> a proper verification was going on. To me the [fact that it was a] computer
> didn’t matter at all, it could also have been Ofer Gabber.868686A
> mathematician who is known to insist on the right details. If I [had] simply
> got a blanket ‘This is all correct’ stamp, I would have been extremely
> suspicious, as I was completely sure that I made some slips.

Scholze had been nervous of a very complicated and technical Theorem, and had
been aware of some nasty details that needed sorting out. He wrote

> The confidence that it is correct comes from … seeing that during the
> process, [the Leaners] seemed to run into all the little nasty details that
> I expected (and sometimes [had] not [expected]).

He ended up by stressing that

> This whole experiment was a genuinely human experience, very similar to
> going through this with a very careful colleague.

Scholze’s description of the proofs themselves being (currently) presented
obscurely by Lean is frustrating (so it can feel like one’s less communicative
colleagues), but there is no reason for that aspect of Lean to not be
improved. For now we have “no sense of the terrain”, only where the peaks are
and whether we have reached them, so what is it that can truly be said to have
been learned? Perhaps now that Lean seems like a viable proof verification
assistant, its design can find a more intuitive way to describe the proof it
has constructed? Often after a first human proof scopes out the terrain, the
next proof gives a more enriching description of the ideas involved, so why
not Lean? This would be useful and move mathematics forward whereas,

> For now, I can’t really see how [Lean] would help me in my creative work as
> a mathematician. Peter Scholze in Nature

##  13\. Myths of objectivity

In confirming that a proof is correct we believe that we can recognize and
establish an objective truth. But can we? It is easy to believe in one’s own
objectivity, or that of an “unbiased machine”, but are such beliefs valid, or
are they self-serving? To help highlight our potential fallacy, we now recall
two famous examples of so-called objectivity, (arguably) objective in their
own time and their original paradigm, but certainly not now.

###  13.1. These self-evident truths

For a long-time in our society biblical guidance was considered to be
objective. Turing [62] remarks that in the time of Galileo, the quotations

> The sun stood still . . . and delayed going down about a whole day
>  – Joshua 10:13
> He laid the foundations of the earth, that it should not move at any time  –
> Psalm 104:5

were considered by many to be an objective refutation of the Copernican
theory.

Social commentators like Donna Haraway [31] explore the abuse of belief in
“objectivity”.878787For recent in-depth discussions on the relationship
between objectivity in mathematics and social ontology, see the theme issue
[10]. For her primary issue, sexism, it is not hard to make the case: for so
long society has tolerated a belief that one’s gender or race quantifiably
affects one’s ability, rather than societal issues, whether they be social
class, upbringing, opportunities, etc. Few today would argue that those
earlier beliefs in objectivity were anything other than self-serving and most
people today believe that one should aim to replace those beliefs by a more
objective understanding. However, Haraway remarks that _any_ proposed
objective standard is really a subjective “power move, not a move towards …
truth”. Indeed to say something is “objectively proven” (that is, objective
within some current paradigm) seems to quash any potential objections.

Poor scholarship continues to support this kind of prejudiced “objectivity”.
For example, in 2005 Larry Summers, while president of Harvard University,
chose to try to explain the low numbers of women in STEM jobs: At that time
some test scores of cognitive abilities for twelfth grade women showed less
variation than for men, indicating fewer women at the top end (as well as at
the bottom end). Identifying factors that might cause these different
statistics, he focussed on “issues of intrinsic aptitude” (as well as “
_lesser factors_ involving socialization and continuing discrimination”). His
excuse for selecting “intrinsic aptitude” to be the most compelling was that
he “wanted to be provocative”.888888Sadly, representative groups censured him
on the grounds of sexism, rather than inadequate scholarship. This led to
their protests being ignored when President Obama selected Summers to be
President of the National Economic Council, thus continuing to add an air of
respectability to his poor scholarship. Today, and even then, one might view
such provocation as “problematic”.

To go beyond highlighting bias, and to not allow the issue to become bad
people’s “bias” versus good people’s “objectivity”, feminists [31] claim that
bias is ingrained into any social construct, arguing that there is a
“collective historical subjectivity”.898989Haraway also notes the desire to
find a legitimate feminist “objectivity”, but this is evidently paradoxical,
in that one is in danger of repeating the same errors that one has been at
pain to identify in others.

Haraway [31] highlights that “parables about objectivity” are told to
mathematics students during their training, even though this is not what
mathematicians do in practice. She argues that there can be no truly “trusted
authorities” but rather an earthwide network of connection, respectful of
different perspectives. Indeed she claims that

> Science has been about a search for translation, convertibility, mobility of
> meanings, and universality,909090All issues touched on in this article, and
> indeed part of what we have referred to as the “community standard”.

but one fails these criteria when work is only being compared to the hegemony.
In the context of this article, Haraway’s work suggests that the community
standards that have served us well in the past should illuminate the path
forward in the rapidly developing age of computer proofs. That these
standards, verifying that the plan and immediate details of a proof appear to
be correct, are the closest we can get to objectivity within any given
paradigm.

###  13.2. Objectivity and infallibility

Can there be infallible proofs? Proofs that are “unconditionally and
eternally” correct? Leibniz and Newton’s infinitesimals were mostly accepted
in the 17th century, yet were replaced by the $\epsilon-\delta$ proofs as
derived by Cauchy through to Weierstrass. Will our current proofs of the
basics of calculus withstand the test of eternity? Are they infallible? Will
they adapt to all the questions that will be asked of them? How can we know?
We have a framework within which they seem to be incontrovertible, but will
that framework seem appropriate in the light of future understandings?

> A triangle. This seems to be extremely simple, and you’d think we …know all
> about it …Even if we prove that it possesses all the attributes we can
> conceive of, some other mathematician, perhaps 1000 years into the future,
> may detect further properties in it; so we’ll never know for sure that we
> have grasped everything that there is to grasp about the triangle. And this
> holds also for bodies, for their extension, for everything! – Descartes
> (16/04/1648) in conversation with Burman.

And what about objective truth within the current framework? From Gödel we
know we can’t come close to verifying all truths since the set of true
statements is far larger than the set of provable statements so we need to
restrict our attention to what is provable.919191Even more we need to restrict
our attention to the (much smaller) family of statements in the complexity
class NP, which loosely means those for which there is a proof or solution
that can be verified on a human time scale, a key issue in theoretical
computer science. This was discussed with a few details at the end of section
2. Even so how do we know whether a given proof is correct? Tarski [60] notes
that “intuitive evidence is far from being infallible, has no objective
character, and often leads to serious errors”, and he views the “subsequent
development of the axiomatic method … as … restrict[ing] the recourse to
intuitive evidence”. This argument suggests that we can only hope to use a
formal proof to obtain sound verification of an objective truth, but can a
proof verifier become infallible and so be the ultimate arbiter of what is
correct? Without doubt a well written program may uncover new problems in
established proofs, and create proofs that are more difficult to challenge,
but that is not quite the same as infallibility.

