prime}}(x)i\gamma_{5}S_{u}^{c^{\prime}a}(-x)]\Bigg{\\}}\mid
0\rangle_{\gamma}\,,$

where $S_{q}(x)$ and $S_{c}(x)$ are represent the full light and massive quark
propagators. Throughout our calculations, we use the x-space expressions for
the $S_{q}(x)$ and $S_{c}(x)$ [23, 24]:

$\displaystyle S_{q}(x)$
$\displaystyle=i\frac{{x\\!\\!\\!/}}{2\pi^{2}x^{4}}-\frac{\langle\bar{q}q\rangle}{12}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{4}\Big{)}-\frac{\langle\bar{q}q\rangle}{192}m_{0}^{2}x^{2}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{6}\Big{)}-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\Big{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\Big{]},$ (10) $\displaystyle S_{c}(x)$
$\displaystyle=\frac{m_{c}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{(\sqrt{-x^{2}})^{2}}\Bigg{]}-\frac{g_{s}m_{c}}{16\pi^{2}}\int_{0}^{1}dv\,G^{\mu\nu}(vx)\Bigg{[}\big{(}\sigma_{\mu\nu}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\mu\nu}\big{)}$
$\displaystyle\times\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+2\sigma_{\mu\nu}K_{0}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}\Bigg{]}.$
(11)

where $\langle\bar{q}q\rangle$ is quark condensate, $m_{0}$ is defined through
the quark-gluon mixed condensate $\langle
0\mid\bar{q}\,g_{s}\,\sigma_{\alpha\beta}\,G^{\alpha\beta}\,q\mid
0\rangle=m_{0}^{2}\,\langle\bar{q}q\rangle$, $v$ is line variable,
$G^{\mu\nu}$ is the gluon field strength tensor,
$\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$ and $K_{i}$’s are
modified Bessel functions of the second kind. The first term of the light and
heavy quark propagators correspond to perturbative or free part and the rest
belong to the interacting parts.

A few remarks should be made here regarding the calculation of the QCD
representation of the correlation function. The correlation function in Eq.
(9) includes two different contributions as perturbative and nonperturbative.
In practice, perturbative contributions, the photon interacts with one of the
quarks, can be computed by the replacing the one of the light or heavy-quark
propagators by

$\displaystyle S^{free}\rightarrow\int d^{4}z\,S^{free}(x-z)\,\hbox
to0.0pt{/\hss}{\\!A}(z)\,S^{free}(z)\,.$ (12)

The remaining five propagators are considered as full propagators.
Nonperturbative contributions, the photon is radiated at long distances, can
be computed by replacing one of the light quark propagators by

$\displaystyle
S_{\alpha\beta}^{ab}\rightarrow-\frac{1}{4}(\bar{q}^{a}\Gamma_{i}q^{b})(\Gamma_{i})_{\alpha\beta},$
(13)

where $\Gamma_{i}$ are full set of the Dirac matrices and surviving
propagators are considered as full quark propagators. It is seen that matrix
elements such as $\langle\gamma(q)|\bar{q}(x)\Gamma_{i}q(0)|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\mu\nu}q(0)\right|0\rangle$
emerge after the calculations given in Eq. (13) are made. These matrix
elements are parameterized with respect to the photon distribution amplitudes
(DAs), which are the key nonperturbative parameters in light-cone sum rules,
whose explicit expressions are given in Ref. [25]. Besides these matrix
elements, in principle, nonlocal operators such as $\bar{q}G^{2}q$ and
$\bar{q}q\bar{q}q$ are expected to show up. However, it is known that the
contributions of such operators are small, which is supported by the conformal
spin expansion [26, 27], and hence we will omit them. When the above-mentioned
calculations are made, the QCD representation of the correlation function is
obtained.

Finally, the structure $q_{\mu}\varepsilon_{\nu}$ is chosen from both
representations and the coefficients of the structure are matched in both
hadronic and QCD representations. Then, Borel transformation and continuum
subtraction are used to suppress the effects of the continuum and higher
states. As a result, the light-cone sum rule for $D\bar{D}^{*}K$ state is as
follows:

$\displaystyle\mu_{D\bar{D}^{*}K}\,\lambda_{D\bar{D}^{*}K}^{2}=e^{\frac{m_{D\bar{D}^{*}K}^{2}}{M^{2}}}\,\,\Delta^{QCD}(M^{2},s_{0}),$
(14)

where $M^{2}$ and $s_{0}$ are free parameters originating from the
applications of the Borel transformation and continuum subtraction procedures.
The $\Delta^{QCD}(M^{2},s_{0})$ function is rather lengthy, explicit
expression of which is not given in the text. In obtaining the final result,
we employ $\frac{1}{M^{2}}=\frac{1}{M_{1}^{2}}+\frac{1}{M_{2}^{2}}$ with
$M_{1}^{2}$ and $M_{2}^{2}$ being the Borel parameters in the initial and
final states, respectively. We set $M_{1}^{2}=M_{2}^{2}=2M^{2}$ as the initial
and final states are the same. We will fix $M^{2}$ and continuum threshold
($s_{0}$) based on the standard prescription of the light-cone sum rule method
in next section.

At the end of this section, we would like to note that the magnetic moment of
the $D\bar{D}^{*}K$ state has been computed from the light-cone sum rules
employing for its hadronic side a single-pole approach [see, Eq. (II)]. In the
case of the multiquark hadrons such approach should be verified by additional
arguments, because a hadronic side of relevant sum rules receives
contributions from two-hadron reducible terms as well. Two-hadron
contaminating terms have to be considered when extracting parameters of
multiquark hadrons [28, 29]. In the case of the multiquark hadrons they lead
to modification in the quark propagator

$\frac{1}{m^{2}-p^{2}}\rightarrow\frac{1}{m^{2}-p^{2}-i\sqrt{p^{2}}\Gamma(p)},$
(15)

where $\Gamma(p)$ is the finite width of the multiquark hadrons generated by
two-hadron scattering states. When these contributions are properly taken into
account in the sum rules, they rescale the residue of the multiquark hadrons
under investigations leaving its mass unchanged. Detailed investigations show
that two-hadron scattering contributions are small for multiquark hadrons (see
Refs. [30, 31, 32, 33, 34, 35, 36, 37, 38]). Therefore, one can safely exclude
the contributions of two-hadron scattering effects in the hadronic side of the
correlation function.

##  III Numerical analysis and conclusions

We assume the following parameters to perform the numerical calculations for
the magnetic moment of the $D\bar{D}^{*}K$ state. The masses of the light
quarks are $m_{u}=m_{d}=0$, $m_{s}=96^{+8}_{-4}\,\mbox{MeV}$, the mass of the
c-quark is $m_{c}=(1.275\pm 0.025)\,$GeV, the condensates of the light quarks
are $\langle\bar{s}s\rangle=0.8\,\langle\bar{u}u\rangle$ with
$\langle\bar{u}u\rangle$ =$\langle\bar{d}d\rangle$=$(-0.24\pm 0.01)^{3}\,$GeV3
[39], the gluon condensate is $\langle
g_{s}^{2}G^{2}\rangle=0.88~{}\mbox{GeV}^{4}$ [40] and the quark-gluon
condensate is $m_{0}^{2}=0.8\pm 0.1$ GeV2 [39]. To proceed the numerical
calculations of the magnetic moment of the $D\bar{D}^{*}K$ state , numerical
values of the mass and residue parameters of this state are also needed. These
values have been computed in Ref. [18]. The obtained results for mass and
residue are given as $m_{D\bar{D}^{*}K}=4.71^{+0.19}_{-0.11}~{}\mbox{GeV}$ and
$\lambda_{D\bar{D}^{*}K}=(4.60^{+1.15}_{-0.69})\times
10^{-4}~{}\mbox{GeV}^{8}$. All necessary terms regarding the DAs of the photon
are borrowed from Ref. [25].

The light-cone sum rules in Eq. (14) is a function of the Borel mass parameter
$M^{2}$ and continuum threshold $s_{0}$. To acquire a reliable light-cone sum
rules result, one should determine proper working intervals for these two free
parameters. The OPE convergence and pole dominance (PC) constraints are widely
used to determine the working intervals of these two parameters. Considering
these limitations, the following working intervals are obtained for these two
free parameters as a result of the numerical analysis,

$\displaystyle 5.0~{}\mbox{GeV}^{2}\leq M^{2}\leq 7.0~{}\mbox{GeV}^{2}$
$\displaystyle 25.2~{}\mbox{GeV}^{2}\leq s_{0}\leq 27.2~{}\mbox{GeV}^{2}.$
(16)

Using the above working intervals for the $M^{2}$ and $s_{0}$, the PC varies
in the intervals $53\%\geq$ PC $\geq 15\%$. At $M^{2}_{max}=7.0~{}$GeV2, the
PC is equal to $15\%$, while at $M^{2}_{min}=5.0~{}$GeV2, it is equal to
$53\%$. In the standard analysis of QCD sum rules, the PC should be larger
than $50\%$ for baryons and mesons. In the case of tetra- and pentaquark
states, it turns out to be as PC $>20\%$. In Refs. [41, 42, 43, 44], it is
indicated that hexaquark spectral densities led to small PC. When we
investigate the OPE convergence, we have acquired that the contribution of the
higher dimensional terms in OPE is less than $\sim 2\%$, thus the convergence
of the sum rules is ensured. In Fig. 1, the $M^{2}$ and $s_{0}$ dependencies
of the magnetic moment of the $D\bar{D}^{*}K$ hexaquark state is shown. As is
seen, the variation of the results with respect to the continuum threshold is
substantial; however, there is much less dependence of the magnetic moment on
the Borel parameters in its working interval.

