Deformation of High Energy Symmetry
in String Theory
Chuan-Tsung Chan
Department of Physics, Tunghai University
April, 02, 2011 at NCTS, NTHU
Cross Strait Meeting on Particle Physics and Cosmology
Based on arXiv:0907.5472 [hep-th] ,arXiv:0907.5347 [hep-th] ,
arXiv:0905.2322 [hep-th],JHEP 0911:081,2009.
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
1 / 32
Based on a series of collaboration with
Borhade,
Chen,
Ho,
Lee,
Teraguchi,
Yang,
Yeh,
Pravina
Wei-Ming
Pei-Ming
Jen-Chi
Shunsuke
Yi
Chi-Hsien
(Tunghai)
(NTU)
(NTU)
(NCTU)
(Nagoya)
(NTU)
(NTU)
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
2 / 32
Outline of This Talk
1
Introduction
2
High Energy Symmetry of String Theory
3
Background (In)dependence of String Theory
4
High Energy Scatterings of String States in Linear Dilaton
Background
5
Conclusion
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
3 / 32
Introduction
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
4 / 32
Introduction
I. Introduction
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
5 / 32
Introduction
WebElements: the periodic table on the world-wide web
http://www.webelements.com/
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
hydrogen
18
helium
1
2
H
He
1.00794(7)
lithium
beryllium
element name
boron
carbon
nitrogen
oxygen
fluorine
4.002602(2)
neon
3
4
Key:
atomic number
5
6
7
8
9
10
Li
Be
symbol
B
C
N
O
F
Ne
6.941(2)
sodium
9.012182(3)
magnesium
2003 atomic weight (mean relative mass)
10.811(7)
aluminium
12.0107(8)
silicon
14.0067(7)
phosphorus
15.9994(3)
sulfur
18.9984032(5)
chlorine
20.1797(6)
argon
11
12
22.989770(2)
potassium
24.3050(6)
calcium
13
Na Mg
19
20
scandium
21
titanium
22
vanadium
23
K
Ca
Sc
Ti
V
39.0983(1)
rubidium
40.078(4)
strontium
44.955910(8)
yttrium
47.867(1)
zirconium
50.9415(1)
niobium
37
38
24
25
iron
26
Cr Mn Fe
51.9961(6) 54.938049(9)
molybdenum technetium
55.845(2)
ruthenium
cobalt
27
nickel
28
Co
Ni
58.933200(9)
rhodium
58.6934(4)
palladium
copper
zinc
16
17
18
P
S
Cl
Ar
26.981538(2)
gallium
28.0855(3)
germanium
30.973761(2)
arsenic
32.065(5)
selenium
35.453(2)
bromine
39.948(1)
krypton
Cu Zn Ga Ge As
29
30
31
32
33
Se
Br
Kr
63.546(3)
silver
65.38(2)
cadmium
69.723(1)
indium
72.64(1)
tin
74.92160(2)
antimony
78.96(3)
tellurium
79.904(1)
iodine
83.798(2)
xenon
35
36
Sr
Y
Zr
41
42
43
44
45
46
47
48
50
51
52
53
I
Xe
87.62(1)
barium
88.90585(2)
lutetium
91.224(2)
hafnium
92.90638(2)
tantalum
95.96(2)
tungsten
[98]
rhenium
101.07(2)
osmium
102.90550(2)
iridium
106.42(1)
platinum
107.8682(2)
gold
112.411(8)
mercury
114.818(3)
thallium
118.710(7)
lead
121.760(1)
bismuth
127.60(3)
polonium
126.90447(3)
astatine
131.293(6)
radon
55
56
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
132.90545(2)
francium
137.327(7)
radium
186.207(1)
bohrium
190.23(3)
hassium
87
88
Hf
174.9668(1)
178.49(2)
lawrencium rutherfordium
103
Ta
W
180.9479(1)
dubnium
183.84(1)
seaborgium
Ru Rh Pd Ag Cd
Re Os
Fr
Ra
Lr
Rf
105
106
107
108
[223]
[226]
[262]
[267]
[268]
[271]
[272]
[270]
cerium
praseodymium
neodymium
promethium
samarium
lanthanum
57
Lanthanoids
58
Db Sg Bh Hs
59
La
Ce
Pr
138.9055(2)
actinium
140.116(1)
thorium
140.90765(2)
protactinium
89
Actinoids
104
Nb Mo Tc
90
Ac
Th
[227]
232.0381(1)
91
Pa
60
61
62