The mathematical philosopher Avigad [5] claims that

> According to the standard view, a mathematical statement is a theorem if and
> only if there is a formal derivation of that statement, or, more precisely,
> a suitable formal rendering thereof.929292So, it seems that the Clausen-
> Scholze result only became a theorem that Avigad would recognize, when Lean
> asserted the proof to be so.

An attractive definition of a theorem, but resting on the assumption of the
infallibility of a “formal derivation of that statement”.939393Avigad goes on
to claim, “When a mathematical referee certifies a mathematical result, then,
whether or not the referee recognizes it, the correctness of the judgement
stands or falls with the existence of such a formal derivation,” a claim to an
extraordinary overview of mathematical process. Obtaining an intuitive proof
and a formal proof (based on the intuitive proof) are attractive goals,
different ways of assuring that there are no easily recognizable errors, but
it is a stretch to believe one can, in this way, assert objective truth.
Avigad’s claims rest on a view of objective truth that does not reflect any
consensus (despite Avigad’s confident phrasing), but rather serves to justify
the focus of a certain sub-community. Indeed this is endorsed by the extreme
claim of his colleague Azzouni [7],

> Formalized proofs have become the norms of mathematical practice.

Not quite sure how this became Azzouni’s “norm” as it is so far from true for
the majority of the mathematical research community. The day may come when
formalized proofs play something other than a peripheral role, but that day
has not yet arrived.

Detlefsen [13] better understands mathematical practice:

> Mathematical proofs are not … generally presented in a way that makes their
> formalizations either apparent or routine. This notwithstanding, they are
> commonly presented in a way that does make their rigor clear … at least by
> the time they’re widely circulated among peers …

That is, the author of a community-accepted rigorous proof rarely concerns
herself with formalization, although the formalizer must surely be concerned
about rigour (but it is then odd of Azzouni to suggest that the formalization
makes the proof significantly more trustworthy). Avigad [5] also argues for
formal proofs since “providing less information only exacerbates the problem:
if even a complete presentation of a formal derivation cannot be checked
reliably, providing strictly less information can hardly provide more
confidence.” So how much explanation is enough? Personally I prefer a clear
one page proof, than a lengthy turgid treatise that fails to appreciate what
is important even if it dots a few i’s,“filling a much-needed gap in the
literature”!

Returning to the theme of the subsection “Robustness and fragility” in section
11, it is interesting that formalizers feel their proofs are more trustworthy
when they are evidently so much more fragile. Moreover if, as Avigad seems to
claim, a proof is not a proof until it can be formalised then one must ask
which formalization is the correct one? Different Leaners might produce very
different formalizations of the proof of Pythagoras’s Theorem so which should
be the accepted one? Do proofs come in “equivalence classes” (that is, proofs
that are the same in disguise are in the same equivalence class)? If so, what
are the criteria for deciding which proof belongs to which class, and if there
is more than one equivalence class then what does that infer about the
fundamental nature of proof? Should machines strive for a “Book” where only
the best proofs are given?

###  13.3. Deus ex machina

literally means “god from the machine”. It describes the viewpoint of some in
the formalized proof community. Believing in infallibility (of their own code)
leads some programmers to not appreciate that their programming might
occasionally be wrong! That only by communicating one’s ideas will their ideas
be accepted. Indeed in [8] Barendregt and Wiedijk confidently assert that a
putative proof is verified if “the small number of logical rules are always
observed”, and so once they believe they have done that they create an
unenticing description of the formalization (see figure 8 of [8]). They need
to appreciate that a proof is a social compact, and so make the extra effort
to inform interested readers.

###  13.4. The Clay Millennium prizes

In 2000 the Clay Foundation announced a prize of a million dollars for the
resolution of any of seven famous mathematical problems. A solution can
receive the prize only two years after it has been published in a refereed
journal, and “has achieved general acceptance in the global mathematics
community”. These rules leave little doubt that the framers only have faith in
community-based proof verification, and even then feel that it takes a while
to be sure.

##  14\. Will machines change accepted proof?

In this article I have asserted that proof verification does little to change
the central tenets of proof as a social construction. Moreover that there is
little added value in learning that a program claims a proof has been verified
(without providing more helpful information to increase the reader’s
understanding). Nonetheless we can expect that efforts will be made to make
those formal proofs more accessible and hopefully useful (for example, they
might eventually remove extraneous ideas from intuitive proofs). Indeed
Patrick Massot [44] recently announced that software tools are being developed
to automatically convert formal proofs into human-readable interactive proofs,
allowing a reader to dig progressively deeper until she reaches a claim that
she believes with no further explanation, whether in a formal or intuitive
proof.

Since the chess program Deep Blue defeated world chess champion Garry Kasparov
in 1997, machine learning programs have become increasingly good at strategy
in board games, by now easily beating the best in the world at both chess and
Go. Indeed in late 2017, DeepMind’s AlphaZero was switched on, played only
games against itself for 24 hours, and achieved a grandmaster level in both
games (and in shogi), making it literally “superhuman”. It has a relatively
shallow search tree (80,000 positions per second in chess compared to 70
million for some other top software), but compensates for that with better
tactics.949494Chess grandmasters found its play “alien” with “insane attacking
chess”, for example sacrificing a queen and bishop to exploit a positional
advantage, something a human would be unlikely to do. Can such “reinforcement
learning” lead to a greater depth and variety of mathematical proofs? Might it
create proofs that are more surprising at times than human ones, like its
chess play?

In October 2022, DeepMind
introduced959595https://www.deepmind.com/blog/discovering-novel-algorithms-
with-alphatensor AlphaTensor, which was set the job of multiplying together
two matrices of given dimensions as efficiently as possible; that is, with the
least number of multiplications. AlphaTensor improved what had been known in
about 20 cases [18] but never by more than 5%, and it is particularly striking
that it does not appear to have deduced any new general theorems, nor
indicated how new types of general theorems might be found.969696In pure
mathematics, finding extensive examples if often a precursor to better
understanding, and thus truly new Theorems. This indicates that this new
technology can surpass human observation that comes directly through
calculation in areas where there are few strong theoretical ideas, but there
is no indication yet that DeepMind’s algorithms will lead to the creation of
new, deep theorems. Perhaps one day it will be able to better identify the
reasons that its tactics are so successful, which might help better understand
what is going on.

No one has yet built a quantum computer that can calculate faster than a
classical computer even in specially selected questions. Nonetheless the
theory suggests that certain parallelizable problems may be _much faster_ to
resolve on a quantum computer, most famously Shor’s quantum factorization
algorithm, which has caused vast resources to be pumped into “post-quantum”
cryptography.979797The security of several important cryptographic protocols
are based on the difficulty of factoring so if factoring becomes easy then
these protocols will no longer be secure. This worries powerful people so
there is a lot of money going into research in this area. For us the question
is whether quantum computing could be adapted to the task of finding proofs
(for example, if one uses a ridiculously large search tree). Carlos Simpson
pointed out to me that we might well run into the problem of obtaining a proof
but having no idea how it was found, and maybe so long as to be uncheckable on
any classical computer.