Figure 1:  Variations of the magnetic moment of $D\bar{D}^{*}K$ state with
$M^{2}$ and $s_{0}$.

After all the above procedures are completed, the results we obtained for the
magnetic moment are given as follows

$\displaystyle\mu_{D\bar{D}^{*}K}$ $\displaystyle=-3.52\pm 1.28\,\mu_{N}.$
(17)

The error in the Eq. (17) is due to all input parameters, extra parameters
such as $s_{0}$ and $M^{2}$, as well as numerical parameters used in
expressions in photon DAs. We see from this result that, the value of the
magnetic moment of $D\bar{D}^{*}K$ state are quite large because of the double
electric charge.

As a byproduct, we have also acquired quadrupole moment ($\mathcal{D}$) of the
$D\bar{D}^{*}K$ state as follows

$\displaystyle\mathcal{D}_{D\bar{D}^{*}K}$ $\displaystyle=-0.030\pm
0.008\,fm^{2}.$ (18)

This result indicates that the charge distribution of the $D\bar{D}^{*}K$
state is non-spherical.

A comparison of our predictions on the magnetic and quadrupole moments of the
$D\bar{D}^{*}K$ state with the estimations of other methods, such as lattice
QCD, quark model, chiral perturbation theory and so on would be interesting.
As we mentioned above $D\bar{D}^{*}K$ state belong to a class of doubly
charged multiquark states that the measurements of their electromagnetic
parameters, like those of the $\Delta^{++}$ baryon, are relatively easy
compared to other multiquark states. These kind of multiquark states have not
been explored so far. We hope our estimations on the electromagnetic
properties of these states together with the results of other theoretical
studies on the spectroscopic parameters of these states will be useful for
their searches in future experiments and will help us define exact inner
structures of these multiquark states.

Let us briefly discuss how the magnetic moment of this state can be measured
experimentally. The possible short lifetime of $D\bar{D}^{*}K$ state does not
allow the employ of spin procession technique to measure the magnetic and
quadrupole moments. An alternative technique based on photon emission off
hadrons [45] can be used in the later case, since the photon carries data on
higher multipole moments of emitting hadrons. The basic scheme of this idea is
that the amplitude for radiative process can be expressed as a power expansion
in the photon energy $\omega_{\gamma}$ as follows

$\displaystyle M\sim
A\,(\omega_{\gamma})^{-1}+B\,(\omega_{\gamma})^{0}+\cdots,$ (19)

where $(\omega_{\gamma})^{-1}$, $(\omega_{\gamma})^{0}$ and dots represent the
electric charge, magnetic moment and higher multipole moments, respectively.
By measuring the cross section or decay width of the radiative process and
excluding from the small contributions of terms linear/higher order in
$\omega_{\gamma}$, one can define the magnetic moment of related hadron.

##  IV Summary

We systematically study the electromagnetic properties of multiquark states.
In this study, inspired by the recent series of studies that showed the likely
existence of a $D\bar{D}^{*}K$ state, we examine the magnetic moment of
$D\bar{D}^{*}K$ hexaquark state in three-meson molecular structure, as well as
having isospin and spin-parity quantum numbers $I(J^{P})=3/2(1^{-})$ via
light-cone sum rules. The magnetic moment obtained for the $D\bar{D}^{*}K$
molecular hexaquark state is quite large due to the double electric charge,
and its magnitude indicates that it is accessible in future experiments. As a
byproduct, the quadrupole moment of the $D\bar{D}^{*}K$ molecular hexaquark
state are also extracted. This value indicates a non-spherical charge
distribution. The magnetic moments of hadrons contain valuable knowledge on
the distributions of charge and magnetization their inside, which can be used
to better understand their geometric shape and quark-gluon organizations. The
results given in this study constitute an estimate of the magnetic moment of
this $D\bar{D}^{*}K$ state and should serve as an inspiration to conduct
experimental examinations of this state.

Data Availability Statement: This manuscript has no associated data or the
data will not be deposited. [Authors’ comment: This is a theoretical research
work, so no additional data are associated with this work.]

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# Thai Winograd Schemas: A Benchmark for Thai Commonsense Reasoning

Phakphum Artkaew
Department of Electrical and Computer Engineering
Tandon School of Engineering
New York University
<EMAIL_ADDRESS>
###### Abstract

Commonsense reasoning is one of the important aspect of natural language
understanding, with several benchmarks developed to evaluate it. However, only
a few of these benchmarks are available in languages other than English.
Developing parallel benchmarks facilitates cross-lingual evaluation, enabling
a better understanding of different languages. This research introduces a
collection of Winograd Schemas in Thai, a novel dataset designed to evaluate
commonsense reasoning capabilities in the context of the Thai language.

Through a methodology involving native speakers, professional translators, and
thorough validation, the schemas aim to closely reflect Thai language nuances,
idioms, and cultural references while maintaining ambiguity and commonsense
challenges. We evaluate the performance of popular large language models on
this benchmark, revealing their strengths, limitations, and providing insights
into the current state-of-the-art. Results indicate that while models like
GPT-4 and Claude-3-Opus achieve high accuracy in English, their performance
significantly drops in Thai, highlighting the need for further advancements in
multilingual commonsense reasoning.

Thai Winograd Schemas: A Benchmark for Thai Commonsense Reasoning

Phakphum Artkaew Department of Electrical and Computer Engineering Tandon
School of Engineering New York University<EMAIL_ADDRESS>

##  1 Introduction

Figure 1: Winograd Schema examples in Thai, with transliteration and
corresponding English version.

Commonsense reasoning is an important challenge in artificial intelligence;
however, most resources and benchmarks are in English, with only a few
translations Davis (2023). The Winograd Schemas Challenge (WSC) has emerged as
a widely adopted benchmark for evaluating the commonsense reasoning
capabilities of language models Levesque et al. (2012). WSC is featured in
both GLUE and SuperGLUE. These benchmarks are widely used to evaluate a
model’s understanding of general language Wang et al. (2018) Wang et al.
(2019). Significant progress has been made in developing commonsense reasoning
models for high-resource languages like English. However, their performance on
low-resource languages remains largely unexplored. This research gap hinders
the development and fair evaluation of multilingual and cross-lingual NLU
systems. In their work, Linzen (2020) notes, “We should ensure that our
architectures, like human learners, are not optimized for English Bender
(2011). To do so, we should develop matched training corpora and benchmarks
for multiple languages. A composite score could reflect average performance
across languages Hu et al. (2020).” Consequently, the development of such
benchmarks or corpora is vital for tasks evaluating the current state of
systems across languages.

There have been several translations of the Winograd Schemas into various
languages, including French Amsili and Seminck (2017), Portuguese De Melo et
al. (2019), Mandarin Bernard and Han (2020), Hebrew Shwartz (2024), Hungarian
Vadász and Ligeti-Nagy (2022), and Russian Shavrina et al. (2020). Before our
translation, no Thai version of the Winograd Schemas existed. This lack
prompted us to take the initiative.

The key contributions of our work are:

  1. 1.

We introduce Thai-WS, the first benchmark dataset for evaluating the Winograd
task and commonsense reasoning capabilities of language models and NLP systems
in the Thai language.

  2. 2.

We evaluated the performance of state-of-the-art models, including GPT,
Claude, and Typhoon, on the Thai-WS dataset to assess their cross-lingual
reasoning abilities.

##  2 Related Work

### Background

The Winograd Schema Challenge, introduced by Levesque et al. (2012) in 2012 as
an alternative Turing Test, consists of pairs of similar sentences with slight
differences that introduce ambiguities resolved through general knowledge and
logical reasoning. A classic example by Winograd (1972) illustrates the
importance of context: "The city councilmen refused the demonstrators a permit
because they feared/advocated violence." The choice of "feared" or "advocated"
shifts the referent of "they" from the city councilmen to the demonstrators.
This is because "feared" aligns with concerns typical of councilmen’s reasons
for refusal, whereas "advocated" suggests demonstrators’ motivations,
reflecting how context dictates meaning.

