QCD Sum Rules for D Mesons at Finite Density

h®s¼G¹ºG

¹ºi

D » ¹cd : mch¹qqi; h¹cci = ? ... CBM @ FAIR

½; ! » ¹uu ¨ ¹dd : mqh¹qqi; ... HADES

Thomas HilgerForschungszentrum Dresden-Rossendorf

TU Dresden

Current-Current Correlation Function

¦(q) = i

Zd4xeiqxh­jT

£j(x)jy(0)

¤j­i

² aj­i 6= 0

² state of minimum energy

[Leinweber:http://www.physics.adelaide.edu.au/ dleinweb/]

Perturbative Approach

¦(q) = i

Zd4xeiqx

h0jT£j(x)jy(0)S

¤j0i

h0jSj0i

S = I¡ i

Z +1

¡1Hint(x)d4x+ : : :

²S-matrix operator in interaction picture

i£P0¹;ª®(x)

¤= @¹ª®(x)

²ground statej0iannihilated by all annihilation

operators

²free equations of motion

Analytic Properties of Π(q)

Re q0

Im q0

sq

0x

fixed q

Lehmann-representation: poles at the entire real axis

¢¦

dispersion relation:

in-medium: ¦(q¹;vº) = ¦(q2;v2;q ¢ v) ´ ¦(q0;!jqj)

¦(q0;!jqj) =

1

¼

Z +1

¡1ds¢¦(s;

!jqj)

s¡ q0

+ polynomials

restriction: j¦(q0)jjq0j!1

<= jq0jm

for some arbitrary but ¯nite and ¯xed m

Operator Product Expansion

T[A(x)B(y)] =X

i

Ci(x¡ y)Oi

h­jT[A(x)B(y)]j­i =X

i

Ci(x ¡ y)h­jOij­i

expansion at operator level!) state independentWilson coe±cents

condensates: parameters characterizing QCD

Oi = 1; ¹ªª;®s¼G¹ºG

¹º; ¹ª¾¹ºGA¹ºt

Aª

+ additional condensates in medium, e.g. hª+ªi

¦(q) ) ¦OPE(q) =X

i

~Ci(q)hOii

QCD Sum Rules*

[*Shifman, Vainshtein, Zakharov: Nucl.Phys.B147(1979)]

hadronic properties

via optical theorem

R =¾e+e¡!hadrons¾e+e¡!¹+¹¡

/ Im¦¹¹(s)

QCD

structure

via OPEsplitting in hadronic parts

semi-local quark

hadron duality

OPE â¦= Im¦¹

¹ ! observable,e.g. Dilepton emission rate or

¦(q0;!jqj) =

1

¼

Z +1

¡1ds¢¦(s;

!jqj)

s¡ q0

¦OPE(q0;!jqj) =

1

¼

0B@

z }| {Z 1

s0

+

Z ¡s0¡1| {z }

+

Z s0

¡s0

1CAds

¢¦(s;!jqj)

s¡ q0

OPE for D-MesonsB£¦e(!2)

¤ ¡M2

¢

=1

¼

Z 1

m2c

dse¡s=M2

Im¦perD+(s)

+ e¡m2c=M

2

µ¡mch ¹ddi +

1

2

µm3c

2M4¡ mc

M2

¶h ¹dg¾Gdi +

1

12h®s¼G2i

+

·µ7

18+

1

3ln¹2m2

c

M4¡ 2°E

3

¶µm2c

M2¡ 1

¶¡ 2

3

m2c

M2

¸h®s¼

µ(vG)2

v2¡ G2

4

¶i

+2

µm2c

M2¡ 1

¶hdyiD0di + 4

µm3c

2M4¡ mc

M2

¶·h ¹dD2

0di ¡1

8h ¹dg¾Gdi

¸¶

B£¦o(!2)