Raw computing power extends our mathematical capacities in many significant
ways and, paraphrasing Kevin Buzzard, the more that people invest in the
possibilities, the sooner interesting things will happen. Venkatesh [63]
suggests that mathematics might “be greatly altered; its central questions and
values … very different from those to which we are accustomed.” This “will
enhance our ability to do mathematics but also will alter our understanding of
what mathematics is”. Indeed it is only a matter of time before we learn how
to uncover tremendous possibilities for mathematics and for proofs revealed by
computing power, software, and brilliant programming ideas.

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# Stringy scaling of n-point hard string scattering amplitudes

Sheng-Hong Lai<EMAIL_ADDRESS>Department of Electrophysics, National
Yang Ming Chiao Tung University, Hsinchu, Taiwan, R.O.C.  Jen-Chi Lee
<EMAIL_ADDRESS>Department of Electrophysics, National Yang Ming Chiao
Tung University, Hsinchu, Taiwan, R.O.C.  Center for Theoretical and
Computational Physics (CTCP), National Yang Ming Chiao Tung University,
Hsinchu, Taiwan, R.O.C.  Yi Yang<EMAIL_ADDRESS>Department of
Electrophysics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan,
R.O.C.  Center for Theoretical and Computational Physics (CTCP), National Yang
Ming Chiao Tung University, Hsinchu, Taiwan, R.O.C.

###### Abstract

Motivated by the recent calculation of the $SL(K+3,\mathbb{C})$ symmetry of
$n$-point Lauricella string scattering amplitudes (SSA) of open bosonic string
theory, we calculate ratios of the solvable infinite linear relations of
arbitrary $n$-point hard SSA ($HSSA$). We discover a general stringy scaling
behavior for all $n$-point $HSSA$ to all string loop orders. For the special
case of $n=4$, the stringy scaling behavior reduces to the infinite linear
relations and constant ratios among $HSSA$ conjectured by Gross Gross  and
later corrected and calculated by the method of decoupling of zero-norm states
CHLTY2 .

##  I Introduction

Symmetry principle was one of the most important discovery of the 20th century
physics. All four fundamental interactions of the universe were based on the
symmetry principles. In particular, the principles of the general coordinate
symmetry and the gauge symmetry have led to Einstein general relativity theory
and Yang-Mills gauge theory, respectively. In string theory, on the contrary,
the situation went the other way around. One is given a set of rules through
quantum consistency of the extended string to fix the forms of interactions or
vertices of the theory. Moreover, in contrast to up to the usual four-point
couplings in quantum field theory, e.g. QCD, in string theory, one encounters
arbitrary $n$-point couplings which correspond to the infinite number of
particles in the spectrum of string. As a result, it is crucial to identify
the huge symmetry structure of string theory and uses it to relate these
infinite number of couplings of particles with arbitrary higher masses and
spins.

One key approach to uncover symmetry of string theory is to explicitly
calculate string scattering amplitudes (SSA). Recently, the author of LLY2
calculated a subset of exact $4$-point SSA with three tachyons and one
arbitrary string states (Note that SSA of three tachyons and one arbitrary
string states with polarizations orthogonal to the scattering plane vanish.)

$\left|r_{n}^{T},r_{m}^{P},r_{l}^{L}\right\rangle=\prod_{n>0}\left(\alpha_{-n}^{T}\right)^{r_{n}^{T}}\prod_{m>0}\left(\alpha_{-m}^{P}\right)^{r_{m}^{P}}\prod_{l>0}\left(\alpha_{-l}^{L}\right)^{r_{l}^{L}}|0,k\rangle$
(1)

where $e^{P}=\frac{1}{M_{2}}(E_{2},\mathrm{k}_{2},0)=\frac{k_{2}}{M_{2}}$ is
the momentum polarization, $e^{L}=\frac{1}{M_{2}}(\mathrm{k}_{2},E_{2},0)$ is
the longitudinal polarization and $e^{T}=(0,0,1)$ is the transverse
polarization on the $\left(2+1\right)$-dimensional scattering plane, and
expressed them in terms of the $D$-type Lauricella functions LLY2 . In
addition to the mass level $M_{2}^{2}=2(N-1)$ with

$N=\sum_{\begin{subarray}{c}n,m,l>0\\\ \\{\text{ }r_{j}^{X}\neq
0\\}\end{subarray}}\left(nr_{n}^{T}+mr_{m}^{P}+lr_{l}^{L}\right),$ (2)

we define another important index $K$ for the state in Eq.(1)

$K=\sum_{\begin{subarray}{c}n,m,l>0\\\ \\{\text{ }r_{j}^{X}\neq
0\\}\end{subarray}}\left(n+m+l\right)$ (3)

where $X=\left(T,P,L\right)$ and we have put $r_{n}^{T}=r_{m}^{P}=r_{l}^{L}=1$
in Eq.(2) in the definition of $K$. Intuitively, $K$ counts the number of
variaty of the $\alpha_{-j}^{X}$ oscillators in Eq.(1). For later use, we also
define $k_{j}^{X}\equiv e^{X}\cdot k_{j}$.

In addition, it was shown that these Lauricella SSA ($LSSA$) can be expressed
in terms of the basis functions in the infinite dimensional representation of
the $SL(K+3,\mathbb{C})$ group Group ; slkc . This is similar to the well-
known spherical harmonics function representation of the $SU(2)$ rotation
group in quantum mechanics. Moreover, it was further shown that there existed
$K+2$ recurrence relations of the $D$-type Lauricella functions. These
recurrence relations can be used to reproduce the Cartan subalgebra and simple
root system of the $SL(K+3,\mathbb{C})$ group with rank $K+2$. On the other
hand, with the Cartan subalgebra and the simple roots, one can easily deduce
the whole Lie algebra of the $SL(K+3,\mathbb{C})$ group. So we have the
following correspondences

$LSSA\Longleftrightarrow SL(K+3,\mathbb{C})\text{ symmetry
}\Longleftrightarrow\text{ Recurrence relations of Lauricella.}$ (4)

As a result, the $SL(K+3,\mathbb{C})$ group with its associated stringy Ward
identities (recurrence relations) can be used to solve solve  all the $LSSA$
and express them in terms of one amplitude. See the recent review paper LSSA .

One important application of Eq.(4) in the hard string scattering limit was to
reproduce LLY2  infinite linear relations with constant coefficients among all
$4$-point hard SSA ($HSSA$) and solve the ratios among them. This high energy
symmetry of string theory GM ; GM1  was first conjectured by Gross Gross  and
later corrected and proved ChanLee ; ChanLee2 ; CHLTY2 ; CHLTY1  by the method
of decoupling of zero norm states (ZNS) Lee ; lee-Ov ; LeePRL . ZNS was also
shown to carry $w_{\infty}$ symmetry charges of $2D$ string ($c=1$, $2D$
quantum gravity) ChungLee1 . See the review papers review ; over . More
importantly, since the decoupling of ZNS or stringy Ward identities persist to
all string loop orders loop , it is conceivable that the infinite linear
relations obtained in the hard scattering limit for arbitrary mass
$M^{2}=2(N-1)$ levels for $4$-point SSA are also valid for all string loop
amplitudes. On the other hand, one notes that these linear relations are not
shared by amplitudes of quantum field theories which depend on polarizations
of the scattering particles spin .