### Other Thai Reasoning Benchmarks

To the best of our knowledge, three public benchmarks are available for Thai
reasoning. First is XNLI, which is a translation of the MNLI benchmark into
Thai, among 14 other languages Conneau et al. (2018). Second is XCOPA,
translated from the COPA benchmark to assess causal commonsense reasoning
Ponti et al. (2020). Third, the recently released M3Exam, is based on
examination questions Zhang et al. (2024). Unlike these benchmarks, the
Winograd Schema Challenge specifically assesses commonsense reasoning through
pronoun reference disambiguation task.

### A Simple Method for Commonsense Reasoning

Trinh and Le (2018) introduced a simple approach to solve the Winograd Schema
hallenge by substituting one of the pronouns in the sentence and allowing the
language model to determine which substitution has a higher probability. This
method effectively frames Winograd Schemas as a binary classification task.
Since then, many language models, such as GPT-2 Radford et al. (2019) and
GPT-3 Brown et al. (2020), have used this method to evaluate their performance
on Winograd Schemas. However, due to limited access to models and
computational power, the evaluation of this version of the Winograd Schema
Challenge was conducted differently, using a prompt-based method, which will
be discussed in the experimental setup section.

##  3 Dataset Construction

###  3.1 Translation

Two professional translators, who were native Thai speakers fluent in English
and had experience translating from English to Thai, were hired. In a pilot
translation phase, one native speaker translated the first 85 sentences. Based
on a qualitative analysis of these 85 sentences, guidelines were provided for
a second native speaker to translate the remaining 200 sentences. In total,
285 sentences were translated from English to Thai. Translation guidelines
were provided, instructing them to adapt names and contexts to suit the Thai
language while preserving the ambiguity and nuances of the original schema.
The translators were also asked to mark any translated names and translations
they were unsure about, so that the validator in the next step could pay extra
attention to those instances. For example, in Figure 1, the names Paul and
George were changed to Mana and Piti, respectively, adapting the names to
better suit the Thai context while preserving the essence of the original
content.

Some phrases could be directly translated, while others required adjustments
to names and contextual adaptations to better suit the Thai language and
culture. However, there were a few instances where the translators highlighted
certain phrases in red, indicating that they are worth mentioning.

For example:

  1. (1)

    1. (i)

Many people start to read Paul’s books and can’t put them down. They are
gripped because Paul writes so well.

    2. (ii)

Many people start to read Paul’s books and can’t put them down. They are
popular because Paul writes so well.

  2. (2)

    1. (i)

Fred covered his eyes with his hands, because the wind was blowing sand
around. He opened them when the wind stopped.

    2. (ii)

Fred covered his eyes with his hands, because the wind was blowing sand
around. He lowered them when the wind stopped.

  3. (3)

    1. (i)

Lily spoke to Donna, breaking her silence.

    2. (ii)

Lily spoke to Donna, breaking her concentration.

Most of the translation problems encountered involve examples like 1, where in
English the word “they” can refer to both people and objects. In this case,
“they” can mean either “people” or “Paul’s books.” However, in Thai, it is
uncommon to use the same pronoun for both objects and people. Therefore, an
alternative pronoun that is acceptable in Thai was chosen. In this instance,
the English equivalent word “those” was used instead.

Similarly, in 2, “them” can refer to either “his eyes” or “his hands.” In
Thai, it is not common to use a pronoun in this way. The translation was
changed to the English equivalent of “those two,” which sounds more natural in
Thai.

In other cases, the meaning had to be adjusted to sound natural in Thai. For
example, 3 “breaking her silence” is not a common expression in Thai, so it
was revised to “The silence was replaced by her voice.”

###  3.2 Validation

Model | Accuracy (English) | Accuracy (Thai)
---|---|---
Typhoon | 64.56% | 54.04%
Claude-3-Haiku | 62.81% | 51.58%
Claude-3-Sonnet | 80.70% | 63.51%
Claude-3-Opus | 92.63% | 77.54%
GPT-3.5 | 71.93% | 51.23%
GPT-4 | 93.68% | 72.28%
Human | 90% | -%
Table 1: Accuracy vs. Model in English and Thai

The translated Winograd Schemas were reviewed with three native Thai speakers,
and final adjustments were made to ensure clarity. A validator was provided
with the translations and tasked with identifying any potential issues. They
were instructed to pay closer attention to the text marked by the translators,
which included changes to names and translations that the translators were
unsure about. Based on the validator’s feedback, final adjustments to the
translations were made, and typographical errors were corrected. Furthermore,
the dataset will be made publicly available111Avaliable at
https://huggingface.co/datasets/pakphum/winograd_th., inviting other native
Thai speakers to verify it and suggest any necessary adjustments.

##  4 Experimental Setup

Large language models were evaluated on both Thai and English Winograd Schemas
to assess Thai natural language processing and enable cross-linguistic
comparison of commonsense reasoning. It is important to note that there are
two versions of the Winograd Schema Challenge (WSC): WSC-273 and WSC-285. In
this study, WSC285 was used.

### Typhoon

The Thai large language model, named Typhoon, is built on the Mistral-7B
architecture Pipatanakul et al. (2023). Mistral-7B features grouped-query
attention for faster responses and sliding window attention for cost-effective
processing of any text length Jiang et al. (2023). Typhoon was pre-trained on
the MC4 and OSCAR datasets, which include approximately 3 TB of Thai text. A
Thai subword tokenizer was added to enhance its performance on Thai language
tasks. In this experiment, the specific model tested was typhoon-
instruct-0219.

### Claude

Anthropic’s large language model Anthropic (2024a), designed to be a helpful
and honest assistant, has its model architecture details kept private. In this
experiment, all the Claude-3 models were tested to compare performance. The
specific models tested were claude-3-haiku-20240307, claude-3-sonnet-20240229,
and claude-3-opus-20240229.

### GPT

GPT or Generative Pre-trained is one of the most popular and powerful large
language models available. While the specific details of GPT-4’s architecture
are not publicly disclosed, it can be assumed that it represents an
advancement over GPT-3. GPT-3 is an autoregressive, decoder-only model with
175 billion parameters Brown et al. (2020). Although the technical report does
not specify its performance in Thai specifically, it asserts that GPT-4
surpasses GPT-3 in the MMLU task for the Thai language Achiam et al. (2023).
In this experiment, gpt-4-0613 and gpt-3.5-turbo-0125 were tested.

### Implementation and evaluation

Language models were evaluated using a prompt structure approach. A system
prompt and user prompt structure were utilized to assess the models. A system
prompt provides context and instructions to language models before a task,
specifying the model’s role, personality, tone, or other relevant information
to enhance its responses Anthropic (2024b). The system prompt was designed to
prepare the model for the Winograd Schemas task, instructing it to respond
with the correct answer. The evaluation focused on accuracy, counting only
answers that exactly matched the correct answer in the corresponding schemas.
Outputs such as “the answer is the city councilmen ” were not counted as
correct; only precise answers like “the city councilmen" were considered
correct. The models were evaluated through their respective APIs. More details
on the evaluation and prompt setup can be found in A.

##  5 Result and Discussion

Table 1 shows the accuracy results when performed on each model. There were
several human baseline for the English Winograd Kocijan et al. (2023). Our
English human baseline is derived from the study conducted by Davis et al.
(2016), which involved human participants evaluating Winograd Schemas in
English. In Davis et al. (2016) research, human accuracy was observed to be
approximately 90%. The human baseline for Thai has not been conducted yet.

### Comparison of models

Typhoon achieved an accuracy of 64.56% in English and 54.04% in Thai, showing
a drop in performance when handling Thai schemas. Claude-3-Haiku scored 62.81%
in English and 51.58% in Thai, similarly reflecting a lower accuracy in the
Thai language. Claude-3-Sonnet demonstrated better performance with 80.70%
accuracy in English and 64.56% in Thai. Claude-3-Opus achieved high accuracy
of 92.63% in English and 77.54% in Thai, showing the best performance among
the Claude-3 variants.

GPT-4 scored the highest in English with 93.33% and performed well in Thai
with 73.68% accuracy. GPT-3.5 showed a significant drop from GPT-4, with
71.58% accuracy in English and 51.58% in Thai.

Overall, the results indicate that all models perform better on the English
version of the Winograd Schema Challenge compared to the Thai version. The
performance drop in Thai suggests that these models may have more difficulty
handling the nuances and linguistic structures of Thai.

##  6 Conclusion

In conclusion, the Thai Winograd Schemas benchmark represents a noteworthy
step in evaluating commonsense reasoning capabilities within the Thai language
context. This novel dataset, meticulously developed and validated by native
speakers and professional translators, aims to preserve linguistic and
cultural nuances unique to Thai. The comprehensive evaluation of state-of-the-
art language models, including GPT-4, Claude-3 variants, and Typhoon, on both
English and Thai versions of the Winograd Schema Challenge offers insights
into their cross-lingual performance. The observed performance drop in Thai
suggests challenges these models may face in handling low-resource languages,
indicating a need for further research and development in multilingual natural
language processing.