¤ ¡M 2

¢

= e¡m2c=M

2

µhdydi ¡ 4

µm2c

2M 4¡ 1

M2

¶hdyD2

0di ¡1

M 2hdyg¾Gdi

¶

Mass-Logarithms

²mass logarithms¡lnm2

¢of light quarks appear

² remnants of large distance behaviour

² to perform a consistent seperation of scales

! absorption into condensates

¦G2

(q) = h:®s¼G2:i

µ¡ 1

24

1

q2 ¡m2c

¡ 1

12

mc

md

1

q2 ¡m2c

¡ 1

24

m2c

(q2 ¡m2c)2

¶

+ h:®s¼

µ(vG)2

v2¡ G2

4

¶:iµq2 ¡ 4

(vq)2

v2

¶µ¡ 1

6q21

q2 ¡m2c

¡ 1

6q2m2c

(q2 ¡m2c)2

¡ 1

9q2

µm2c

(q2 ¡m2c)2

+1

q2 ¡m2c

¶ln

µm2d

m2c

¶¡ 2

9q2

µm2c

(q2 ¡m2c)2

+1

q2 ¡m2c

¶ln

µ¡ m2

c

q2 ¡m2c

¶¶

Absorption of Divergences

h¹qqi= h: ¹qq :i +3

4¼2m3q

µln¹2

m2q

+ 1

¶¡ 1

12mqh: ®s¼G2 :i + :::

h¹qg¾GAtAqi= h: ¹qg¾GAtAq :i ¡ 1

2mq ln

¹2

m2q

h: ®s¼G2 :i + :::

h¹q°¹iDºqi= h: ¹q°¹iDºq :i +9

4¼2m4qg¹º

µln¹2

m2q

+5

12

¶¡ g¹º

48h: ®s¼G2 :i

+1

18

³g¹º ¡ 4

v¹vºv2

´µln¹2

m2q

¡ 1

3

¶h: ®s¼

µ(vG)2

v2¡ G2

4

¶:i + :::

h¹qiD¹iDºqi= h: ¹qiD¹iDºq :i ¡ m5q

2¼2g¹º

µln¹2

m2q

+ 1

¶¡ mq

16g¹º

µln¹2

m2q

¡ 1

3

¶h: ®s¼G2 :i

+mq

36

³g¹º ¡ 4

v¹vºv2

´µln¹2

m2q

+2

3

¶h: ®s¼

µ(vG)2

v2¡ G2

4

¶:i + :::

² result in MS up to O(®s)

² def. of physical condensate:h­j¹ªO [D¹]ªj­i = h­j:¹ªO [D¹]ª:j­i

¡iZd4ph­jTr

hO³¡ip¹ ¡ i ~A¹

´Sª(p)

ij­i

Analyzing the Sum Rules

f ´Z s+0

s¡0

ds s¢¦e¡s2=M2

=X

§m§F§e¡m

2§=M

2

g ´Z s+0

s¡0

ds¢¦e¡s2=M2

=X

§§F§e¡m

2§=M

2

¢m(n) ¼¡ 1

2

dgdn

¯̄¯0m2(0) + dg0

dn

¯̄0

f(0)n

m(n) ¼m(0) ¡ 1

2m(0)

dfdn

¯̄¯0m2(0) + df 0

dn

¯̄0

f(0)n

for small densities

¢¦(s) =½¼P§ §F§±(s § (M§ ¡M­))

Im¦per(s)

! pole + continuum

dm§dM

= 0 ¢m =1

2

gf 0 ¡ fg0f 2 + gg0

=1

2(m+ ¡m¡)

m =

s¢m2 ¡ ff 0 + (g0 )2

f2 + gg0=

1

2(m+ +m¡)

upon Borel transformation

² mass splitting driven by odd OPE

² mass shift determined by even OPE

² complicated dependence on vacuum mass

Numerical Results

D§

D¡

D+

Vacuum Medium

²D§-pattern at nuclear saturation density

n = 0:17fm¡3 in linear density approximation

for the condensates:

¢m ¼ ¡50 MeV (robust)

±m ¼ ¡50 MeV (not robust)

² strong dependence of the mass splitting on the

chiral condensate and the odd mixed quark-gluon

condensate