More recently, it was demonstrated short ; long  that by using the string on-
shell recursion relation of SSA bcfw4 ; bcfw3 ; stringbcfw , one can show that
all $n$-point SSA can be expressed in terms of the Lauricella functions. This
result extends the previous exact $SL(K+3,\mathbb{C})$ symmetry of the
$4$-point $LSSA$ of three tachyons and one arbitrary string states in Eq.(4)
to the whole tree-level open bosonic string theory. Motivated by this
remarkable result short ; long , one is tempted to believe that for the higher
point ($n\geq 5$) $LSSA$, there exist hard scattering kinematics regimes for
which the infinite linear relations persist and the ratios can be solved
accordingly.

In general, the explicit calculation of higher point ($n\geq 5$) SSA is very
lengthy short ; long . In this letter, we will generalize the method of
decoupling of ZNS and calculate the solvable infinite linear relations and
ratios of arbitrary $n$-point $HSSA$. The case of $n=4$ corresponds to the
previous Gross conjecture. In addition, we will use saddle point method to re-
calculate these infinite linear relations and ratios of a class of $n$-point
$HSSA$ to justify our results.

For each $n$ with $n\geq 5$, we discover new ratios among $n$-point $HSSA$.
Moreover, we discover the reduction of both the number of kinematics variables
dependence on the ratios and the number of independent $HSSA$. These stringy
scaling behaviors reminiscent of Bjorken scaling bs  and the Callan-Gross
relation cg  in deep inelastic scattering of electron and proton in the quark-
parton model of QCD are stringy phenomena with energy much higher than the
Planck energy ($\gg M_{Planck}^{2}\sim 1/\alpha^{\prime}$), while the QCD
scaling behavior is considered to be a low energy ($\ll M_{Planck}^{2}$)
string theory.

Nevertheless, there is another view to study hard scattering of QCD in string
theory, namely, string realization of QCD through gauge/string duality or
string on warped spacetime geometry ps ; bt ; mul . In this picture, it was
shown that the hard string scattering is power-law and thus QCD-like. However,
since the scattering energy is of order of the string scale, it involves
physics beyond the supergravity approximation, and one encounters the
difficulty of solving string theory on curved spacetime. We believe that the
stringy scaling behavior discovered in this paper on flat spacetime background
and its possible extention to warped spacetime geometry may find application
on these issues, in particular, the realization of QCD scaling behavior in
string theory.

The stringy scaling behavior also reminds us of renormalization group (RG)
approach to the critical phenomena of phase transition Wilson  such as
ferromagnet, percolation, chaos, polymers and wind fetch. However, although
both theories are fixed-point theories ($D=26$ for the string case), the
stringy scaling or reduction of number of kinematics variables shows up only
in the hard scattering limit. See more discussion following Eq.(32).

On the other hand, there exists mechanism of (Spontaneously) broken symmetry
in the Landau-Wilson theory of phase transition where the low-temperature
phase has lost some symmetry. For the string theory side, there are two
different concepts of broken symmetry. The first one was provided by the no-
ghost theorem which says that the string spectrum is ghost-free provided that
the intercept $a=1$ and $D=26$ or $a\leq 1$ and $D\leq 25$ GSW . For the case
of $D=26$ comparing to the latter case, there are enhanced
$SL(K+3,\mathbb{C})$ symmetries due to the existence of the type II zero-norm
states. The second one was discovered more recently that the exact
$SL(K+3,\mathbb{C})$ symmetry of $LSSA$ at $D=26$ was broken down to the
$SL(5,\mathbb{C})$ symmetry in the Regge limit LSSA  and to the
$SL(4,\mathbb{C})$ symmetry in the nonrelativistic limit LSSA .

##  II Stringy scaling by decoupling of ZNS

The use of method of decoupling of ZNS (or stringy Ward identities) to
calculate infinite linear relations for the $4$-point $HSSA$ was first adapted
in the ”physical state basis” ChanLee ; ChanLee2  with the Virasoro
constraints imposed. It was then extended to the general mass levels
$M^{2}=2(N-1)$ in the ”oscillator state basis” without imposing the
constraints CHLTY1 ; CHLTY2 . We will generalize the method to the higher
point ($n\geq 5$) $HSSA$ cases as will describe in this section.

In the old covariant first quantized open bosonic string spectrum, there are
two types of physical ZNS GSW . While type I states have zero-norm at any
spacetime dimension, type II states have zero-norm only at $D=26$. The
starting point is to apply the $n$-point $l$-loop stringy on-shell Ward
identities JCLee

$\left\langle V_{1}\chi V_{3}\cdots V_{n}\right\rangle_{l-loop}=0$ (5)

in the hard scattering limit. In Eq.(5), $V_{j}$ can be any string vertex and
the second vertex $\chi$ is the vertex of a ZNS. In the hard scattering limit
of scattering processes on the scattering plane, the space part of momenta
$k_{j}$ ( $j=3,4,\cdots,n$) form a closed $1$-chain with $(n-2)$ sides due to
momentum conservation and it can be shown that at each fixed mass level
$M^{2}=2(N-1)$ only states of the following form CHLTY1 ; CHLTY2

$\left|N,2m,q\right\rangle=\left(\alpha_{-1}^{\hat{T}}\right)^{N-2m-2q}\left(\alpha_{-1}^{L}\right)^{2m}\left(\alpha_{-2}^{L}\right)^{q}\left|0;k\right\rangle$
(6)

are leading order in energy. In Eq.(6) we have defined
$e^{P}=\frac{1}{M_{2}}(E_{2},\mathrm{k}_{2},\vec{0})=\frac{k_{2}}{M_{2}}$ the
momentum polarization, $e^{L}=\frac{1}{M_{2}}(\mathrm{k}_{2},E_{2},\vec{0})$
the longitudinal polarization and the transverse polarization
$e^{\hat{T}}=(0,0,\vec{\omega})$ where

$\omega_{i}=\cos\theta_{i}\prod\limits_{\sigma=1}^{i-1}\sin\theta_{\sigma}\text{
with }i=1,\cdots,r,\text{ }\theta_{r}=0\text{ }$ (7)

are the solid angles in the transverse space spanned by $24$ transverse
directions $e^{T_{i}}$. Note that $\alpha_{-1}^{\hat{T}}=\alpha_{-1}\cdot
e^{\hat{T}}$ etc. Most importantly, to the leading hard scattering limit in
energy, one can identify $e^{P}\simeq$ $e^{L}$ ChanLee ; ChanLee2 . This is
different from the Regge regime where $e^{P}\ncong$ $e^{L}$ bosonic . It is a
straightforward calculation to show that, at each fixed mass level $N$, the
decoupling of ZNS leads to ratios among $n$-point $HSSA$ CHLTY1 ; CHLTY2