## Limitations

While cultural nuances are preserved as much as possible during the
translation process, it is acknowledged that complete preservation is not
always achievable due to differences between Thai and English. Additionally,
there is a limitation regarding the Thai human baseline for Thai Winograd
Schemas, as such a baseline has not yet been established. Direct evaluation on
large language models like GPT-4 and Claude-3 cannot be performed due to lack
of access and insufficient computational power. Therefore, an alternative
approach using prompt-based evaluation is adopted.

## Acknowledgments

Thank you to the two translators, Chanikarn Inthongpan and Korakoch Rienmek,
who worked on translating these Winograd schemas. It was challenging to find
individuals skilled enough for this specific task, which demands a deep
understanding of language. I also extend my gratitude to my friends and
professors who supported me throughout the writing process.

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##  Appendix A Prompt Evaluation

To ensure consistency and reproducibility in evaluating each model, specific
settings were implemented alongside the prompt structure. The system prompt
was devised to provide clear instructions: “You will be provided with a
sentence and a snippet containing a pronoun enclosed in asterisks (**). Your
task is to determine the correct referent of the pronoun from the given
options. Respond only with one of the provided choices, exactly as it is
written. For example, if the options are ‘The city councilmen’ and ‘The
demonstrators’, respond only with ‘The city councilmen’ or ‘The
demonstrators’.” Additionally, the temperature parameter was set to 0 to
minimize randomness, and when possible, the seed value was fixed at 77 to
further enhance reproducibility. For the user’s prompt, the model was given a
Winograd Schema Challenge, formatted with the scenario and its options
separated by a new line, adhering to the same standards for consistency and
replicability.

An example user prompt format would be: “The city councilmen refused the
demonstrators a permit because they feared violence.\nSnippet: they feared
violence\nOptions:\nThe city councilmen\nThe demonstrators.”

Only exact matches to the correct choice were counted as correct answers. For
instance, responses such as “the answer is The city councilmen” were not
counted as correct; only precise answers like “The city councilmen” were
considered correct. This process was executed through the model API, ensuring
that each evaluation followed the same method and settings for all models.

The code used to reproduce all of our experimental results is available at the
following GitHub repository: https://github.com/PhakphumAdev/Thai-Winograd.


# Transient response under ultrafast interband excitation of an intrinsic
graphene

P.N. Romanets  F.T. Vasko<EMAIL_ADDRESS>Institute of Semiconductor
Physics, NAS of Ukraine, Pr. Nauky 41, Kiev, 03028, Ukraine

###### Abstract

The transient evolution of carriers in an intrinsic graphene under ultrafast
excitation, which is caused by the collisionless interband transitions, is
studied theoretically. The energy relaxation due to the quasielastic acoustic
phonon scattering and the interband generation-recombination transitions due
to thermal radiation are analyzed. The distributions of carriers are obtained
for the limiting cases when carrier-carrier scattering is negligible and when
the intercarrier scattering imposes the quasiequilibrium distribution. The
transient optical response (differential reflectivity and transmissivity) on a
probe radiation and transient photoconductivity (response on a weak dc field)
appears to be strongly dependent on the relaxation and recombination dynamics
of carriers.

###### pacs:

72.80.Vp, 78.67.Wj, 81.05.ue

##  I Introduction

The transient response of photoexcited carriers under ultrafast interband
pumping has been studied during the last decades in bulk semiconductors and
heterostructures (see Ref. 1 for review). The unusual transport of carriers in
graphene is caused by a neutrinolike energy spectrum in gapless semiconductor,
which is described by the Weyl-Wallace model 2 , and a substantial
modification of scattering processes. Recently, the properties of graphene
after ultrafast interband excitation attract special attention. The
experimental results in relaxation dynamics of photoexcited electrons and
holes were published in 3 ; 4 ; 5  and 6  for epitaxial and exfoliated
graphene, respectively. The relaxation of nonequilibrium optical phonons,
which are emitted by carriers after photoexcitation, is studied in Ref. 7. The
theoretical consideration of the carrier relaxation and generation-
recombination processes caused by optical phonons is performed in 8 ; 9 . The
quasielastic energy relaxation of carriers due to acoustic phonons was
considered in 10 ; 11  for low energy carriers (at low temperatures or under
mid-IR excitation). In particular, an interplay between energy relaxation and
generation-recombination processes determines the relaxation dynamics of
photoexcited carrier distribution. 10  To the best of our knowledge, both this
interplay and the relaxation dynamics at low temperatures are not considered
so far. Thus, the investigation of the transient response of carriers under
these conditions is timely now.

In this paper, we consider the transient response of an intrinsic graphene in
case of ultrafast interband excitation in passive region, where the carrier
energies are smaller than the optical phonon energy. Such a regime can be
realized under the pumping in the mid-infrared (IR) spectral region or at low
temperatures, when the peak of photoexcited carriers formed after the process
of optical phonon emission, remains a narrow one. Describing the
photoexcitation process, we restrict ourselves by the collisionless regime,
when a pulse duration, $\tau_{p}$, is shorter than the momentum relaxation
time. Considering the low-temperature transient dynamics of photoexcited
carriers, one takes into account the intraband quasielastic energy relaxation
due to acoustic phonons and generation-recombination interband transitions due
to thermal radiation. The carrier-carrier scattering is described within two
limiting regimes: ($i$) when the Coulomb interaction is unessential, and
($ii$) when intercarrier scattering imposes the quasiequilibrium distribution
of carriers. With the obtained transient distribution of carriers, we analyze
a time-dependent response on the probe field, i.e. we consider the transient
reflection and transmission in THz and mid-IR spectral regions. The transient
photoconductivity is also analyzed below, because the energy relaxation
corresponds to a nanosecond scale (the radiative recombination remains
essential up to microsecond).

Since the electron-hole energy spectrum and scattering processes are symmetric
in an intrinsic graphene, the phenomena under consideration are described by
the same distribution functions for electrons and holes, $f_{pt}$. Such
distribution is governed by the general kinetic equation 12 :

$\frac{\partial f_{pt}}{\partial
t}=\sum_{k}J_{k}\left\\{f_{t}|p\right\\}+G\\{f|pt\\},$ (1)

where the collision integrals $J_{k}\left\\{f_{t}|p\right\\}$ describe the
relaxation of carriers caused by the carrier-carrier scattering ($k=cc$), the
acoustic phonons ($k=ac$), and the thermal radiation ($k=r$), respectively.
The photogeneration rate, $G\\{f|pt\\}$, describes the interband excitation of
electron-hole pairs by the mid-IR ultrafast pulse. Below Eq. (1) is solved
with the initial condition $f_{pt\to-\infty}=f_{p}^{(eq)}$, where
$f_{p}^{(eq)}$ is the equilibrium distribution. The transient response on a
probe radiation is described by the dynamic conductivity due to interband
transitions. The transient response on a weak dc field (photoconductivity) is
considered with the use of the phenomenological model of momentum scattering
suggested in 13 .

The analysis carried out below is organized as follows. The photoexcitation
process under the inerband pumping is described in Sec. II. The transient
evolution distributions are given in Sec. III for the cases ($i$) and ($ii$).
Section IV presents a set of results of transient reflectivity and
transmittivity, and also the transient photoconductivity. The discussion of
the assumptions used and concluding remarks are given in the last section.
Appendix contains the microscopical evaluation of the interband
photogeneration rate under ultrafast interband excitation.

##  II Ultrafast excitation

In the framework of the Weyl-Wallace model (spin- and valley-degenerate linear
energy spectrum of carriers which is determined by the characteristic velocity
$v_{W}$), the interband photoexcitation is caused by the in-plane electric
field, $w_{t}{\bf E}\exp(-i\Omega t)+c.c.$ where $\bf E$ is the field
strength, $\omega$ is the frequency, and $w_{t}$ is the envelope form-factor.
Eq. (1) is transformed to the collisionless form on the initial intervals,
when scattering mechanisms are not essential: $\partial f_{pt}/\partial
t=G\left\\{{f|pt}\right\\}$. Using the boundary condition of Eq. (1), one can
rewrite this equation in the integral form
$f_{pt}=f_{p}^{(eq)}+\int_{-\infty}^{t}dt^{\prime}G\\{f|pt^{\prime}\\}$. The
photogeneration rate here is evaluated in Appendix as follows:

$\displaystyle
G\left\\{f|pt\right\\}=\left(\frac{eEv_{W}}{\hbar\Omega}\right)^{2}w_{t}\int\limits_{-\infty}^{0}d\tau
w_{t+\tau}$
$\displaystyle\times\cos\left[\left(\frac{2v_{W}p}{\hbar}-\Omega\right)\tau\right]\left(1-2f_{pt+\tau}\right),$
(2)

where the Pauli blocking factor $(1-2f_{pt+\tau})$ is responsible for the
coherent Rabi oscillations of the excited carriers.