Ir
192.217(3)
meitnerium
109
Pt
Au Hg
195.078(2) 196.96655(2)
darmstadtium roentgenium
92
U
231.03588(2) 238.02891(3)
[145]
neptunium
150.36(3)
plutonium
In
Tl
Sn Sb
Pb
Bi
Te
Po
204.3833(2)
207.2(1)
208.98038(2)
[209]
ununtrium ununquadium ununpentium ununhexium
54
86
At
Rn
[210]
ununseptium
[222]
ununoctium
110
111
112
113
114
115
116
117
118
[276]
[281]
[280]
[285]
[284]
[289]
[288]
[293]
—
[294]
europium
gadolinium
terbium
dysprosium
Mt
63
Ds Rg Uub Uut Uuq Uup Uuh Uus Uuo
64
65
Nd Pm Sm Eu Gd Tb
144.24(3)
uranium
200.59(2)
ununbium
49
34
Rb
Lu
40
manganese
15
Si
85.4678(3)
caesium
Cs Ba
39
chromium
14
Al
151.964(1)
americium
157.25(3)
curium
158.92534(2)
berkelium
93
94
95
96
97
[237]
[244]
[243]
[247]
[247]
Np Pu Am Cm Bk
66
holmium
67
Dy Ho
162.500(1)
californium
98
Cf
[251]
164.93032(2)
einsteinium
erbium
68
thulium
69
ytterbium
70
Er Tm Yb
167.259(3)
fermium
168.93421(2)
mendelevium
173.054(5)
nobelium
99
100
101
102
[252]
[257]
[258]
[259]
Es Fm Md No
Element symbols and names: symbols, names, and spellings are recommended by IUPAC (http://www.iupac.org/). Names are not yet proposed for the elements beyond 111 - those used here are IUPAC’s temporary systematic names (Pure & Appl. Chem., 1979, 51, 381–384). In the USA and some other countries, the spellings
aluminum and cesium are normal while in the UK and elsewhere the usual spelling is sulphur.
Atomic weights (mean relative masses): Apart from the heaviest elements, these are IUPAC 2007 values (Pure & Appl. Chem., 2007, in press). Elements with values given in brackets have no stable nuclides and are represented by integer values for the longest-lived isotope known at the time writing.
The elements thorium, protactinium, and uranium have characteristic terrestrial abundances and these are the values quoted. The last significant figure of each value is considered reliable to ±1 except where a larger uncertainty is given in parentheses.
Periodic table organisation: for a justification of the positions of the elements La, Ac, Lu, and Lr in the WebElements periodic table see W.B. Jensen, “The positions of lanthanum (actinium) and lutetium (lawrencium) in the periodic table”, J. Chem. Ed., 1982, 59, 634–636.
Group labels: the numeric system (1–18) used here is the current IUPAC convention. For a discussion of this and other common systems see: W.C. Fernelius and W.H. Powell, “Confusion in the periodic table of the elements”, J. Chem. Ed., 1982, 59, 504–508.
©2007 Dr Mark J Winter [WebElements Ltd and University of Sheffield]. All rights reserved. For updates to this table see http://www.webelements.com/nexus/Printable_Periodic_Table. Version date: 21 September 2007.
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
6 / 32
Introduction
Symmetry Dictates Physics!
The essence of the physics behind the periodic table is based on
THREE symmetry principles:
Dynamics: U(1) gauge symmetry ⇒ Coulomb Potential.
Geometry: SO(3) rotational symmetry ⇒ angular momentum
conservation.
Statistics: Pauli exclusion principle ⇒ atomic shell structure.
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
7 / 32
Introduction
Symmetry and Dynamics
1.
General Covariance
⇒
(Equivalence principle)
2.
Gauge Invariance