$\frac{\mathcal{T}^{\left(N,m,q\right)}}{\mathcal{T}^{\left(N,0,0\right)}}=\frac{\left(2m\right)!}{m!}\left(\frac{-1}{2M}\right)^{2m+q}$
(8)

where $\mathcal{T}^{\left(N,m,q\right)}$ is the $n$-point $HSSA$ of any string
vertex $V_{j}$ with $j=1,3,\cdots n$, and $V_{2}$ is the high energy state in
Eq.(6); while $\mathcal{T}^{\left(N,0,0\right)}$ is the $n$-point $HSSA$ of
any string vertex $V_{j}$ with $j=1,3,\cdots n$, and $V_{2}$ is the leading
Regge trajectory string state at mass level $N$. Note that we have omitted the
tensor indice of $V_{j}$ with $j=1,3,4$ and keep only those of $V_{2}$ in
$\mathcal{T}^{\left(N,2m,q\right)}$.

###  II.1 Examples

For $n=4$, Eq.(8) implies the ratios are independent of $1$ scattering angle
$\varphi$ which reproduces Gross conjecture Gross . For $n=5$, Eq.(8) implies
the ratios are independent of $3$ kinematics variables ($2$ angles and $1$
fixed ratio of two infinite energies) or, for simplicity, $3$ scattering
”angles”. For $n=6$, there are $5$ scattering ”angles”. See the $r=1$ special
case of Eq.(21) at the end of this section. To illustrate the infinite linear
relations, historically, the first example of Eq.(8) was calculated for $n=4$
and $M^{2}=4$, $6$ in the ”physical state basis” with the Virasoro constraints
imposed. At mass level $M^{2}=4,$ there are two physical positive-norm states,
one symmetric spin-$3$ state and one anti-symmetric spin-$2$ state. The
decoupling of ZNS leads to $3$ linear relations among $4$ leading order
$4$-point $HSSA$ JCLee ; ChanLee ; ChanLee2

$\displaystyle\mathcal{T}_{LLT}^{5\rightarrow 3}+\mathcal{T}_{(LT)}^{3}$
$\displaystyle=0,$ (9) $\displaystyle 10\mathcal{T}_{LLT}^{5\rightarrow
3}+\mathcal{T}_{TTT}^{3}+18\mathcal{T}_{(LT)}^{3}$ $\displaystyle=0,$ (10)
$\displaystyle\mathcal{T}_{LLT}^{5\rightarrow
3}+\mathcal{T}_{TTT}^{3}+9\mathcal{T}_{[LT]}^{3}$ $\displaystyle=0,$ (11)

which can be solved to get the ratios ChanLee ; ChanLee2

$\mathcal{T}_{TTT}:\mathcal{T}_{LLT}:\mathcal{T}_{(LT)}:\mathcal{T}_{[LT]}=8:1:-1:-1.$
(12)

Eq.(12) calculated in the ”physical state basis” is consistent with the result
of Eq.(8) calculated in the ”oscillator state basis” in Eq.(6). It was also
demonstrated that the next to leading order amplitudes are in general not
proportional to each other ChanLee ; ChanLee2 .

In general scattering processes out of the scattering plane, the general high
energy states at each fixed mass level $M^{2}=2(N-1)$ can be written as

$\left|\left\\{p_{i}\right\\},2m,2q\right\rangle=\left(\alpha_{-1}^{T_{1}}\right)^{N+p_{1}}\left(\alpha_{-1}^{T_{2}}\right)^{p_{2}}\cdots\left(\alpha_{-1}^{T_{r}}\right)^{p_{r}}\left(\alpha_{-1}^{L}\right)^{2m}\left(\alpha_{-2}^{L}\right)^{q}\left|0;k\right\rangle$
(13)

where $\sum_{i=1}^{r}p_{i}=-2(m+q)$ with $r\leq 24$. With
$\left(\alpha_{-1}^{T_{i}}\right)=\left(\alpha_{-1}^{\hat{T}}\right)\omega_{i}$,
we easily obtain

$\displaystyle\left(\alpha_{-1}^{T_{1}}\right)^{N+p_{1}}\left(\alpha_{-1}^{T_{2}}\right)^{p_{2}}\cdots\left(\alpha_{-1}^{T_{r}}\right)^{p_{r}}\left(\alpha_{-1}^{L}\right)^{2m}\left(\alpha_{-2}^{L}\right)^{q}\left|0;k\right\rangle$
$\displaystyle=\left(\omega_{1}^{N}\prod_{i=1}^{r}\omega_{i}^{p_{i}}\right)\left(\alpha_{-1}^{\hat{T}}\right)^{N-2m-2q}\left(\alpha_{-1}^{L}\right)^{2m}\left(\alpha_{-2}^{L}\right)^{q}\left|0;k\right\rangle,$
(14)

which leads to the ratios of $n$-point $HSSA$

$\frac{\mathcal{T}^{\left(\left\\{p_{i}\right\\},2m,2q\right)}}{\mathcal{T}^{\left(\left\\{0_{i}\right\\},0,0\right)}}=\frac{\left(2m\right)!}{m!}\left(\frac{-1}{2M}\right)^{2m+q}\prod_{i=1}^{r}\omega_{i}^{p_{i}}$
(15)

where $\mathcal{T}^{\left(\left\\{0_{i}\right\\},0,0\right)}$ is the $HSSA$ of
leading Regge trajectory state at mass level $M^{2}=2(N-1)$. These ratios are
valid to all string loop orders.

We are now ready to discuss the stringy scaling behavior of these $n$-point
$HSSA$. First, we would like to count the number of independent kinematics
variables of the $n$-point $HSSA$. For the simple case with $n=4$ and $r=1$ in
Eq.(13) or Eq.(6), one has two variables, $s$ and $t$, or energy $E$ and the
scattering angle $\varphi$, and Eq.(8) means that all $HSSA$ at each fixed
mass level are proportional to each other and the ratios are independent of
the scattering angle $\varphi$. For the general $n$-point $HSSA$ with $r\leq
24$ in Eq.(13), we have $k_{j}$ vector with $j=1,\cdots,n$ and $k_{j}$ $\in
R^{d-1,1}$. We can count the number of independent kinematics variables to be
$n\left(d-1\right)-\frac{d\left(d+1\right)}{2}$.