Introducing the dimensionless intensity,
$I_{ex}=(eE\tau_{p}v_{W}/\hbar\Omega)^{2}$, we consider below the linear
regime of excitation which takes place if $I_{ex}\ll 1$ and $f_{pt}\ll 1$, so
that the Pauli factor can be neglected (if $\hbar\Omega$ comparable to the
equilibrium temperature $T$ one has to use the equilibrium Pauli factor in Eq.
(2)). Using the Gaussian form-factor
$w_{t}=\sqrt[4]{2/\pi}\exp\left[-\left(t/\tau_{p}\right)^{2}\right]$ with the
pulse duration $\tau_{p}$, 14  one obtaines the photoexcited distribution in
the form

$f_{pt}^{(ex)}\approx
I_{ex}\int\limits_{-\infty}^{t}{dt^{\prime}}w_{t^{\prime}}\int\limits_{-\infty}^{0}{d\tau
w_{t^{\prime}+\tau}\cos\left({\frac{{2v_{W}\delta p}}{\hbar}\tau}\right)}.$
(3)

Here $\delta p=p-p_{\Omega}$ is centred in the characteristic momentum
$p_{\Omega}=\hbar\Omega/2v_{W}$. The evolution of photoexcited distribution,
$f_{pt}^{(ex)}/I_{ex}$, is shown in Fig. 1. The distribution is dependent on
$t/\tau_{p}$ and $\delta p/\Delta p$, where $\Delta p=\hbar/2v_{W}\tau_{p}$
determines the width of distribution which is proportional to $\tau_{p}^{-1}$.
For $t\gg\tau_{p}$, the integrations in Eq. (3) can be exactly performed and
we obtain steady-state distribution after the photoexcitation pulse,
$f_{p}^{(ex)}=f_{pt\to\infty}^{(ex)}$, as the Gaussian peak of width
$\propto\Delta p$:

$f_{p}^{(ex)}=\sqrt{\frac{\pi}{2}}I_{ex}e^{-(\delta p/\sqrt{2}\Delta p)^{2}}.$
(4)

Thus, at $t\geq 2\tau_{p}$ (see Fig. 1) one can omit the photogeneration rate
in Eq. (1) using instead the initial condition:

$f_{pt=0}=f_{p}^{(eq)}+f_{p}^{(ex)},$ (5)

which is given as a sum of the equilibrium and photoexcited contributions. The
condition (5) can be used directly in case of weak intercarrier scattering. In
case of optical excitation, with a subsequent emission of cascade of $2{\cal
N}$ optical phonons of energy $\hbar\omega_{0}$, the photoexcited distribution
can be written in the form (5) where $\delta p$ is centred in
$p_{\overline{\omega}}=(\hbar\Omega-2{\cal N}\hbar\omega_{0})/2v_{W}$ and
$\Delta p$ is included an additional broadening during the cascade emission.

Figure 1: Temporal evolution of photoexcited distribution $f_{pt}^{(ex)}$
normalized to $I_{ex}$ versus dimensionless momentum and time, $\delta
p/2\Delta p$ and $t/\tau_{p}$.

Under an effective intercarrier scattering, one needs to calculate the initial
temperature and concentration of carriers. The photoexcited concentration and
energy of carriers, which are described by the peak of distribution (4) are
given by

$\left|\begin{array}[]{*{20}c}\Delta n_{ex}\\\ \Delta
E_{ex}\end{array}\right|=\frac{4}{L^{2}}\sum\limits_{\bf
p}\left|\begin{array}[]{*{20}c}1\\\
{v_{W}p}\end{array}\right|f_{p}^{(ex)}\simeq\frac{I_{ex}(\overline{\omega}/v_{W})^{2}}{2\overline{\omega}\tau_{p}}\left|\begin{array}[]{*{20}c}1\\\
\hbar\overline{\omega}/2\end{array}\right|,$ (6)

where $L^{2}$ is the normalization area. One obtains $\Delta
E_{ex}/n_{ex}=\hbar\overline{\omega}/2$, for the Gaussian shape of pulse, i.e.
the averaged energy per generated particle is equal to the excitation energy.
In case of optical excitation, with $\cal N$ optical phonons emitted, the
energy per photoexcited particle, $\Delta E_{ex}/\Delta n_{ex}$, agrees
closely with $\hbar\omega-{\cal N}\hbar\omega_{0}$ (see above).

Figure 2: Initial maximum distribution (a) and effective temperature (b),
$f_{ex}$ and $T_{ex}$, versus pumping ($\Delta n_{ex}/n_{T}\propto I_{ex}$ for
$\hbar\overline{\omega}=60$ meV and 120 meV (solid and dashed curves,
respectively).

If $\tau_{p}\ll\tau_{cc}\ll\tau_{ac,r}$ , where $\tau_{cc}$, $\tau_{ac}$, and
$\tau_{r}$ correspond to the intercarrier scattering, the energy relaxation,
and the generation-recombination processes, respectively [the Coulomb-
controlled case ($ii$)], the dominanting carrier-carrier scattering imposes
the quasi-equilibrium distribution

$f_{pt}=\left[\exp\left(\frac{v_{W}p-\mu_{t}}{T_{t}}\right)+1\right]^{-1}$ (7)

with the effective temperature $T_{t}$ and the quasichemical potential
$\mu_{t}$. If $\tau_{cc}\ll t\ll\tau_{ac,r}$, the initial values
$T_{ex}=T_{t\to 0}$ and $f_{ex}=f_{p=0t\to 0}$ are determined from the
concentration and energy conservation requirements:

$\frac{2}{\pi}\left(\frac{T_{ex}}{\hbar
v_{W}}\right)^{2}\int\limits_{0}^{\infty}dxxf_{x}\left|{\begin{array}[]{*{20}c}1\\\
{T_{ex}x}\\\ \end{array}}\right|=\left|{\begin{array}[]{*{20}c}{n_{T}+\Delta
n_{ex}}\\\ {E_{T}+\Delta E_{ex}}\\\ \end{array}}\right|,$ (8)

where the function $f_{x}$ is introduced according to $f_{x}\equiv
f_{ex}/[e^{x}(1-f_{ex})+f_{ex}]$. Using $\Delta n_{ex}$ and $\Delta E_{ex}$
given by Eq. (6) and solving the transcendental system (8) one obtains the
initial values $f_{ex}$ and $T_{ex}$. The calculations here and below are
performed for the nitrogen temperature, $T=$77 K, the excitation energies
$2v_{W}p_{\overline{\omega}}=$120 meV (CO2 laser) and 60 meV (as an example of
interband excitation with subsequent optical phonon emission), and the
broadening energy $\hbar/\tau_{p}\simeq$6.6 meV, which corresponds to the
pulse duration $\simeq$0.1 ps. In Fig. 2 we plot $f_{ex}$ and $T_{ex}$ versus
the pumping level which is proportional to $\Delta n_{ex}/n_{T}$. Fast
increase of $T_{ex}$ and fast decrease of $f_{ex}$ take place for $\Delta
n_{ex}/n_{T}<1$, while a linear increase of these values are realized if
$\Delta n_{ex}/n_{T}>1$.

##  III Energy relaxation and recombination

In this section we analyze the transient evolution of $f_{pt}$ caused by the
energy relaxation and recombination processes. We consider the cases ($i$) and
($ii$), when the initial condition is given by Eq. (5) and written through
$f_{ex}$ and $T_{ex}$ plotted in Fig. 2.

###  III.1 Weak intercarrier scattering

If the carrier-carrier scattering is ineffective [case ($i$)], the
distribution $f_{pt}$ is governed by the kinetic equation (1) without the
$cc$-contribution

$\displaystyle\frac{\partial f_{pt}}{\partial
t}=\frac{\nu_{p}^{\ss(qe)}}{p^{2}}\frac{d}{dp}\left\\{p^{4}\left[\frac{df_{pt}}{dp}+\frac{f_{pt}(1-f_{pt})}{p_{T}}\right]\right\\}$
$\displaystyle+\nu_{p}^{(r)}[N_{2p/p_{T}}(1-2f_{pt})-f_{pt}^{2}]$ (9)

and with the initial condition (5) used instead of generation rate. Here we
substituted the explicit expressions of the collision integrals for the
quasielastic acoustic scattering approximation (written in the Fokker-Planck
form) and for the generation-recombination processes, see discussion in 10 .
The Planck distribution $N_{2p/p_{T}}$ is written through $p_{T}=T/v_{W}$
while the energy relaxation rate $\nu_{p}^{(qe)}=v_{qe}p/\hbar$ and the rate
of radiative transitions $\nu_{p}^{(r)}=v_{r}p/\hbar$ are written through the
characteristic velocities $v_{qe}\propto T$ and $v_{r}$ 15 .