(Hilbert-Einstein action)
⇒
(local phase transformation)
3.
?
Relativity (Gravity)
Standard Model
(Yang-Mills action)
⇒
Quantum Gravity
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
8 / 32
Introduction
What is String Theory?
A perturbative prescription
1. Fundamental degrees of freedom: shape of the string ⇒ vibration
frequency ⇒ mass, direction ⇒ spin
internal charges can be engineered by
D-brane (open channel) or compactification (closed channel)
2. Dynamics: area and topology of the world-sheet
gluing of clothes and pants ⇒ Feynman Rules
3. Symmetries: irrelevant oscillations ⇒ gauge symmetry
freedom of defining local coordinate ⇒ duality
Global Symmetry?
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
9 / 32
Introduction
A Hint from Supersymmetry:
- Unification of bosons and fermions
symmetry charges carry spin 1/2. (Q ∼
√
P)
fermionic nature implies finite trucation.
2↔
3
2
↔1↔
1
2
↔0
True Unification of Space-time and Matter:
-Unification of graviton and gauge bosons (and fermions)
(basic) symmetry charges carry spin 1.
bosonic nature implies infinite multiplet!
⇒ ∞ # of particles + symmetry!
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
10 / 32
Introduction
Infinite Gauge Symmetry of Open String Theory
(1) First Quantized String Theory: canonical transformation in phase
space (G. Veneziano)
(2) World-sheet description: deformed conformal symmetry (M. Evans
& B. Ovrut)
(3) First Quantized String Theory: old covariant quantization ⇒
zero-norm states (J. C. Lee)
(4) Witten’s Open String Field Theory: δΨ = Q · Λ + gs [Ψ, Λ]
Chern-Simon’s action in loop space
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
11 / 32
Introduction
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
12 / 32
High Energy Symmetry of String Theory
II. High Energy Symmetry of
String Theory
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
13 / 32
High Energy Symmetry of String Theory
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
14 / 32
High Energy Symmetry of String Theory
General Pattern of High-Energy Scattering Amplitudes (HESA)
among Stringy Excitations (4-point functions)
0
A(s, t, α0 ; ni , i ) = C(ni , i )R(s, t, α0 ; ni i )e−α f (s,t)
C(ni , i ) is a numerical coefficient, which gives rise to linear
proportional relations among HESA.
R(s, t, α0 ; ni i ) is a rational function of Mandelstam variables,
which describes the subleading energy dependence and filter out
the leading states.
0
exp − α2 [−(s + t) ln(s + t) + s ln(s) − t ln(−t)] is the universal
tachyonic tail (surpression of high-energy divergence).
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
15 / 32
High Energy Symmetry of String Theory
Main Results for HESS in flat space-time:
1
At any fixed mass level, only one independent high-energy scattering amplitude!
(n)
n
TTT
···T ≡< V1 V2 (TT · · · T )V3 V4 >
2
n
All other leading amplitudes are proportional to TTT
···T and the proportional
constants are independent of
(i) scattering angle (ii) scattering process (iii) string loop-orderχ
(n)
(n)
< V1 V2 (J2 , k ) V3 V4 >χ ∝< V1 V2
J̃2 , k V3 V4 >χ
3
The proportional constants can be calculated by algebraic means and originate
from zero-norm states.
4
Master formula for inter-level relation: n = n1 + n2 + n3 + n4