In the hard scattering limit, for the kinematics parameter space $\mathcal{M}$
defined by

$\omega_{j}\left(\text{kinematics parameters with
}E\rightarrow\infty\right)=\text{fixed constant \ }(j=2,\cdots,r)\text{,}$
(16)

we can count the dimension of $\mathcal{M}$ to be

$\text{dim}\mathcal{M}\text{ }\mathcal{=}\text{
}n\left(d-1\right)-\frac{d\left(d+1\right)}{2}-1-\left(r-1\right)=\frac{\left(r+1\right)\left(2n-r-6\right)}{2}.$
(17)

where $r=d-2$ is the number of transverse directions $e^{T_{i}}$. In sum, the
ratios among $n$-point $HSSA$ with $r\leq 24$ calculated in Eq.(15) are
constants and independent of the scattering ”angles” in $\mathcal{M}$. For
$n=6$ and $r=3$, as an example, $\mathcal{M}$ is defined by

$\theta_{j}\left(8\text{ kinematics parameters}\right)=\text{fixed constant, \
}j=1,2,$ (18)

and we have $\dim\mathcal{M=}$ $6$. For this case, the ratios

$\frac{\mathcal{T}^{\left(\left\\{p_{1},p_{2},p_{3}\right\\},m,q\right)}}{\mathcal{T}^{\left(\left\\{0,0,0\right\\},0,0\right)}}=\frac{\left(2m\right)!}{m!}\left(\frac{-1}{2M}\right)^{2m+q}\left(\cos\theta_{1}\right)^{p_{1}}\left(\sin\theta_{1}\cos\theta_{2}\right)^{p_{2}}\left(\sin\theta_{1}\sin\theta_{2}\right)^{p_{3}}$
(19)

are independent of kinematics parameters in the space $\mathcal{M}$. For
example, for say $\theta_{1}=\frac{\pi}{4}$ and $\theta_{2}=\frac{\pi}{6}$, we
get the ratios among $6$-point $HSSA$

$\frac{\mathcal{T}^{\left(\left\\{p_{1},p_{2},p_{3}\right\\},m,q\right)}}{\mathcal{T}^{\left(\left\\{0,0,0\right\\},0,0\right)}}=\left(-\frac{1}{M}\right)^{2m+q}(2m-1)!!\left(\frac{1}{2}\right)^{p_{2}+p_{3}}\left(\sqrt{3}\right)^{p_{3}}\text{.}$
(20)

We conclude that, in the hard scattering limit, the number of scattering
”angles” dependence on ratios of $n$-point $HSSA$ with $r\leq 24$ reduces by
dim$M$. For a given ($n,r$), we list some numbers of dim$\mathcal{M}$
calculated in Eq.(17) as following

$\begin{tabular}[c]{lllll}$\text{dim}\mathcal{M}$&$r=1$&$r=2$&$r=3$&$r=4$\\\
$n=4$&$1$&&&\\\ $n=5$&$3$&$3$&&\\\ $n=6$&$5$&$6$&$6$&\\\
$n=7$&$7$&$9$&$10$&$10$\end{tabular}\ \ \ \ \ \ \ \ .$ (21)

We see from Eq.(15) that form of the ratios calculated for a given $r$ are
valid for arbitrary $n$-point $HSSA$ and to all string loop orders. Note that
for a given $n$,$\ \max(r)=n-3$, we have dim$\mathcal{M}\left(r=n-4\right)=$
dim$\mathcal{M}\left(r=n-3\right)=\frac{\left(n-3\right)\left(n-2\right)}{2}.$

The reduction of both the number of scattering ”angles” dependence on ratios
and independent SSA in the hard scattering limit suggests a very interesting
phenomenon of stringy scaling behavior.

##  III String amplitudes in the hard limit

To do a consistent check of the ratios calculated in Eq.(15) by the decoupling
of ZNS, in this section, we will use saddle point method to calculate the
$n$-point $HSSA$. Since the ratios in Eq.(15) are independent of the choices
of $V_{j}$ with $j=1,3,\cdots n$, we choose them to be tachyons and $V_{2}$ is
chosen to be Eq.(13). On the other hand, since the ratios are independent of
the loop order, we choose to calculate $l=0$ loop. In the hard scattering
limit, $p=E\rightarrow\infty$, we define the $26$-dimensional momenta in the
CM frame to be

$\displaystyle k_{1}$ $\displaystyle=\left(E,-E,0^{r}\right),$ $\displaystyle
k_{2}$ $\displaystyle=\left(E,+E,0^{r}\right),$ $\displaystyle\vdots$
$\displaystyle k_{j}$
$\displaystyle=\left(-q_{j},-q_{j}\Omega_{1}^{j},-q_{j}\Omega_{2}^{j},\cdots,-q_{j}\Omega_{r}^{j},-q_{j}\Omega_{r+1}^{j}\right)$
(22)

where $j=3,4,\cdots,n$, and

$\Omega_{i}^{j}=\cos\phi_{i}^{j}\prod\limits_{\sigma=1}^{i-1}\sin\phi_{\sigma}^{j}\text{
with }\phi_{j-1}^{j}=0,\text{ }\phi_{i>r}^{j}=0\text{ and
}r\leq\min\left\\{n-3,24\right\\}$ (23)

are the solid angles in the $\left(j-2\right)$-dimensional spherical space
with $\sum_{i=1}^{j-2}\left(\Omega_{i}^{j}\right)^{2}=1$. In Eq.(22), $0^{r}$
denotes the $r$-dimensional null vector. The condition $\phi_{j-1}^{j}=0$ in
Eq.(23) was chosen to fix the frame by using the rotational symmetry. Note
that the space part of $k_{j}$ ( $j=3,4,\cdots,n$) form a closed $1$-chain
with $(n-2)$ sides in the $25$-dimensional space due to momentum conservation.
In the hard scattering limit $E\rightarrow\infty$, the independent kinematics
variables counted in the last section can be chosen to be some
$\varphi_{i}^{j}$ and some fixed ratios of infinite $q_{j}$ defined in
Eq.(22). We consider the $n$-point $HSSA$ in $26D$ spacetime with $n-1$
tachyons and $1$ state defined in Eq.(13). With the change of variables
$z_{i}=\frac{x_{i}}{x_{i+1}}$ or $x_{i}=z_{i}\cdots z_{n-2}$, the $n$-point
$HSSA$ can be calculated to be

$\mathcal{T}^{\left(\left\\{p_{i}\right\\},m,q\right)}=\int_{0}^{1}dx_{n-2}\cdots\text{
}\int_{0}^{x_{4}}dx_{3}\int_{0}^{x_{3}}dx_{2}ue^{-Kf}=\left(\prod_{i=3}^{n-2}\int_{0}^{1}dz_{i}\text{
}z_{i}^{i-2-N}\right)\int_{0}^{1}dz_{2}ue^{-Kf}$ (24)

where

$\displaystyle f$ $\displaystyle=-\underset{i<j}{\sum}\frac{k_{i}\cdot
k_{j}}{K}\ln\left(x_{j}-x_{i}\right)$ (25)
$\displaystyle=-\underset{i<j}{\sum}\frac{k_{i}\cdot
k_{j}}{K}\left[\ln(z_{j}\cdots z_{n-2})+\ln\left(1-z_{i}\cdots
z_{j-1}\right)\right],\text{ }K=-k_{1}\cdot k_{2},$ (26) $\displaystyle u$
$\displaystyle=\left(k^{T_{1}}\right)^{N+p_{1}}\left(k^{T_{2}}\right)^{p_{2}}\cdots\left(k^{T_{r}}\right)^{p_{r}}\left(k^{L}\right)^{2m}\left(k^{\prime
L}\right)^{q}.(k^{\prime L}=\frac{\partial k^{L}}{\partial x_{2}})$ (27)