The boundary conditions are imposed by both the condition $f_{p\to\infty
t}=0$, which is transformed into the requirement

$p^{4}\left(\frac{\partial f_{pt}}{\partial
p}+\frac{f_{pt}}{p_{T}}\right)_{p\to\infty}<{\rm const},$ (10)

and Eq. (9) at $p=0$ which is transformed into the initial condition
$f_{p=0t}=1/2+f_{p=0}^{(ex)}\exp[-(v_{r}/v_{W})Tt/\hbar]$. According to Eq.
(4) one obtains
$f_{p=0}^{(ex)}=\sqrt{\pi/2}I_{ex}\exp[-(\Omega\tau_{p})^{2}/2]\ll 1$ and one
can neglect the second contribution in this initial condition, so that
$f_{p=0t}=1/2$. Numerical solution of the Cauchy problem given by Eqs. (5),
(9), and (10) is obtained below by the use of the iteration procedure. 16

Figure 3: Distribution $f_{pt}$ versus carrier energy $pv_{W}$ for different
delay times (marked) and excitation conditions: (a)
$\hbar\overline{\omega}=$120 meV and $I_{ex}=$0.26, (b)
$\hbar\overline{\omega}=$120 meV and $I_{ex}=$0.05, and (c)
$\hbar\overline{\omega}=$60 meV and $I_{ex}=$0.21.

In Fig.3 we demonstrate the evolution of the distribution $f_{pt}$ at 77 K for
the cases when carriers are excited around the energies 60 meV and 30 meV. The
delay times are marked in panels a-c and the pumping levels are determined
through the initial peak value, given by $\sqrt{\pi/2}I_{ex}$, see Eq. (4)).
Under mid-IR pumping with pulse duration $\tau_{p}=$0.1 ps and the spot sizes
$\sim$0.5 mm the above-used pumping levels correspond to the pulse energies
$\sim$85 pJ and $\sim$17 pJ for Figs. 3a and 3b, respectively, see 17  for
experimental details. Under optical pumping ($\hbar\Omega\sim$1.6 eV) and
subsequent emission of phonon cascade, the pumping level in Fig. 3c
corresponds to the pulse energy $\sim$12 nJ (duration and size are the same as
above). One can see that the transient evolution of distribution occurs in two
stages: energy relaxation and recombination. During the first stage (about
$t\lesssim$50 ns, which is dependent on position and maximum value
$f_{p}^{(ex)}$; compare with Figs. 3a-c) the initial peak is tranformed into
the quasiequilibrium high-energy tail (with the equilibrium temperature caused
by the energy relaxation) which is connected to the low-energy equilibrium
distrbution. During the next stage (up to 1 $\mu$s) the high-energy tail
shifts to the lower energies and transforms into the equilibrium distribution
due to effective radiative recombination in low-energy region.

###  III.2 Coulomb-controlled case

In the carrier-carrier scattering case ($ii$), one has to describe the
transient evolution of the effective temperature $T_{t}$ and the maximum
distribution $f_{t}=f_{p=0t}$, that replaces the chemical potential. Since the
intercarrier scattering change neither the concentration,
$n_{t}=(4/L^{2})\sum_{\bf p}f_{pt}$, nor the energy of carriers,
$E_{t}=(4/L^{2})\sum_{\bf p}v_{W}pf_{pt}$, the balance equations for $n_{t}$
and $E_{t}$ take forms: 18

$\frac{d}{{dt}}\left|{\begin{array}[]{*{20}c}{n_{t}}\\\ {E_{t}}\\\
\end{array}}\right|=\frac{4}{{L^{2}}}\sum\limits_{\bf
p}{\left|{\begin{array}[]{*{20}c}{J_{r}\\{f_{t}|p\\}}\\\
{v_{W}p\left[{J_{ac}\\{f_{t}|p\\}+J_{r}\\{f_{t}|p\\}}\right]}\\\
\end{array}}\right|}.$ (11)

Further, we transform the balance equations, expressing the left-hand side of
(11) through $T_{t}$ and $f_{t}$ as follows:

$\displaystyle\frac{d}{dt}\left(T_{t}^{2}A_{t}^{(1)}\right)=R_{t}^{(1)},$ (12)
$\displaystyle\frac{d}{dt}\left(T_{t}^{3}A_{t}^{(2)}\right)=R_{t}^{(2)}+Q_{t}.$

Here the coefficients $A_{t}^{(1,2)}$ are written as
$A_{t}^{(q)}=\int_{0}^{\infty}dxx^{q}f_{xt}$, where the quasiequilibrium
distribution is given by $f_{xt}=f_{t}/\left[e^{x}(1-f_{t})+f_{t}\right]$, so
that $A_{t}^{(q)}/T_{t}^{l}$ are only depend on $f_{t}$. After substitution of
the collision integrals $J_{r}$ 10 ; 18  and integration, the generation-
recombination contributions to Eq. (12) are obtained in the form

$R_{t}^{(q)}=\frac{2v_{r}T_{t}^{q+2}}{v_{W}\hbar}\int\limits_{0}^{\infty}dxx^{q+2}f_{xt}^{2}\left[\frac{e^{2x}(1-f_{t})^{2}}{(e^{x2T_{t}/T}-1)f_{t}^{2}}-1\right].$
(13)

Similarly, the energy relaxation contribution is written by the use of
$J_{ac}$ as follows

$Q_{t}=\frac{T-T_{t}}{T}\frac{v_{qe}T_{t}^{4}}{v_{W}\hbar}\int\limits_{0}^{\infty}dxx^{4}e^{x}f_{xt}^{2}\frac{1-f_{t}}{f_{t}}.$
(14)

The initial conditions for the system (11) are written as $T_{t=0}=T_{ex}$ and
$f_{t=0}=f_{ex}$.

Figure 4: Temporal evolution of effective temperature, $T_{t}$ (a), and
maximum distribution, $f_{t}$ (b), for different excitation conditions: (1)
$\hbar\overline{\omega}=$60 meV and $I_{ex}=$0.21, (2)
$\hbar\overline{\omega}=$60 meV and $I_{ex}=$0.1, (3)
$\hbar\overline{\omega}=$120 meV and $I_{ex}=$0.052, and (4)
$\hbar\overline{\omega}=$120 meV and $I_{ex}=$0.026. Figure 5: Energy per
carrier (a) and concentration (b) versus time. Solid and dotted curves
correspond to the cases ($i$) and ($ii$) , respectively; excitation conditions
(1) - (4) are the same as in Fig. 4.

Numerical solution of the nonlinear system (11) is performed using the
iteration procedure. In Fig. 4 we plot $T_{t}/T$ and $f_{t}$ versus time.
Temperature relaxes to the equilibrium one during the energy relaxation times
($\lesssim$ 100 ns) while $f_{t}$, which is determined by the chemical
potential $\mu_{t}$, relaxes to 1/2 over 1 $\mu$s (the recombination time
scale), in analogy with the case ($i$). Notice, that after the fast energy
relaxation, one obtains $f_{t}>$1/2 [dotted line in Fig. (4b)], i.e. the low-
energy electron-hole pairs appear to be unstable. 19

Fig. 5 shows the plot of temporal evolutions of the energy per particle and
concentration, $E_{t}/n_{t}$ and $n_{t}$ [see the definitions before Eq.
(11)], for the cases ($i$) and ($ii$). The relaxation processes to the
equilibrium (at nitrogen temperature,
$E_{t\to\infty}/n_{t\to\infty}\simeq$14.5 meV and $n_{t\to\infty}\simeq
5.3\cdot 10^{9}$ cm-2) occur during the same scales as in Figs. 3 and 4. The
temporal dependencies of $n_{t}$ obtained for both cases are in good agreement
(the carreir-carrier scattering does not change concentration) while $E_{t}$
demonstrates a different evolution for cases ($i$) and ($ii$) at $t<$50 ns.
This is because of drift and decrease of photoexcited peak during the energy
relaxation time, see Fig. 3.

##  IV Transient response

Here we turn to consideration of the response of photoexcited carriers on a
probe radiation (reflection and transmission in the THz and mid-IR spectral
regions) and on a weak dc electric field (photoconductivity). The transient
electrodynamics of graphene is described using the time-dependent dynamic
conductivity, $\sigma_{\omega t}$, which is caused by the collisionless
interband transitions, see Appendix B. The transient photoconductivity is
calculated by the use of the phenomenological model of momentum relaxation
suggested in 13 .