α0
n n2 n3 n4
3
n
TT11···T
=
(−2E
sin
φ)
exp
−
[s
ln(s)
+
t
ln(t)
+
u
ln(u)]
1 T2 ···T2 T3 ···T3 T4 ···T4
2
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
16 / 32
Background (In)dependence of String Theory
III. Background
(In)dependence of String
Theory
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
17 / 32
Background (In)dependence of String Theory
Einstein-Hilbert action for pure gravity
SEH
=
⇒ SEH
=
Z
p
1
expand gµν = ηµν + hµν ,
d 4 x −gR,
16πG
Z
1
d 4 x(∂h∂h + h∂h∂h + h2 ∂h∂h + · · · )
16πG
Similarly, in gauge theory with SSB
1 2
v
2
2
d x − F + |Dφ| − U φ; µ , λ , expand φ = √ ,
4
2
1 2 1 2 2
1
− F + M B + e2 v χB 2 + e2 χ2 B 2
4
2
2
√
1
λ
µ4
+ (∂χ)2 − µ2 χ2 − λµχ3 − χ4 +
,
2
4
4λ
r
1
µ2
√ (v + χ) exp (iθ) , v =
, Fµν ≡ ∂µ Bν − ∂ν Bµ .
λ
2
Z
Sgauge
=
we get Lgauge
=
where φ
≡
4
A non-perturbative formulation of string theory has to provide a starting point for
expansion around arbitrary backgrounds.
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
18 / 32
Background (In)dependence of String Theory
Strategy: Push the theory to special limit and look for
simplification!
⇒High-energy expansion of string scattering amplitudes.
String theory is a (infinitely) higher-spin gauge theory with
spontaneous symmetry breaking.
Assuming that we are focusing on the transplanckian kinematic region
(so that the vaccum expectation values can be ignored), the symmetry
pattern among leading scattering amplitudes should reflect the global
symmetry of Goldstone bosons.
⇒A higher-spin generaliztion of equivalence theorem.
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
19 / 32
High Energy Scatterings of String States in Linear Dilaton
Background
IV. High Energy Scatterings of
String States in Linear Dilaton
Background
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
20 / 32
High Energy Scatterings of String States in Linear Dilaton
Background
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
21 / 32
High Energy Scatterings of String States in Linear Dilaton
Background
Special Features of String Theory in the Linear Dilaton
Background
Deformation of the conformal symmetry
T =−
1
∂X · ∂X + V · ∂ 2 X ,
α0
⇒ [Lm , Ln ] = (m − n)Lm+n +
V = ∂Φ.
D + 6α0 V 2
m(m2 − 1)δm+n
12
Consequences:
modified on-shell condition: e.g., “photon” α0 k (k + iV ) = 0, (k + iV ) = 0.
modified inner product for quantum state: < k 0 |k >= (2π)D δ (D) (k 0∗ − k − iV ).
modified vertex operators (extra factor).
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
22 / 32
Vertex operators for low-lying stringy excitations:
0
|T (k )i : e−α k ·V τ eik ·X :
ζ · Ẋ + iα0 V −α0 k ·V τ ik ·X
√
|P(ζ, k )i :
e
e
:
2α0
i
h
ν
i
0
µν
µ
0 µ
0 ν
√
· Ẍ e−α k ·V τ eik ·X :
|M(µν , k )i :
Ẋ + iα V
Ẋ + iα V −
0
2α0
2α
Oscillator basis for low-lying stringy excitations:
|T (k )i ≡ |0, k i, with α0 k · k + iV = 1.
µ
|P(ζ, k )i ≡ ζµ α−1
|0, k i, with α0 k · k +iV = 0, and ζ· k +iV = 0.
µ
µ ν
|M(µν , k )i = µν α−1
α−1
+ µ α−2
|0, k i,
√
2α0 µν k ν + iV ν + µ = 0,
√
gives µν η µν + 2α0 µ 2k µ + 3iV µ = 0.
L1 |M(µν , k )i = 0,
L2 |M(µν , k )i = 0,
with α0 k · k + iV = −1.
gives
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
23 / 32
Spectrum of Zero-norm States at άm2 = 1:

v2 L L
v2 P P
 − iv − 2 α−1 α−1 + iv − 2 α−1 α−1

√
|ZNSI (L)i = 
P L
iv L  |0, k i
iv P
2
+ 2 + v α−1 α−1 − √ α−2 +
2 + √ α−2
2
2
√
24
X
2 2 Ti
iv
Ti
Ti
L
P
α α +
α
|ZNSI (Ti )i =
uPTi 2α−1 α−1 +
|0, k i
2 − iv −1 −1 2 − iv −2
i=1


3v 2 L L
3v 2 P P
 5 − 6iv − 2 α−1 α−1 + 1 − 2 α−1 α−1 

 P


Ti
Ti
2 αP αL
|ZNSII i =  + 24
α
α
+
6iv
+
3v
 |0, k i
i=1
−1
−1
−1
−1



 10
7iv L
7iv P
+ √ − √ α−2 + √ α−2
2
2
2

Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
24 / 32
High Energy Scatterings of String States in Linear Dilaton
Background
Main Results in our calculations (hep-th/0905.2322)
1
We solved the Virasoro constraints up to the first massive level
and obtained the spectrum as a continuous function of moduli
parameter (V0 ).
2
We checked the on-shell stringy Ward identities hold true up to the
first massive level with the generalized on-shell conditions and
modified energy-momentum conservation law.
3
We examined the deformation of the high-energy stringy
symmetry from V0 = 0 to V0 → ∞ limit.
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
25 / 32
High Energy Scatterings of String States in Linear Dilaton
Background
T
(P
T)
(P
T
)
M
T
P)
M
M)
(T
(T
。
。
P
M
P)
P
(T
T)
T
(P
。
T)
。(P
M
)
(P
P
P
P
M
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
26 / 32
High Energy Scatterings of String States in Linear Dilaton
Background
T →P
P → MLL
P → MLT
P → MTT
Flat Background (V0 = 0, E → ∞)
3√
φ
2 2 α0 E tan
2
√
φ
−1
2 2 α0 E tan
2
1
cos
φ
−
2 2
φ
cos2 2
3√
φ
2 2 α0 E tan
2
P → ML
0
P → MT
0
T → MLL
T → MLT
T → MTT
0
2
2α E tan
0
T → MT
0
1
0
2 2 α EV0
3√
φ
2 2 α0 E tan
2 √
0
2i α V0 1 + sin2 φ
2
sin φ
2
2
φ
2
√
cos φ
φ
2 α0 E
tan
φ
2
2
cos 2
3 0 2
2 φ
2 α E tan
2
T → ML
Linear Dilaton Background (V0 /E → ∞, E → ∞)
3√
φ
2 2 α0 E tan
2
φ
−1 03
2
2 2 α 2 EV0 cot
2
−1
2
2
sec
φ
2
2α02 E 2 V02
3
2 0
2
2 α 2 E V0 tan
3
0
2
2 α E tan
2
φ
2
φ
2
3
2 2 iα0 EV0 1 + sin2 φ
2
cos2 φ
2
√
φ
2 φ
2 α0 E sec
tan
2
2
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
27 / 32
High Energy Scatterings of String States in Linear Dilaton
Background
~
SO(3) algebra= ~L ≡ ~r × P(set
~ = 1)



 [Lx , Ly ] = iLz
 [Lx , Ly ] = yPz − zPy
 [Px , Py ] = 0
[Ly , Lz ] = iLx ,
[Ly , Lz ] = zPx − xPz ⇒
[Px , Lz ] = −iPy



[Lz , Lx ] = iLy
[Lz , Lx ] = xPy − yPx
[Py , Lz ] = iPx
Rescale z → ξz 0 , and take ξ → ∞ limit.
Squash the sphere!
x2
R2
+
y2
R2
+
z2
R2
=1⇒
x2
R2
+
y2
R2
+
z 02
2
R
ξ
= 1.
Lx → −Py , Ly → Px , Lz 0 = Lz .
SO(3): isometry → E(2) Euclidean Group!
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
28 / 32
High Energy Scatterings of String States in Linear Dilaton
Background
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
29 / 32
Conclusion
V. Conclusion
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
30 / 32
Conclusion
Summary of This Talk
Concrete Realization of Gross’ Conjectures on the High-Energy
Symmetry of String Theory.
Connection between High-Energy Stringy Symmetry and infinite
stringy gauge symmetry.
Deformation of High-Energy Stringy Symmetry as a function of
moduli parameters.
Stringy Generalization of Equivalence Theorem for SSB?
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
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Thank You!!
Chuan-Tsung Chan (Department of Physics, Tunghai
Deformation
University)
of the High Energy Symmetry in String Theory
32 / 32