In Eq.(27), we have defined

$k=-\sum_{i\neq 2,n}\frac{k_{i}}{\left(z_{i}\cdots z_{n-2}-z_{2}z_{3}\cdots
z_{n-2}\right)}=\sum_{i\neq 2,n}\frac{k_{i}}{x_{i}-x_{2}}$ (28)

and $k^{X}$ is the momentum $k$ projected on the $X$ polarization. We also
define
$k_{\perp}=\left|k_{\perp}\right|\sum_{i=1}^{r}e^{T_{i}}\omega_{i}=\left|k_{\perp}\right|e^{\hat{T}}$.
We will show that the new defined vector $k$ is the key quantity to connect
the stringy scaling variables $\omega_{i}$ or $\theta_{i}$ introduced in
Eq.(7) and the kinematic variables $\Omega_{i}^{j}$ or $\varphi_{i}^{j}$
introduced in the CM frame defined in Eq.(23).

To do the integration in Eq.(24), we define the saddle point
$\tilde{z}_{i}=\left(\tilde{z}_{2},\cdots,\tilde{z}_{n-2}\right)$ to be the
solution of

$\frac{\partial f}{\partial z_{2}}=0,\cdots,\frac{\partial f}{\partial
z_{n-2}}=0.$ (29)

Note that Eq.(29) implies

$\displaystyle\tilde{k}^{L}$ $\displaystyle=\frac{\tilde{k}\cdot
k_{2}}{M}=\frac{k_{12}}{M}\left.\frac{\partial f}{\partial
x_{2}}\right|_{z_{i}=\tilde{z}_{i}}$
$\displaystyle=\frac{k_{12}}{M}\left.\frac{\partial z_{j}}{\partial
x_{2}}\frac{\partial f}{\partial z_{j}}\right|_{z_{i}=\tilde{z}_{i}}=0,\text{
}\left|\tilde{k}\right|=\left|\tilde{k}_{\perp}\right|\text{. }$ (30)

To proceed, let us introduce the following key identity

$\tilde{k}^{2}+2M\tilde{k}^{\prime L}=0,$ (31)

which can be interpreted as a mathematical identity of the geometric
parameters of a closed $1$-chain with $n-2$ sides. For $n=4$, one can easily
solve the saddle point $\tilde{z}_{2}=\sec^{2}\frac{\varphi}{2}$ to prove
Eq.(31) analytically. For the cases of $n=5$ and $6$, we are able to prove the
identity numerically. Indeed, we will see that the identity is a result of the
calculation of decoupling of ZNS in Eq.(15). Eq.(31) is crucial to show the
stringy scaling behavior of $HSSA$. On the other hand, Eq.(31) can be used to
express $\omega_{i}$ (or $\theta_{i}$) defined in Eq.(7) in terms of
independent kinematics variables defined in Eq.(22) if one can analytically
solve the saddle point
$\tilde{z}_{i}=\left(\tilde{z}_{2},\cdots,\tilde{z}_{n-2}\right)$.
Unfortunately, it turns out to be nontrivial except for the case of $n=4$. For
$n=6$ and $r=3$, see Eq.(18) as an example, we can only formally express
$\theta_{i}$ in terms of kinematics variables defined in Eq.(22) as

$\theta_{1}=\arctan\frac{\sqrt{\left(\tilde{k}_{\bot}^{T_{2}}\right)^{2}+\left(\tilde{k}_{\bot}^{T_{3}}\right)^{2}}}{\tilde{k}_{\perp}^{T_{1}}}\text{,
}\theta_{2}=\arctan\frac{\tilde{k}_{\perp}^{T_{3}}}{\tilde{k}_{\perp}^{T_{2}}}.$
(32)

It is important to note from Eq.(19) that the ratios of $6$-point $HSSA$
depends only on $2$ variables $\theta_{1}$ and $\theta_{2}$ instead of $8$.
This stringy scaling behavior is reminiscent of the model-independent
relations or scaling relations among critical exponents through RG analysis of
Widom’s hypothesis in statistical mechanics where a set of $6$ exponents
$\alpha$, $\beta$, $\gamma$, $\delta$, $\nu$ and $\eta$ are related by $4$
scaling relations, and the number of independent exponents reduces from $6$ to
$2$.

In view of Eq.(27) and Eq.(30), all up to $(2m)$-order differentiations of $u$
function in Eq.(27) at the saddle point vanish except

$\left.\frac{\partial^{2m}u}{\partial
z_{2}^{2m}}\right|_{z_{i}=\tilde{z}_{i}}=\frac{\left(2m\right)!}{\left(-2M\right)^{2m+q}}\left|\tilde{k}\right|^{N+2m}\omega_{1}^{N}\prod_{i=1}^{r}\omega_{i}^{p_{i}}$
(33)

where we have used the identity Eq.(31). With the above inputs, we can
calculate the $n$-point $HSSA$

$\displaystyle\mathcal{T}^{\left(\left\\{p_{i}\right\\},2m,2q\right)}$
$\displaystyle\simeq\left(\prod_{i=3}^{n-2}\int_{0}^{1}dz_{i}\text{
}z_{i}^{i-2-N}\right)\int_{0}^{1}dz_{2}\left(\frac{\partial^{2m}\tilde{u}}{\partial
z_{2}^{2m}}\frac{\left(z_{2}-\tilde{z}_{2}\right)^{2m}}{\left(2m\right)!}\right)e^{-Kf}$
$\displaystyle\simeq\frac{1}{\left(2m\right)!}\frac{\partial^{2m}\tilde{u}}{\partial
z_{2}^{2m}}\left(\prod_{i=3}^{n-2}\tilde{z}_{i}^{i-2-N}\right)\int_{0}^{\infty}dz_{2}\left(z_{2}-\tilde{z}_{2}\right)^{2m}e^{-Kf\left(z_{2}\right)}$
$\displaystyle=2\sqrt{\pi}e^{-K\tilde{f}}\left|\tilde{k}\right|^{N-1}\left(\prod_{i=3}^{n-2}\tilde{z}_{i}^{i-2-N}\right)\omega_{1}^{N}\frac{\left(2m\right)!}{m!}\left(\frac{-1}{2M}\right)^{2m+q}\prod_{i=1}^{r}\omega_{i}^{p_{i}}$
(34)

where
$f\left(z_{2}\right)=f\left(z_{2},\tilde{z}_{3},\cdots,\tilde{z}_{n-2}\right)$.
Eq.(34) gives the ratios of $n$-point $HSSA$ in Eq.(15). If we confine the
$n$-point $HSSA$ on a scattering plane, i.e. $r=1$ and $\omega_{1}=1$, the
ratios in Eq.(15) reduces to the ratios of $n$-point $HSSA$ in Eq.(8). For the
special $n=4$ case, one reproduces the ratios obtained in CHLTY1 ; CHLTY2  for
the Gross conjecture on $4$-point $HSSA$. Thus Gross conjucture is just a
special $4$-point case of ratios of $n$-point $HSSA$ with $r=1$ in Eq.(8).