###  IV.1 Reflection and transmission

To calculate the transient reflectance and transmittance of the graphene sheet
placed at $z=0$ on the in-plane electric field ${\bf E}_{zt}\exp(-i\omega t)$
propagated along $0Z$, we apply the wave equation, see 20  and references
therein. The induced current density, $\sigma_{\omega t}E_{z=0}$, is located
around $z=0$ and direction of in-plane field ${\bf E}_{zt}$ is not essential
due to the in-plane isotropy of the problem. Separating the incident
radiation, $E_{in}e^{ik_{\omega}z}$, with the wave vector
$k_{\omega}=\omega/c$, we write the field distribution outside of the graphene
sheet in the form:

$E_{zt}=\left\\{\begin{array}[]{*{20}c}E_{in}e^{ik_{\omega}z}+E_{t}^{(t)}e^{-ik_{\omega}z},&{z<-0}\\\
E_{t}^{(t)}e^{i\overline{k}_{\omega}z},&{z>+0}\end{array}\right.,$ (15)

where $\overline{k}_{\omega}=\sqrt{\epsilon}\omega/c$ is the wave vector in
the substrate with the dielectric permittivity $\epsilon$. The transmitted and
reflected electric fields, $E_{t}^{(t)}$ and $E_{t}^{(r)}$, are determined
from the boundary conditions at $z\to 0$ as follows:

$\frac{E_{t}^{(t)}}{E_{in}}=\frac{2}{1+A_{\omega
t}},~{}~{}~{}~{}\frac{E_{t}^{(t)}}{E_{in}}=\frac{1-A_{\omega t}}{1+A_{\omega
t}}.$ (16)

Here we introduce the dimensionless factor $A_{\omega
t}=\sqrt{\epsilon}+(4\pi/c)\sigma_{\omega t}$. The reflection and transmission
coefficients, $R_{\omega t}=|E_{t}^{(r)}|^{2}/E_{in}^{2}$ and $T_{\omega
t}=|E_{t}^{(t)}|^{2}/E_{in}^{2}$, are written through $A_{\omega t}$ according
to

$R_{\omega t}=\left|\frac{1-A_{\omega t}}{1+A_{\omega
t}}\right|^{2},~{}~{}~{}~{}~{}T_{\omega
t}=\frac{4\sqrt{\epsilon}}{\left|1+A_{\omega t}\right|^{2}}.$ (17)

Using $\sigma_{\omega t}$ determined by Eqs. (B3) and (B4), we consider below
the differential changes in reflectivity and transmissivity, $(\Delta
R/R)_{\omega t}=(R_{\omega t}-R_{\omega}^{(eq)})/R_{\omega}^{(eq)}$ and
$(\Delta T/T)_{\omega t}=(T_{\omega t}-T_{\omega}^{(eq)})/T_{\omega}^{(eq)}$,
which are written through the equilibrium reflection and transmission
coefficients, $R_{\omega}^{(eq)}$ and $T_{\omega}^{(eq)}$.

Figure 6: (a) Spectral dependencies of differential reflectivity, $(\Delta
R/R)_{\omega t}$, for different delays (marked) at the excitation conditions:
(a) $\hbar\overline{\omega}=$120 meV and $I_{ex}=0.052$ in the case ($i$), (b)
$\hbar\overline{\omega}=$120 meV and $I_{ex}=$0.052 in the case ($ii$), and
(c) $\hbar\overline{\omega}=$60 meV and $I_{ex}=$0.21 in the case ($ii$).

The evolution of the differential reflectivity for the cases ($i$) and ($ii$)
are shown in Figs. 6a and 6b, 6c, respectively. If the Coulomb scattering is
not effective [case ($i$)], the distribution of carriers relaxes during the
energy relaxation time scale (around 10 ns, cf. with Fig. 3), when a quenching
of photoexcited peak takes place (if $\hbar\omega$ is comparable with the peak
energy). In case ($ii$) any peculiarities of the spectrsal dependencies at
stort times are absent because the initial distribution is transformed into
the quasiequilibrium one during times $\sim\tau_{cc}\to 0$. The further
evolution of $(\Delta R/R)_{\omega t}$ is limited by the generation-
recombination process and extended up to microseconds. In the THz spectral
region ($\hbar\omega\geq$10 meV is considered here because we neglect the
intraband relaxation), the differential reflectivity increases and changes a
sign. In the high-energy region, $(\Delta R/R)_{\omega t}$ decreases
monotonically with $\omega$ and $t$ and does not exceed $\sim 10^{-4}$ for the
near-IR spectral region. Beside of this, the response is approximately
proportional to the pumping intensity, $I_{ex}$, and $(\Delta R/R)_{\omega t}$
increases with increasing of the photoexcitation energy,
$\hbar\overline{\omega}$ (cf. Figs. 6b and 6c).

Figure 7: (a) Differential transmissivity, $(\Delta T/T)_{\omega t}$, versus
$\hbar\omega$ and $t$ for cases ($i$) and ($ii$) [panels (a) and (b),
respectively] at the same excitation conditions: $\hbar\overline{\omega}=$120
meV, $I_{ex}=$0.052.

In Fig. 7 we plot the differential transmissivity for the cases ($i$) and
($ii$) under the same excitation conditions. Once again, in the high-frequency
region the differential transmissivity decreases slowly (during a microsecond
time scale) and $(\Delta T/T)_{\omega t}$ does not exceed $\sim 10^{-4}$ for
the near-IR spectral region. In the THz spectral region, $(\Delta T/T)_{\omega
t}$ increses and changes the sing in the same manner as $(\Delta R/R)_{\omega
t}$ (cf. Figs. 6 and 7). The dependencies on the excitation parameters
($I_{ex}$ and $\hbar\overline{\omega}$) are also similar to the reflectivity.
Additionally, in case ($i$) a fast (at $t<$10 ns) quenching of the
photoexcited peak contribution in the spectral region $\sim\hbar\Omega$ takes
place.

###  IV.2 Photoconductivity

Finally, we consider the transient photoconductivity, i.e. the response of the
photoexcited carriers to the weak dc electric field. Since the momentum
relaxation is governed by elastic scattering mechanisms, 13  one can use the
following expression for the dc conductivity $\sigma_{t}$:

$\sigma_{t}=\sigma_{0}\left[2f_{p=0t}-\frac{l_{c}}{\hbar}\int_{0}^{\infty}dpf_{pt}\frac{\Psi^{\prime}(pl_{c}/\hbar)}{\Psi(pl_{c}/\hbar)^{2}}\right].$
(18)

Here $l_{c}$ is the correlation length characterizing the disorder scattering
and the function $\Psi(z)=e^{-z^{2}}I_{1}(z^{2})/z^{2}$ is written through the
first order Bessel function of imaginary argument, $I_{1}(z)$. The normalized
conductivity, $\sigma_{0}$, is introduced for the case of short-range
scattering, when $l_{c}=0$. The distribution $f_{p=0t}$ is shown in Fig. 4b
for the case ($ii$) while $2f_{p=0t}=1$ for the case ($i$). If $l_{c}=0$, one
obtains $\sigma_{t}/\sigma_{0}=1$, i.e. there is no transient
photoconductivity for the case ($i$); for the case ($ii$) one obtains
$\sigma_{t}/\sigma_{0}=2f_{t}$ and the transient photoconductivity is clear
from Fig. 4b.

Figure 8: Temporal evolution of conductivity for excitation conditions (1)-(4)
which are the same as in Fig. 4 for the correlation length $l_{c}=$30 nm and
10 nm. Solid and dashed curves are correspondent to the cases ($ii$) and
($i$).

If $l_{c}\neq 0$, the transient evolution of conductivity is shown in Fig. 8.
For the definiteness, it was assumed that $l_{c}$=10 and 30 nm and variations
of $\sigma_{t}$ are increased with $l_{c}$ essentially due to contribution of
high-energy carriers. Similar to Sec. IVA, one can separate two stages of
evolution: the fast decrease of $\sigma_{t}$ due to energy relaxation (up to
$\sim 30\div 50$ ns for the conditions considered) and the slow quenching of
$\sigma_{t}$ due to carrier recombination. If $t>1~{}\mu$s, the conductivity
approaches to the equilibrium values: $\sigma_{t\to\infty}/\sigma_{0}=$1.445
if $l_{c}$=30 nm and $\sigma_{t\to\infty}/\sigma_{0}=$1.035 if $l_{c}$=10 nm.
Since the transient conductivity can be measured for the subnanosecond time
scale 21 , such a scheme can be used for verification both energy relaxation
and recombination mechanisms.

##  V Concluding remarks

To summarize, we have considered both the interband ultrafast photoexcitation
and the relaxation dynamics of the carriers in an intrinsic graphene. In
contrast to the measurements 3 ; 4 ; 5 ; 6  and calculations 8 ; 9  performed,
where the evolution corresponds to the subpicosecond time scales due to the
opticlal phonon contribution, here we consider the slow relaxation of the low-
energy carriers. The distribution of carriers at $T=$77 K is obtained for the
limiting cases with negligible or dominating intercarrier scattering when the
energy relaxation and generation-recombination processes are caused by the
quasielastic acoustic phonon scattering and thermal radiation, respectively.
The initial distribution is obtained in the framework of the linear, with
respect to pumping, approximation for the collisionless regime of the
interband transitions. The transient optical response on the probe radiation
(transmission and reflection) as well as on the weak dc field (transient
photoconductivity) appears to be strongly dependent on the relaxation and
recombination dynamics of carriers.

Next, we discuss the assumptions made. The main restrictions of the results
presented are the consideration of the low-energy carriers, when the
interaction with optical phonons is unessential, and the single generation-
recombination mechanism (due to thermal radiation) is taken into account.
These conditions are realized at low temperatures under the mid-IR ultrafast
excitation 17  of the clean sample (e.g. suspended graphene 22 ). Such an
approach can be used for the case of optical interband excitation, when the
low-energy initial distrbution, with a phenomenological broadening, is formed
after the cascade process of optical phonon emission. The consideration is
restricted by the radiative recombination (the Auger processes are forbidden
due to the symmetry of electron-hole states 23 ), with the characteristic time
scales up to microseconds. Any visible contribution of other generation-
recombination mechanism (e.g., because of disorder-induced interband
transitions with acoustic phonons, or under intercarrier scattering) leads to
fast decrease of photoresponse. Such a regime requires an additional
investigation but the quasielastic energy relaxation stage is described by the
presented results.

The rest of assumptions are rather standard. The consideration in Sec. III is
limited by the simple cases ($i$) and ($ii$), with and without the
intercarrier scattering. The main peculiarities of the response under
consideration are similar for both cases but the complete description of the
nonequilibrium carriers had been performed neither under optical excitation,
nor under high dc field, see 18 ; 24  and Refs. therein. The description of
the momentum relaxation in Sec. IV is based on the phenomenological model of
Ref. 13. The utilization of the quasielastic energy scattering and the
collisionless interband photoexcitation appear to be rather natural. The
listed assumptions do not change either the character of the response or the
numerical estimates.

In closing, the peculiarities of the transient optical response (transmission
and reflection) as well as of the transient photoconductivity appear to be
useful tool in order to verify the relaxation and generation-recombination
mechanisms of carriers. Thus, in addition to the recently obtained
experimental results 3 ; 4 ; 5 ; 6 ; 7  measurements under mid-IR excitation
and at low-temperature will be useful for characterization of graphene.

##  Appendix A Generation rate

Below we describe the interband carrier excitation under ultrafast mid-IR
pumping ${\bf E}_{t}\exp(-i\Omega t)$ for the collisionless case, when
$\tau_{p}$ is shorter than relaxation times. The photogeneration rate into the
$\alpha$-state is based on the general expression (see 1  and Sec. 54 in Ref.
12)

$\displaystyle G_{\alpha
t}=2Re\left(\frac{e}{\hbar\Omega}\right)^{2}\int\limits_{-\infty}^{0}d\tau
e^{\lambda\tau-i\Omega\tau}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (19)
$\displaystyle\times\left\langle\alpha\left|\left[e^{i\hat{h}\tau/\hbar}\left[\left({\bf
E}_{t+\tau}\cdot{\bf\hat{v}}\right),\hat{\rho}_{t+\tau}\right]e^{-i\hat{h}\tau/\hbar},\left({\bf
E}_{t}\cdot{\bf\hat{v}}\right)^{+}\right]\right|\alpha\right\rangle,$

where $\hat{\rho}_{t}$ is the density matrix, $\hat{\bf v}$ is the velocity
operator, and $\lambda\to+0$. Since the collisionless regime of
photoexcitation, we calculate (A1) with the use of the free states $|l{\bf
p}\rangle$ and the energy $\varepsilon_{lp}$ where $l=\pm 1$ stands for $c$\-
or $v$-bands and $\bf p$ is the 2D momentum. Neglecting the nondiagonal
components of the density matrix $\hat{\rho}_{t}$ and using the distribution
functions $f_{l{\bf p}t}$, one obtains the generation rate

$\displaystyle G\\{f|1{\bf
p}t\\}=\left(\frac{e}{\hbar\Omega}\right)^{2}\int\limits_{-\infty}^{0}d\tau
e^{\lambda\tau-i\Omega\tau}e^{i(\varepsilon_{1p}-\varepsilon_{-1p})\tau/\hbar}$
$\displaystyle\times\langle 1{\bf p}|({\bf
E}_{t+\tau}\cdot{\bf\hat{v}})|-1{\bf p}\rangle\langle-1{\bf p}|({\bf
E}_{t}\cdot{\bf\hat{v}})^{+}|1{\bf p}\rangle~{}~{}~{}~{}~{}$ (20)
$\displaystyle\times\left(f_{-1{\bf p}t+\tau}-f_{1{\bf
p}t+\tau}\right)+c.c.~{},$

moreover $G\\{f|-1{\bf p}t\\}=-G\\{f|1{\bf p}t\\}$ according to the particle
concervation law. Next, we separate the envelope form-factor $w_{t}$ using
${\bf E}_{t}={\bf E}w_{t}$ and take into account the in-plane isotropy of the
problem, when one arrives to the averaged matrix element
$\overline{\left|\left\langle+1{\bf p}\left|({\bf
E}\cdot{\bf\hat{v}})\right|-1{\bf p}\right\rangle\right|^{2}}=(Ev_{W})^{2}/2$.
As a result, we obtain the in-plane isotropic generation rate $G\\{f|pt\\}=\pm
G\\{f|\pm lpt\\}$ in the following form:

$\displaystyle
G\\{f|pt\\}=\left(\frac{eEv_{W}}{\hbar\Omega}\right)^{2}\frac{w_{t}}{2}\int\limits_{-\infty}^{0}d\tau
w_{t+\tau}e^{\lambda\tau-i\Omega\tau}$ $\displaystyle\times
e^{i(2v_{W}p)\tau/\hbar}\left(f_{-1{\bf p}t+\tau}-f_{1{\bf
p}t+\tau}\right)+c.c.~{}.$ (21)

Finally, using the electron-hole representation and replacing the filling
factor here by $(1-2f_{pt})$, we arrive to Eq. (2).

##  Appendix B Dynamic conductivity

The response of graphene on the in-plane probe field ${\bf E}\exp(-i\omega t)$
is described by the dynamic conductivity 20 ; 25

$\displaystyle\sigma_{\omega t}\approx i\frac{2(ev_{W})^{2}}{\omega
L^{2}}\sum\limits_{\bf p}(1-2f_{pt})~{}~{}~{}~{}$ (22)
$\displaystyle\times\left(\frac{1}{\hbar\omega+2v_{W}p+i\lambda}-\frac{1}{\hbar\omega-2v_{W}p+i\lambda}\right)$

with $\lambda\to+0$. The parametric time dependency of $\sigma_{\omega t}$ is
valid if the time scales under consideration exceed $\omega^{-1}$. It is
convenient to separate the time-independent contribution,
$\overline{\sigma}_{\omega}$, described the undoped graphene in the absence of
photoexcitation, when $f_{pt}$ vanishes. Using the energy conservation law one
obtains ${\rm Re}\overline{\sigma}_{\omega}=e^{2}/4\hbar$. In the framework of
the Weyl-Wallace model, the ${\rm Im}$-contribution into
$\overline{\sigma}_{\omega}$ appears to be divergent at $p\to\infty$. It is
convenient to approximate ${\rm Im}\overline{\sigma}_{\omega}$ as a sum of
$\propto\omega^{-1}$ and $\propto\omega$ terms, which correspond to the
contributions of the virtual interband transitions and the ion background,
correspondingly. As a result, we obtain:

${\rm
Im}\overline{\sigma}_{\omega}\approx\frac{e^{2}}{\hbar}\left(\frac{\varepsilon_{m}}{\hbar\omega}-\frac{\hbar\omega}{\varepsilon_{i}}\right),$
(23)

where the characteristic energies, $\varepsilon_{m}\simeq$0.1 eV, and
$\varepsilon_{i}\simeq$ 6.8 eV are correspondent to the recent measurements of
the graphene optical spectrum. 26

Next, substituting the time-dependent distribution $f_{pt}$ obtained in Sec.
III into the dynamic conductivity (B1) one transforms the real and imagional
parts of $\sigma_{\omega t}$ as follows

$\displaystyle{\rm Re}\sigma_{\omega
t}=\frac{e^{2}}{4\hbar}\left[1-2F\left(p_{\omega},t\right)\right],~{}~{}~{}~{}~{}~{}~{}$
(24) $\displaystyle{\rm Im}\sigma_{\omega t}={\rm
Im}\overline{\sigma}_{\omega}-\frac{e^{2}}{\pi\hbar}{\cal
P}\int\limits_{0}^{\infty}\frac{dyy^{2}}{1-y^{2}}F(p_{\omega}y,t).$

Here $\cal P$ means the principal value of integral. We also introduced the
function $F(p,t)=f_{pt}$ for the case ($i$) and

$F\left(p_{\omega}y,t\right)=\frac{f_{t}}{\exp[(\hbar\omega/T_{t})y](1-f_{t})+f_{t}}$
(25)

for the case ($ii$), when $\sigma_{\omega t}$ is determined both the effective
temperature and the carrier concentration, $T_{t}$ and $f_{t}$.

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