To illustrate the calculation in Eq.(34), the ratios of $n$-point $HSSA$ for
the case of $r=2$ can be calculated to be

$\displaystyle\frac{\mathcal{T}^{\left(p_{1},p_{2},m,q\right)}}{\mathcal{T}^{\left(N,0,0,0\right)}}$
$\displaystyle=\frac{\left(2m\right)!}{m!}\left(\frac{-1}{2M}\right)^{2m+q}\frac{\left(2K\tilde{f}_{22}\right)^{m+q}}{\left(\sum_{i\neq
2,n}\frac{k_{i}^{T_{1}}}{\tilde{x}_{i}-\tilde{x}_{2}}\right)^{2m+2q+p_{2}}\left(\sum_{i\neq
2,n}\frac{k_{i}^{T_{2}}}{\tilde{x}_{i}-\tilde{x}_{2}}\right)^{-p_{2}}}$
$\displaystyle=\frac{\left(2m\right)!}{m!}\left(\frac{-1}{2M}\right)^{2m+q}\frac{\left(\frac{\sum_{i\neq
2,n}\frac{k_{i}^{T_{2}}}{\tilde{x}_{i}-\tilde{x}_{2}}}{\sum_{i\neq
2,n}\frac{k_{i}^{T_{1}}}{\tilde{x}_{i}-\tilde{x}_{2}}}\right)^{p_{2}}}{\left(\frac{\sum_{i\neq
2,n}\frac{k_{i}^{T_{1}}}{\tilde{x}_{i}-\tilde{x}_{2}}}{\sqrt{2K\tilde{f}_{22}}}\right)^{2m+2q}}$
(35)

where $\tilde{f}_{22}$ is defined in the following

$f_{2}\equiv\frac{\partial f}{\partial z_{2}}\text{,
}f_{22}\equiv\frac{\partial^{2}f}{\partial z_{2}^{2}}\text{,
}\tilde{f}=f\left(\tilde{z}_{2},\cdots,\tilde{z}_{n-2}\right)\text{,
}\tilde{f}_{22}=\left.\frac{\partial^{2}f}{\partial
z_{2}^{2}}\right|_{\left(\tilde{z}_{2},\cdots,\tilde{z}_{n-2}\right)}.$ (36)

On the other hand, the decoupling of ZNS of Eq.(15) gives

$\frac{\mathcal{T}^{\left(p_{1},p_{2},m,q\right)}}{\mathcal{T}^{\left(N,0,0,0\right)}}=\frac{\left(2m\right)!}{m!}\left(\frac{-1}{2M}\right)^{2m+q}\omega_{1}^{p_{1}}\omega_{2}^{p_{2}}=\frac{\left(2m\right)!}{m!}\left(\frac{-1}{2M}\right)^{2m+q}\frac{(\tan\theta_{1})^{p_{2}}}{(\cos\theta_{1})^{2m+2q}}.$
(37)

The saddle point calculation in Eq.(35) and the ZNS calculation in Eq.(37) can
be identified for any $p_{2}$, $m$ and $q$ if

$\left(\sum_{i\neq
2,n}\frac{k_{i}^{T_{1}}}{\tilde{x}_{i}-\tilde{x}_{2}}\right)=\sqrt{2K\tilde{f}_{22}}\cos\theta_{1}\text{,
}\left(\sum_{i\neq
2,n}\frac{k_{i}^{T_{2}}}{\tilde{x}_{i}-\tilde{x}_{2}}\right)=\sqrt{2K\tilde{f}_{22}}\sin\theta_{1},$
(38)

which implies the identity

$\left(\sum_{i\neq
2,n}\frac{k_{i}^{T_{1}}}{\tilde{x}_{i}-\tilde{x}_{2}}\right)^{2}+\left(\sum_{i\neq
2,n}\frac{k_{i}^{T_{2}}}{\tilde{x}_{i}-\tilde{x}_{2}}\right)^{2}=2K\tilde{f}_{22}.$
(39)

The identity derived in Eq.(39) is the $r=2$ special case of the identity in
Eq.(31).

##  IV Conclusion

In this paper, we extend the previous calculation of ratios among $HSSA$ for
$n=4$ with $r=1$ in Eq.(21), which were conjectured by Gross Gross  and
calculated in ChanLee ; ChanLee2 ; CHLTY2 ; CHLTY1 , to obtain new ratios for
general $n$-point $HSSA$ with $r\leq 24$. Moreover, we discover the reduction
of both the number of kinematics variables dependence on the ratios and the
number of independent SSA in the hard string scattering limit. These stringy
scaling behaviors are reminiscent of deep inelastic scattering of electron and
proton where the two structure functions $W_{1}(Q^{2},\nu)$ and
$W_{2}(Q^{2},\nu)$ scale, and become not functions of kinematics variables
$Q^{2}$ and $\nu$ independently but only of their ratio $Q^{2}/\nu$. That is,
the structure functions scale as bs

$MW_{1}(Q^{2},\nu)\rightarrow F_{1}(x),\text{ \ \ }\nu
W_{2}(Q^{2},\nu)\rightarrow F_{2}(x)$ (40)

where $x$ is the Bjorken variable and $M$ is the proton mass. Moreover, due to
the spin-$\frac{1}{2}$ assumption of quark, Callan and Gross derived the
relation cg

$2xF_{1}(x)=F_{2}(x).$ (41)

These scaling behaviors in the hard scattering limit of quark-parton model in
QCD seems to persist in some way in the $HSSA$ of string theory although the
latter are stringy phenomena with energy much higher than the Planck energy,
while the QCD scaling behavior is considered to be a low energy ($\ll
M_{Planck}^{2}$) string theory. Nevertheless, there seems to be interesting
similarities between the two scaling behaviors. For given $n$-point $HSSA$
with $r\leq 24$, the number of independent kinematics variables on the ratios
of $HSSA$ reduced by dim$\mathcal{M}$ as was calculated in Eq.(21). Moreover,
the number of independent $n$-point $HSSA$ for a given $r$ also reduces for
each fixed mass level as can be seen in Eq.(15).

We believe that, comparing to hard QCD scaling, hard string theory in general
has not been well studied yet in the literature oleg . More new phenomena of
stringy scaling proposed in this letter remain to be uncovered.

This work is supported in part by the Ministry of Science and Technology
(MoST) and S.T. Yau center of National Yang Ming Chiao Tung University (NYCU),
Taiwan.

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